Composition Methods in Homotopy Groups of Spheres. (AM-49), Volume 49 9781400882625

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ANNALS O F MATHEMATICS STUDIES Edited by Robert C. Gunning, John C. Moore, and Marston Morse 1. Algebraic Theory of Numbers, by

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COMPOSITION METHODS IN HOMOTOPY GROUPS OF SPHERES BY

Hirosi Toda

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 1962

Copyright © 1 9 6 2 , by Princeton University Press All Rights Reserved L. C. Card 62-12617

Printed in the United States of America

CONTENTS

1

INTRODUCTION...................................................... Part I CHAPTER I.

General Compositions and Secondary Compositions........

5

CHAPTER II.

Generalized Hopf Invariant and SecondaryCompositions. .

16

CHAPTER III.

Reduced Join and Stable G r o u p s ....................... 25

CHAPTER IV.

Suspension Sequence Mod 2 ............................. 3^

CHAPTER V.

Auxiliary Calculation of

CHAPTER VI.

Some Elements Given by Secondary Compositions.......... 51

CHAPTER VII.

*n + ]z(Sn >2 ') for1 < k < 7 . . .

2 -Primary Components of

39

8 < k < 13. . . .

61

Part II CHAPTER VIII. Squaring Operations................................... 82 CHAPTER IX.

Lemmas for Generators of

CHAPTER X.

2 -Primary Components of

*n+ii(Sn ;2 ) .................. 89 for

k = 1 U and 15. .95

CHAPTER XI.

Relative J-Homomorphisms............................. 1 1 2

CHAPTER XII.

2 -Primary Components of

for 1 6 < k < 1 9 • . . 135

CHAPTER XIII. Odd Components....................................... 1 7 2 CHAPTER XIV. T a b l e s ............................................... 1 8 6 BIBLIOGRAPHY

192

COMPOSITION METHODS IN HOMOTOPY GROUPS OF SPHERES

INTRODUCTION

The i-th homotopy group k^CX) of a topological space X is con­ sidered as the set of the homotopy classes of the mappings from i-sphere S 1 into X preserving base points. One of the main problems in homotopy theory is to determine the homotopy groups n^(Sn ) of spheres, since this is the first fundamental difficulty in the computations of the homotopy groups of polyhedra and topological spaces. The group n^CS11) is trivial if il, n = 1 . The first se­ quence of non-trivial groups is n = 1 ,2 ,... . The group is infinite cyclic and the homotopy class of a mapping of Sn into itself is characterized by the Brouwer degree of the mapping. The second example of non-trivial groups appeared in Hopf's work [8 ], in which he gave a homo­ morphism H of It2 n - i ^ n ^ i11^ 0 the group Z of integers. If n is even, then the image H(*2 n_ 1 (Sn )) of Hopf’s homomorphism contains 2Z, in par­ ticular if n = 2 ,1+ or 8 then H is onto and the so-called Hopf fibre map h : S2 n_ 1 — > Sn has Hopf invariant 1 , i.e H(h) = i- It is remark­ able that the groups ^ n ^ 11) and *1 ^ - 1 (S2n) are the only examples of infinite groups [1 3 ]• The first example of a non-trivial finite group n^(Sn ) was pre­ sented by Freudenthal in [7] as the result: » Z2 = Z/2Z for n > 3 . In the paper [7], Freudenthal's suspension homomorphism E : it ^ S 11) — > * 1 + 1 (Sn + 1 ) 2

was defined and played an important role- The group * (S ) is infinite J o 2 cyclic and generated by the class r\2 of the Hopf map : SJ — > S . The generator of the group ffn+i(s n ) rin = En “ 2 t\2 , where En " 2 is the (n-2)-fold interation of E. The above suspension homomorphism E is an isomorphism if i < 2 n- 1 , and this provides the concept of stable group

«k - ^

- W

sn>

which is isomorphic to if n > k + 1 . The second sequence of the groups irn+2 (Sn ) « Z2,

n ^ 2 , was

determined by G- Whitehead [2 7 ], and the generator of *n+2 ^ n ) is the composition tj OTln+1 • general, the composition operator 0 :

x

— > itjCS11)

Is defined simply by taking the homotopy class of the composition — > S1 — > Sn of mappings f and g-

g 0 f :

2

COMPOSITION METHODS IN HOMOTOPY GROUPS OF SPHERES We see in these examples, together with the beautiful isomorphism «i-1(Sn’1) + ni (S2n'1) — > ni (Sn ),

n = 2,^,8

given by the correspondence (a,p) — > Ea + 7 ° p for the class 7 of Hopf fibre map h (see [17]), that the suspension homomorphism and the composi­ tion operator are fundamental tools for calculating the generators of the homotopy groups of spheres. The purpose of this book is to compute the groups ^or k 19> by means of the suspension homomorphisms, the compositions and secondary operators derived from compositions- The book will be divided into two parts. In the first part, we shall compute the 2-primary compo­ nents of for k < 13 by use of purely composition methods and without use of cohomological operations, topology of Lie groups and other methods. It seems that our composition methods are not sufficient to de­ termine the groups 7tn+1 ^ (Sn ), and also it was seen that purely cohomo­ logical methods as in [23] are not sufficient to determine the groups of the stable case- But, combining these two sorts of methods the groups Ttn+ii+(Sn ) can be determined- Thus, in the second part, we shall no more insist on using only composition methods but apply cohomological methodsSome of the results on the stable groups implied from the algebraic struc­ ture of the cohomological operations will be used without proofs, because the theory of the cohomological operations and its applications is quite different from our geometrical situation and is still in the process of de­ velopment The readers are assumed to know familiar concepts of algebraic to­ pology, and the recent book "Homotopy Theory" of Hu will give some good in­ formation for readers who are not familiar with homotopy problems. The first chapter is a general consideration of the set jc(X-* Y) of the homotopy classes of maps between topological spaces X and Y, re­ lated with suspensions, reduced joins, compositions and secondary compo­ sitions. For a triple (a , & , ? ) having vanishing compositions; composition

of elements a 0En3 = 0

of^k+n^™ ^ x *j ^x and p°7 = 0, thesecondary

)

(a, E np, E n 7 }n C *1+n+1(Sm )

is defined; it is a coset of the subgroup a o En tt^+1(S^) + n j+r+ ](Sm )°En+17 of ^i+n+i (S111)• Roughly speaking, the secondary composition is also a sort of composition, namely, it is represented by a composition f °Eng : gi+n+1— ^ j,n-£ — ^ gmn-fold suspension Eng of a mapping g : Sl+1— > K and a mapping f : EnK — > Sn such that K = Sn U e^+1 is a cell complex having 3 as the class of the attaching map of e^+1, the restriction of f on EnSk = Sk+n represents a and such that g maps the upper-hemisphere of S1+1 into K like the suspension of a represen­ tative of 7 and lower-hemisphere of S^+1 into S^ •

INTRODUCTION

3

Chapter II contains the sequence

> *. (Sn ) £->

«i+1(Sn+1 ) —> n1+1(s2n+1 ) A.>

n^(sn+1 ) ...

of James [10], which Is exact if n is odd and If i < 3n- 3 > and which is an exact sequence of the 2-primary components if i > 2n. The above H is a generalization of the Hopf homomorphism, first given by G- Whitehead in [26] for I < 3n- 3 * Several propositions on the relations of secondary compositions with H and A are proved in this chapter. The anti-commutativity : a ° p = (-1 p0a, aeG^, ^€^j' ^ 6 composition operator was proved in [5 ] by use of reduced joins. Chapter III is an application of results about the reduced join to secondary composi­ tions. The stable secondary composition < a,

p,

7 > e G 1+j+k+1/ ( a ° G j+k+1

for the triple (a, p, 7) e G. x Gj x Gk of ap = p°7 = 0 is defined as the suspension limit of the usual secondary composition. For this stable operation we derived in [25] anti-commutativity < a,

p,

, > =

< r,

p,

a >

and the Jacobi identity (-1 ) l k < a ,

p,

7 > + (-1 ^

< P

7 , a > + (-1 )kj' < 7 , a ,

P > s

0.

Chapter IV is a preliminary for the computations of the 2-primary components of and it derives an exact sequence ... > * n. E . > n n+1 .+ 1 H . > * .+2n+1 1 — > Ait n - > ... which is adapted from James' sequence, and where it ^ coincides with the 2-primary component of Tt^(Sn ) if I 4 n, 2n-iChapters V, VI, and VII are the computations of it for k < 13. The computations are done by induction both on k and n- The first example of a homotopy group of spheres which differs from Z or Z2 is itg(S^) ~ Z 12 (it g ~ Z^ )• From the above exact sequence we obtain the following group extension: 0 — > Z2

* I -^> Z2 — > 0.

Then a generator of it | is given by the secondary composition {rj^, 2 i iv> generates jt^(sM) and the structure of the group ex­ tension is determined by properties of the secondary composition (cf [20]). Chapter VIII is a discussion on Steenrod's squaring operations {Sq1}, in particular the functional squaring operations ([18]) for Sq2, Sq\ and Sq^ give generators of it ], n and it Furthermore some discussions on Sq1^ are contained In this chapter. Chapter IX is devoted to proving a lemma useful for obtaining a generator of G ] and it will be applied in Chapter X to compute the group n * n+ 14 '

COMPOSITION METHODS IN HOMOTOPY GROUPS OF SPHERES

In Chapter XI, we shall define some modifications of G. Whitehead’s J-homomorphism [2 6 ]; these will give much information about iterated sus­ pension homomorphisms E^ : *j_(Sn) Iti+ic(Sn+ )• Chapters X and XII are the computation of the groups

*n+k

^or

1U < k £ 19*

To complete the table of ^ Chapter XIV, we shall use Chapter XIII about the computation of the odd primary components of *n+k (Sn ) for k £ 1 9 , and in which we shall not bother to compute the odd components further. The computation of the odd components of **or larger k will be carried out elsewhere.

PART CHAPTER I

General Compositions and Secondary Compositions Denote by

In

the unit n-cube in euclidean n-space, i.e.,

In ={(t],...,t ) | 1 1 , . ..,t Denote by

Sn

real numbers,

0 < t^ < i for i = i,••.,n}•

the unit n-sphere in euclidean

Sn = {(tn,•••,tn+1) | t 1 ,•..,tR+1

(n+i)-space, i- e-,

real numbers,

t2 + . ••+ t2 +1 =i}.

Define a continuous mapping (i.i)

: In — > Sn

as follows. The center cQ = (?,•••,4) of In is mapped by \|f to the point e* = (-1 ,0 ,...,o) of Sn . Each point x of the boundary in of In is mapped by \|r to the point eQ = (1 ,0 , . .,o) of Sn . Then, each seg­ ment -— — > is mapped by a uniform velocity onto a great semi-circle co * eQ eQ whose tangent at the point eQ is parallel to the vector ( - 1 ) x c'qX^ C ( - 1 ) x In . has the following properties. (i -2 ). *n (in ) = eQ and \|;n is a homeomorphism of in (t) = (cos 2 nt, sin 2 «t). injection i : Sn — > Sn+1 ^n (t 1 , • •-,tn ) = (sr ••-,sn+1 s^ +1 ^ 0 (resp. si + 1 < o).

In - in

onto

Sn - eQ

^n+1 (t 1, •. •,tR,i ) = i U n (t 1, •••,tR )) for given by i(s1 ,...,sn+1 ) = (s1 ,•••,sR+ 1 ,0 ). )■If t± :> i (resp. tj_ < \ ), then Ift^ is replaced by 1 -t^,then s^ +1 is

. an Let

replaced by ~ s ±+}' ^ and are interchanged then so are s^ +1 and sj+1. Throughout this book, we associate with each topological space X a point xQ of X, which is called as the base point of XForthe sphere Sn , we take the point eQ as the base pointEvery mapping and homotopy of topological spaces will have to pre­ serve the base points, i.e., a mapping f : X — > Y and a homotopy H : X x I 1 — > Y will satisfy the conditions f(xQ ) = y Q and H(xQ,t) = y Q for all tel 1 • Denote by *(X — > Y) the set of the homotopy classes of the map­ pings f : (X,x0 ) — > (Y,y0 ). (f) €jt(X — > Y) indicates the homotopy class of f . In particular, if X = Sn , then n(X — > Y) coincides with the n-dimensional homotopy group *n (Y) °? Y, with respect to the base point y Q. 5

6

CHAPTER I.

Consider two elements ae*(X — > Y) and Pe*(Y — > Z) and let f : X — > Y and g : Y — > Zbe representatives of a and p respective­ ly. Then the composition g of : X — > Z of f and g represents an ele­ ment of jt(X — > Z) which is independent of the choice of therepresenta­ tives f and g, and the class of gof is denoted by p°aejr(X — >Z) and called as the composition of a and p• The formula p°a = f (p ) =g^(a) defines mappings

and induced by

f* : *(Y — > Z) — > *(X — > Z) g# : n(X — > Y) — > it(X — > Z)

f and g respectively. Obviously the composition operator is associative: (7°p) oa = ro(poa )

(f£"f1 )* = f * o f 2 and (8 a °Si K = g2*°g1#The reduced join A # B of two spaces A and B, with the base points aQ and b Q, is obtained from the product A x B by shrinking the subset A v B = A x bQ U aQ x B to a single point which is taken as the base point of A # B-Denote by 0^ B : A x B — > A # B the shrinking map which defines A # B. Two reduced joins (A # B) #C and A #(B # C) are identified with thecorrespondence 0A# B,C ^0A,B^a'b ^ 0 — > 0A,B # C^a' 0B, C ^ ,c^, and denoted simply by A # B # Cand thus

For two spheres a mapping (1 -3 )

Sm

and

Sn,

we identify Sm# Sn

with

sm+n

by

0^ n : Sra x Sn — > Sm+n

given by the formula V n ^ m (x)' tn (y)) = *m+n(x'y)

for

e Im x In - Im+n .

This identification allows us to identify (Sm # Sn ) # Sp 0ra+n p( 0ra n (a>b),c) =

with

sm # (Sn # Sp ) as above, since the equality

0m,n+p^a' 0n,p^-t>,c^, (a,b,c) e Smx Snx Sp , holds. For two mappings f : A — > A' and g : B — > B' (of course f(a0 ) = ao and g(bQ ) = b^), their reduced .join f # g : A # B — > A 1 # B* is naturally defined from the product f x g : A x B— > A' x B' off andg by(f#g)° 0^B = 0A r B 1 °

)•

^he

following

propertiesare

checkedeasily.

0).

{ ] -k)

If f ^ f 1 : (A, aQ ) — > (A’, a£ ) and g ~ g ' : (B, bQ ) — > (B1, b^), then f # g ~ f 1 # g* : (A#B,a0#bQ ) — > (A'#B',a'#b’ ). i). (f#g ) # h = f # (g#h). ii). (f2 of,)# (g2 °g,) - (f2 #g2 ) • (f,#g l ). Let

f and g be representatives of aejt(A — > A' ) and respectively, then the class of f # g is independent of the choice of f and g, by o) of (1 -4 ), and the class is denoted by a # Pejt(A#B — > A'#B’ ) and called as the reduced join of a and p. It 3 err(B — > B 1 )

GENERAL COMPOSITIONS AND SECONDARY COMPOSITIONS

7

follows from (l.4 ) ) follows from (1.k

(K5) °'5)

i1 ) ). . (a#&) ( a # p#) 7# =7 a= #a (P#r # ( P)#•r ) . (a'c a) # (p'o p) = (a'#fi') (a'#e>) ° (a#P). (crfg ). ii). (a'*

The reduced join X # Sn of a space X and the n-sphere Sn is denoted by by EnX EnX= =X X# #Sn and called as the n-fold suspension of X. E E^ ^ isis denoted simply simply by by EX EX == XX ## SS11.. By By identification identification XX ## Sra+n= Sra+n= XX # Sm Sm# # Sn ,Snwe , we have En (EmX) - Era+nX. For a given given mapping mapping ff :: XX — —>> Y, Y, its its reduced reduced join join with with the the iden­ iden­ tity of Sn is called called as as the then-fold n-fold suspension suspension of f and of denoted f and denoted by by Enf : EnX — > E nY • (Ef = E ]f). The equality equality En En (Eraf) (Eraf) =Em+nf Em+nfholds. holds. ItItfollows follows from (1.*0 and ((11 ..55 )) 0). (1.6) (1.6) 1). ii).

f 2 g :XX ——>> YY implies implies Enf Enf ~~ Eng Eng,, and andthe thecorrespondence correspondence {f} — > {Enf } defines a mapping En : int(X — > Y ) — > ir(E ir(EnX nX — —>> E1 E1 ^^) .) . En (EmcO (Ema) ==E En+ma n+ma for aex(X for — > Y). aex(X — > Y). En (g°f)==Eng Eng ooEnf Enf and and EnEn(f3oa) (f3oa)= = Enp Enp 0 0Ena Ena . .

The suspension suspension EX EX ofof X X is is also also defined defined by by aa mapping mapping d^. : X x I 1 — > EX given by setting == 0X 0X Sn Sn ° °^1X ^1X XX ^ ^where wherelx lx denotes the identity of X- dx shrinks the subset X x I 1 U x Q x I 1 to the base point of EX, and this property property characterizes characterizes dxdx- - The The suspension suspension Ef : EX —> —> EY EY of of aa mapping mapping ff :: XX —> Y—>isY also is also givengiven by the by formula the formula Ef(dx (x,t)) = dy(f(x),t),

xeX,

ttel1 el1 .

Consider the set it (EX — > Y) of homotopy classes. For given mappings f,g f,g :: (EX, (EX, xxQQ)) —— > (Y,yQ > (Y,yQ ), ),define define mappings f ++ gg and and -f ::(EX, (EX, xxQQ)) — —>> (Y,yQ (Y,yQ)by ) by thethe formulas formulas ( f(dx (x,2t)), (x, 2 t )), ° < t < i i (f+g)(dx (x,t)) = I and ((-f)(dx - f )(dx ((x,t)) x,t)) = f(dx (x,i-t)). / g(dx (x,2t-i )), )), ii < t < 11 , Then these operations + and and - -are are compatible compatible with with the the homotopy homotopy and thus andin­ thus in­ duces those in tt(EX — > Y)- By By the the operations, operations, itit(EX (EX — —>> Y) Y) forms forms aa group, group, yv similar to the fundamental group ^ ( Y q ) of the mapping space Y Q of the mappings : (X,xQ ) —> (Y,y )• In fact, n(EX — > Y) and n1 (Yq) are ca­ nonically isomorphic if X is locally compact. We have easily 7 ).• —>> X, X, (i .7) For mappings f f: :Y Y—> —>Z, Z, and g ::WW — f * :: jt(EX —> Y) —> it(EX — > Z)Z) and and Eg* Eg* :: iitt(EX (EX —> —> Y) — > it*(EW (EW —> —> Y) Y) are are homomorphisms. Thus{f}° {f}° (a(a+ +a ’ a )’)= ={f} {f} o a + [f} 00 a 1 and (a + a') 0 E {g ] = a°E{g} °E{g}+ a' + a'o o E { g } . The group *(EnX *(EnX —> Y)is abelian if n >> 2. In the case X = Sn Sn,, we we denote dx by ( 1 •8 )

dndn:Sn : Sn X XX’—> X ’—> Sn +Sn ' +'

which is defined by theformula

^n (^n (t > • • •>tR ), ),tt))== tn+1 tn+1(t (t1, 1,•••••, •,tn tn,,tt ))••

8

CHAPTER I.

The group operations of it(EX — > Y) and nn+ ^ (Y) coincide in this case. the space of the loops Denote by flY = [i : (I1, i1 ) — > (Y,y0 in Y with the base point y Q • For a given mapping f X — > Y, define a mapping flf : ftX — > QY by the formula i

(nf(i))(t) = f(i(t)),

e

nx,

t € i1.

If f ~ g, then fif ^ Qg and we obtain a correspondence n : n (X — > Y) — > n(ax — > ftY) byn{f) = {nf} . Obviously ft(g o f) = fig ° nf and ft(3 ° a) = ft(3 ) °ft(a). Considerthe space ft(EX) of loops in EX- For a point x of X,define a loop i(x) in EX by the formula i(x)(t) = d^(x,t). Then we obtain an injection i : X — > ft(EX), which will be called as the canon­ ical injection. The following diagram is commutative f X -> Y (1 -9) i 1 JLUL)— > n(EY) SI ( E X ) For a mapping

F

EX — > Y,

denote that

ftQF = ftF i : X — > ft(EX) — > ftYftQF is defined directly by the formula ftQF(x)(t) = F(d^(x,t))-> ftQF is a homeomorphism between compact subsets of The correspondence F {EX — > Y) and [X — > ftY} and thus induces a one-to-one correspondence nr

(1.10)

tt(EX

— > Y) ^ it(X — > ftY).

This ftQ is an isomorphism if we give a multiplication in it(X — > ftY) which is induced by the loop-multiplication of ftY- This mul­ tiplication coincides with the above 'V1 if X is a suspension EX'• The following diagram is commutative. «(X — > Y) (1-11)

i* n(X

For mappings their homotopy classes lations hold. (1 •12 )

->

n (E X — > E Y )

-> n(EY*f) ■

f : X — > Y, g : EY — > Z and h : Z — > W and for a = ff}, 3 = {g} and y = {h}, the following re-

ftQ(g o Ef) = ftng fto n(3 o Ea) = ftr

f,

nQ(h ° g) = nh

a,

fig(r ° P ) = ^7

noS’ nQe.

Denote by 0 e i t(X — > Y ) the class of the trivial map e Y. x - > y0 Let n > 0 be an integer. Consider elements a e rr(EnY 3 e A (X — > Y ) and y e *(W — > X) which satisfy (1-13)

a

• EnP = 0

and

-> Z),

3 ° y = 0.

b : (X, xQ ) -> (Y, y n ) and Let a : CeP y , J — > (Z, Z q ) , (W, w Q ) — > (X, x n ) be representatives of a , 3 and y respectively.

GENERAL COMPOSITIONS AND SECONDARY COMPOSITIONS

9

by the the assumption assumption (1 (1--1133); ); there there exist exist homotopies Then by homotopies A^ : (EnX, (EnX, xQ xQ))—— >> (Z, zQ) zQ) and and :: (W, (W, wQ) wQ)—— >> (Y, (Y, yQ) yQ),, °° ££ tt< >(Z, (Z, zQ zQ)) by by the the formula (En+1W, wQ) formula (1

-U)

( a(EnB2t_1(w))for H(d(w, (w,tt))) == Z) as the set of all the homotopy classes of the mappings H given as above. Lemma 1.1: {a, Enp, is a double coset of the subgroups *(En+1X — > Z) Z) .o En En+1 and aa °° En — >Z ). IfIf n(En+1X + , r7 and En (rt(EW (*(E W — — >> Y)) Y ) ) in inJt(En+1¥ it(En+1W — >Z). n(En+1W If n > o, WW == EW1 EW1 or or ZZ = siZ', it(En+1¥ — > Z) Z) Is is abelian, abelian, in in particular if flZ’, then {a, EnP, En?}n is a coset of the subgroup n(E +\ + EnP, Enr)n it(En+1 — >Z) ° 0 En En+1r a o En (jc(EW (n(EW — )). a o — > Y Y)). Here we use the following notations. Let A and B be subsets of j (Y —— >> Z) and the set jtt (Y and n(X n(X —— >>YY)respectively, ) respectively, then then AA 0 0 B denotes the set of all cc o p for B- Let and A2 A2 bebe subsets of for aae eA,A,p pe eBLet AA11 and *(EX for € A 1, * (EX — > Y). Then A ] + Ag denotes the set of all a 1 + a2

€

A 2 *'

Proof: It is easily verified that any mapping of (1.1^) is homo­ topic to a mapping of (i -1^) representatives a,b (i -1^) for for fixed representatives a,b and and cc- Let Let HH be given as above above and andlet be given by by use use of of other other null-homotopies A£ let H* H* be and B£ ofof a a° °Enb and bb °° c, c, respectively. Considermappings mappings PP :: Enb and respectively. Consider (En+1X, x0x0 )— (EW, wQ wQ))——>(Y, >(Y, y0 y0)) given givenby by ) —> >(Z, (Z,zQ) zQ) and and GG::(EW, ( A2t-1 ()U), t)’ U i Ui t' £ (A2t_1 P(dR (x,t)) = S A ]_2t(x), j_2t(x), o i t £ i ,

j

( ®2t-1(w) ,(w) ’ , 1 , G(dw w,t)) = Bl_2t (w) , G(dW ( (w’t)} B1_2t(w)

ii £ 1 t £^ l, 1 o^ t i i

where R = EnX, x ee RR and and ww ee WW- Then Then it it is is verified verified that that H* H* is is homoto­ pic to F Fo oEn+1c En+1c+ +(H (H ++ aa °° EnG EnG °° a), a), where where aa is is aa homeomorphism homeomorphism of En+1W = W # Sn+1 on itself given by the formula a(0 (w, \|rn+1(t1,...,tR,t))) n+1(t1,...,tR,t)) )= 0 (w, ^n+1(t,t],•••,t )|rn+1 (t, t], •••,t )) ))•• Since the correspondence vn+1(t1,•••,tn,t) vn+1 (t1, •••,tn,t ) \jrn+1 (t, 1 1, •••,tn ) is a homeomorphism of Sn with the degree (-1 )n, then a is homotopic to the identity or a reflection of En+1W. Therefore {H1}= F ° En+1r + ((H) _+ a o En {G) ) and H and H* H ’ belong to the same double coset. Conversely, any element of it r(En+1¥ W — > Z), which belongs to the same double coset of {H}, is represented by a mapping F ’ ° En+1c + (H + a o EnG ’ 0 a) for some mappings F* : (En+1X, x ) — > (Z, zQ) and G 1 : (EW, wQ ) — >(Y, y0)• By setting

CHAPTER I. j F'(dR (x,2-2t) ) , At (x) = i L ( A 2 t (x),

5 < t < 1, o < t < i,

j G'(dw (w, 2t-1)) , i £ t £ 1, Bt (w) M L

(B2t(w),

0 < t 1 i,

of a ° E b and b ° c. Let H' be we have null-homotopies A^ and constructed by use of A£ and B£ then H' is homotopic to F ' ° En+1c + (H + a ° EnG'). So, any element belonging to the same double coset of {H} is represented by a mapping of (1 .1k ). Consequently we have proved that {a, Enp, En7 }n is a double coset of *(En+ 1X — >Z) ° En+17 and a ° En (^(EW — > Y ))• The second part of the lemma follows easily. Let n > m > o . The mapping of (1 .ik ) may be regarded that it is constructed from the mappings a, En~rab, En-mc and the homotopies A^, En*mBt . It follows then la,

(1 -1 5 ) For the case

En7)n

n = 0 , we write

i). ii). III). IV).

E V

C (a, Em (En~“f3 ), Em (En'm7)}m • {a, 3 , y )0 simply as (a, P, y) C jt(EW — >X). If one of a , p or y jls o, then

{a, Enp, En; = 0• l a , En( 3 , En7)n ■ > En+18 C I L a o Enp = 3 o 7 = 0 , then (a, Enp, En y » 5 )}n . a o Enp = p o 7 o 6 = o, then [ a , EnP, En (7 » 5 )>n C IL Ca, En (p o : ), En 5 )n . 1 L a ° Enp ° En 7 = 7 » 5 = 0 , then Ca ° Enp, En7 , EnS)n c {a, En (p < 7 ) , En6 }n . I£ P o En7 = 7 o 5 = 0 , then a ° {P, En7, En 5 ) C (a ° 0, En7j

r,no

Proof: o ) • If a or p is o, then we may chose a or b as the trivial map and A^ as the trivial homotopy. It follows that the class of the mapping H of (1 .1k-) belongs to a ° En (it(EW — > Y ))■ Thus {a, EnP,

E °- The case 7 = 0 is proved similarly. i). According to the definition, consider the mapping H of (i.i1*), whose class belongs to {a, Enp, given from a,b,c,At and B^ • Let d be a representative of 5 . Similarly, by use of the mappings a,b,c ° d and the homotopies A^ and B^ 0 d, we obtain a mapping H 1 whose class belongs to {a, Enp, En (7 ° s )^n such that the equality H 1 = H 0 En+1d holds. Thus i) is proved- The proof of ii), iii), and iv) is similar, and left to the reader. Proposition 1 .3 : En+1p, En+17)n+l = Enp, En7)n and -E{a, Enp, En 7 )n C (Ea, En+1fs, En+,7 )n+1 (n > 0 ). Proof: Let a,b and c be representatives of a e tf(En+1Y — >Z), P e tt(X — >Y) and y e it(W — >Z) respectively. An arbitrary element of [ a , En+1P, En+17)n+1 is represented by a mapping H : En+2W — >Z which is given by / . a(E B„t_ ,(dR (w,s ))), i < t < l, E ( a VT,(a-R (v}s),t)) = < ^ m R ( A 1_2t(En+1c(dR (w,s))), 0 E W be a homeomorphismgiven bya (d-g^ (dR (w, s ),t )) = d-^ (dR (w, t ), s )•Then, as is seen in the proof of the lemma 1.1, {Ho a} = -{H}- By (1 -12) and by the definition of nQ, we have that a0a(EnB2t_1(w)), 4 < t < 1, fiQ(H o a)(dR (w,t )) = a,A, ot.(Enc(w)) ■ 0 < t < i'‘ ‘(^l -2t Thus ftQ{H o a } belongs to {ft0a, Enp, En y ) n ' Conversely, an arbitrary eleof {ft0Q;, En3, En7)n is represented by a mapping ftQ(H ° a) as above, since is a homeomorphism. Then the equality of this proposition is proved. Let l be the class of the canonical injection i : Z — >ft(EZ). By (1 .1 1 ) and by iv) of Proposition 1 .2,

-ECa, Enp, Enr)n =

°

Ellp’eI17 )n ) c-n"1[t °a, Enp, En7)n

1° t o a , En+1P, En+17)n+1 = {Ea, En+V

=

En+17 )n+l .

Then the proof of Proposition 1-3 is established. Proposition 1.b : If a ° En3 = £ 0 7 = 7 0 5 = 0, then (a, EnP, Enr)n ° En+1S = (-1 )n+1(a o En {p,/; b) ) . Proof: Let a,b,c and d be representatives of a € n(EnY — >Z), and b e it(V — >W), respectively. An arbi­ € it(X — >Y), 7 ejt(W — >X) trary element of {a , Enp, ° En+1S is represented by a mapping H En+1V — >Z given by

a(EnB2t - i ( ^ d ( s ) ) ) ,

i < t < 1,

H(dg(s,t)) = | A 1-2t

(^nd (s ))) >

o < t < i,

where A. and B. are null-homotopies of a 0 E b and b ° c , respecn tively, and s e S = E V- Let C^ be anull-homotopy of c ° d. Consider a homotopy : (En+1V, vQ ) — >(Z, zQ ) given by the formula a(B2t_1(End(s))) Hu (ds (s,t)) =

for

En+1V is an identification defining En+1V. The map­ ping K represents an element of {p, y, 5 ). Since the correspondence ^n+1 (t1/...,tn ,t) ^n+1(1-t,1 1,...,tn ) is a homeomorphism of Sn+1 with the degree ( - O n+\ then {HI = {a ° EnK ° a) is an element of ( - O n+1(a ° En

CHAPTER I.

(3 , y, 5))- Conversely an arbitrary element of (-1 )n+1(a ° En {3 , y, 5}) is represented by the above composition a ° EnK ° a for suitable homotopies Bt and Ct . By use of the homotopy H ]_u , it is proved that {a o EnK ° a) € {a, En3, En 7 ) 0 En+ 1 6 . Then the proof of Proposition ^ .k is completed. Proposition 1 .5 : Assume that a o p = p o 7 = 7 o 5 = 6 o £ = 0 , {3 , 7f 5 } ando € (3, y, S} Ee .Then there exist elements X {a, 3, 7 }, |i € (3,7 , 8 } and v£ { 7 , 6 ,e) such that \ o E 5 = a ° (i= 0, 3 °v = i a ° E e = 0 and the sum {\, E5, Ee} + {a, \i, Ee} + {a, 3 , v} contains the zero element. Briefly says, {{a, 3, y), E 6 , Ee} + {a, (3, y, &}, Ee} + {a, 3 A y , 6 , e}} = 0 . o

€ e

a o

This proposition is a generalization of ii) of Theorem ^ . 3 in [25], obtained by changing spheres by topological spaces, and the proofsare same. Proposition 1 .6 : Let a, a ’ e n (E1^ — >Z), 3, 3* € ir(X — >Y) and 7 , 7 * e jt(W — >X). Then or if if

Ca, Enp, Enr)n + (a, EnP , Enr ’)nD Ca, E11?, En (r + W = EW', (a, Enp, Enr)n + la, EnP ', Enr)n = Ca, En (p + n > 1 or 7 = E r 1(X = EX', W = EW'), Ca, E11^, En7)n + (o', Enp , Enr)n D (a + a ' , n > 1 or p = EP' and y = Ey'(Y = EY', X = EX', Proof:

since

First remark that the group

n > i or We shall of the subgroup [a, En 3 • En 7 }R +

r'))n

If

i

p'), En7)n uPfr, Enr)n

W = E W 1).

it(En+1W — >Z)

is abelian,

W = EW' • prove the last relation. {a + a', En 3 , Eri7 }n is a coset (a + a 1)o En+ 1 n(W — >X) + ir(En+1X — > Z ) ° En+ 1 7 . {«’,E n 3 Ell7 }n is a coset of the subgroup

a . En+1it(W— >X) + a' ° En+1it(W— >X) + n(En+1X — >Z) . En + V which con­ tains (a + a') ° En+1( W — >X) + n(En+1X — >Z) • E ^ V Thus it is suffi­ cient

to prove that

Ca, EnP, En 7 )n + Ca', EnP,

s-nd (a + a', E1^,

En 7 )n

have a common element. According to the definition of the secondary composition, we con­ struct mappings H and H* : (En+ 1 W, w Q ) — >(Z, zQ ) by (1-1^), which repre­ sent elements of {a, En 3 , and {a*, En 3 , ErV ) n > respectively from re­ presentatives a,a',b and c of a, a 1, 3 and 7 , respectively, and by use of null-homotopies At of a « Enb, A£ of a 1 0 Enb and Bt of b °c. Let 0 : En_1W x S2— >En+1W = En~1 # S2 be a mapping defining En+1W = E ^ ' V

where

En_,W = W'

if

n = o.

Let

a : En+1W — >En+1W

be a homeo-

morphism given by a(0 (w, ijr (s,t ))) = 0 (w, \|r (t,s)), then a is homotopic to the reflection of En+1W and thus {HQ 0 a} = ~fHp} for arbitrary HQ : (En+1 W, w Q ) — >(Z, z0 ) • Define a mapping H'’ : (En+ W, w Q ) — >(Z, z Q ) by H(0 (w, \|r2 (2 s,t)))

for

0 < s £ i,

GENERAL COMPOSITIONS AND SECONDARY COMPOSITIONS

Then it is verified directly that H ’ 1 ° a = (H « a ) + (H1 ° a) and that H * 1 is constructed by (1-1*0 by use of the mappings a + a 1, b, c and homotopies At + A£, B^. It follows that (H) + {H *) = (H") € (a + a', E11?, En r}n

and the class CH* T} is a required common element. Thus the last relation of Proposition 1 . 6 is proved. The proof of the first two relations is simi­ lar and omitted. A cone CX over a base space X will mean a space given from the product X x I 1 by shrinking its subset X x (1 ) U x Q x I 1 to a single point. Denote by d-£ : X x I 1 — > CX the identification defining CX- We shall identify the space X with the base d-£(X x (o)) by the correspon­ dence x dx (x, o). Consider an element p of jt(X — >Y) and let f : (X, x Q) — > (Y, y Q ) be a representative of p. A space which is obtained from the dis­ joint union of CX and Y by identifying X with its image under f, will be denoted by Y Uf CX or Y Up CX. The homotopy type of Y CX is independent of the choice of re­ presentatives f of p . An element a of jt(Y U« CX — >Z) will be called as an extension ------ 1 — p of a € «(Y — >Z), if the restriction g |Y of a representative g of a represents a- An element y of n(EW — > Y CX) is called a coextension of 7 € jt(W — >X), if 7 is represented by a mapping h : EW — >Y CX which satisfies the condition / d£(C(w),i-2 t )

if

h(dw (w,t)) = | e y

if

C : W — >X is a representative of y . An extension a of a exists if and only if extension 7 of 7 exists if and only if P ® 7 = o. We shall use the following Identification

where

(1 .1 6 )

En ( Y Up CX) = Eft Upl C(EnX),

a o

p=0 .

P 1 = Epp,

given by the correspondence 0 (d-£(x, t), ^n (t 1 > •••»tR )) d£(0*(x, i|rn (t1, ...,tn )), t ), where Z = EnX and 0 and 0 ’ are mappings which define En (Y CX) = (Y Up CX) # Sn and EnX = X # Sn respectively. Proposition 1 .7 : Assume that a ° Enp = P ° y = 0 for a e jc(EnY — >Z), P e it(X — >Y) and y e jt(W — >X). Consider an extension a e jt(En (Y Up CX) — >Z) of a and a coextension y e jt(EW — >Y CX) of 7 . Then the set [a ° En7}of the compositions a ° E n 7 coincides with the secondary composition (— 1 )n Cee^ EnP, ■ Proof: For the above mapping h, we set h(dy(w, t))= ^ 2 ^_ 1 (w) for i £ t £ 1 , then Bt is a null-homotopy of b ° c, where b = f is a representative of p . We consider also that arepresentative- g of a satisfies the formula

g(d-£(x, t)) = ^ ( x )

^ or a null~k°ni0' t:opy

Aco­

CHAPTER I.

n

A Q = a ° Enb = (g| EnY ) ° Enf . Then it is verified directly that g o Enh = H ° a 1 for a mapping H given by (1.1b ) and for a homeomorphism a of En+1W in the proof of Lemma 1.1. Then the class a ° Eny of g o Enh be­ longs to (-1 )n {a, Enp, En7}n - Conversely, any element of (-1 )n {a, Enp, En7)n is represented by a mapping of the form g ° Enh = H ° a"1. Thus Proposition 1.7 is proved. Proposition 1.8: a , p and 7 are same as the above proposition. Let p ejr(En+1X — >Z U CE1^ ) be a coextension of Enp, Then the set of n+1 T1 T"1 all the compositions p ° E 7 coincides with -i^ [ a , E l , E' 7 ^n > where i is the in.jection of Z into Z CE^. Proof: Consider a homotopy Hg : (En+1W, w Q )— > (Z CE^Y, zQ ) given bythe formula (by the same notations of (1 .114 )). (A . H (d(w, t))

=

\

(Enc(w))

for i < t
X Uy CW) — of 5 e *(V — > W ).

Proposition 1.9: a , p and 7 are same as Proposition 1.7. Let p e rt(X CW — > Y ) be an extension of p. Then there exists an ele­ ment X of jr(En+1W — >Z) such that (E1^)**, = a 0 Enp . The set {A,} of such elements forms a coset of jr(En+1X — >Z) ° En+17 which is a subset of [ot, Enp, ER7}n • Furthermore, any element X of [ a , Enp, ER 7 }n satisfies therelation (Enp)* X = a 0 Enp for some choice of p. Proof: The element a ° Enp is represented by amapping G : En (X CW) — >Z satisfying the formulas G(x) = a(Enb(x)) for x e EnX and G(d^(w, t)) = a(EnBt (w)) for w e EnW = V, where a,b and c are re­ presentatives of a , p and 7, respectively, and is a null-homotopy of b 0 c . Since a ° Enp - 0, the restriction G| EnX is null-homotopic, and by the homotopy extension theorem, we have a homotopy G : En (X U CW) n 7 — > Z such that GQ = G and G ](E X) = zQ. Then there exists uniquely a mapping H ] : En+1W — >Z such that H ] 0 Enp - G 1• Thus the class X of H 1 satisfies the condition (Enp)*A, = a ° Enp • Furthermore, any element satisfying (E p)*X = a 0 E p is represented by a mapping H 1 given in this way. Denote that A^ = G^| EnX, then A^ is a null-homotopy of a ° Enb . Consider a homotopy Hs : En+1W — >Z given by the formula

HQ

repre

GENERAL COMPOSITIONS AND SECONDARY COMPOSITIONS Gs (d^-(w, (2t-l+s )/(l+s )) H (dv (w, t)y - s ' A 1_2t(Eno(w) ))

if

(1-

if

0 £ t £ (l-s)/2,

s

)/2

1

< t < 1,

Then Hs is a homotopy between PL and HQ = H of (1 .1 k ). Thus X = (EL1 = {H} belongs to [ a, En3, E 7} . In the above discussion the homo­ topy is fixed but we may use arbitrary null-homotopy A^ of a o Enb . Then, as is seen In the proof of Lemma 1.1, the set {A,} is a coset of jt(En+1X — > Z ) o En+17 • Furthermore, by changing {a,} rims over the whole of {a, Enp, Consequently Proposition 1.9 is proved.

CHAPTER II

Generalized Hopf Invariant and Secondary Composition. In the following, we shall devote to the consideration on the homo­ topy groups of spheres Denote by of

Sn .

Obviously,

= *(s£Usm ).

Ln

£

the homotopy class of the

in 0 a

= a = a °i

identity mapping

a € ^(S™).

for

It follows from (1 .7 ) (2 .1 )

a ° (P1 + P 2 ) = a ° P 1 + a ° P 2 = a 1 0 Ep + a2 o Ep, and The

defined as in 0m n

(1*3).

and

in particular,

k(a o Ep) = (k a) ° Ep

(a1 + a2 ) o Ep k(a o p )= a °

for an integer

reduced join a#Pe ITp+q(ST11+n) of

(kp)

k.

a€Jtp(Sm ) and P€Jt^(Sn )

the previous chapter by use of the special mappings In particular, from the definition of

is

0^ ^ and

E11 it follows that

Ena = a # tn • Also the secondary composition (a, En3 ,En, } n e *j_+n+ 1 (Sm)/ (a . En « 1 + 1 (Sk ) + « j + n + 1 ( s ' W + V ) P€^j(Sk )

and

7 €iri (SJ*)is defined if the condition (1 .1 3 )

where we use the mapping

di+n

of (1 .8 ) in place of

d

of

of ac«k+n(Sm: is satisfied,

(1 .1 h ).

Of course, all the discussion in the previous chapter may be ap­ plied in this particular case. Now we introduce from [9] the concept of reduced product of and some necessary results. (Sm )n

unit

Denote by

is a free semi-group with the set eQ . Each point of

(S™)^

the reduced product of Sm - eQ

of generators and the

is represented by a product

x 1,...,xt e Sm . For fixed positive integer 16

t,

Sm .

x 1...xt

of

denotes the set of

17

GENERALIZED HOPF INVARIANT AND SECONDARY COMPOSITIONS

all elements Smx... xSm

x 1 ... x^.

Then the topology of (Sra)t is given from the product

of t-times Sm under the identification: Smx...xSm ->(Sm )t (Sm )t “ (Sm)t- 1

(x1, ...,xfc)-> x^-.x^.

is 311 °Pen tm-cell.

of

By giving the

weak topology, (S™)^ is a CW-complex and (Sra)t is its tm-skeleton.

We iden­

tify (Sra)1 with Sra in the natural way. Consider a mapping (Sq-)t -^(Sm )t

f : (S^, e0 )->(Sra, eQ ), then a mapping (f)t :

(t = 1, 2, ...,)

f (y 1)• •-f (yt )•

Obviously

is given by the formula (f )t (y1. . yt ) =

(f’)s =

The canonical injection of ( S ^

such a way that

x^-.x^

for> s 1 t * i : Sm-> ft(Sm+1 ) is extended

to

the whole

is mapped to a loop in Sm+1 which is

represented by a suitably weighted sum of the loops resulted mapping is injective, denoted by

i(x1 ),..., i(xt ).

i :

The

ft(Sra+1 ) and called

also by a canonical injection. The following diagram is homotopically commutative

(S*1^ (2.2)

— i— > u(sq + 1 )

4(sm )oo

n! ' _ i _ >ft(3“+1)

.

As one of the main results on reduced product space, the canonical injection induces isomorphisms of homotopy groups: « (n(Sm+1 ))

for all

i* : jt^( (Sm )co) ->

q.

That is to say,

and

+1 n(STn+ ) have the same singular homo­

topy type, and thus we have Lemma 2 . 1 .

i* : n(K->(Sm )oo)jt(K-> ft(Sm+1 ))

arbitrary finite cell complex (or more generally Define a one-to-one mapping (2 .3 )

CW-complex)

£!, = l'I « nQ : n(EK -> Sm+1) n (K

(Sm )m )•

£21 : n^+1 (Stn+1 ) =

mutative.

i’ :

(S111)^,

the following diagram is com­

T71 yy»I1 it(K -> Sm ) — — > *(EK -> Sm+1 )

(2.If)

i>\

I

fl, (S“ ).)

Consider a mapping

(2 - 5 )

)m )

k. For the injection

formula

K.

a1 by

In particular, we have isomorphisms for all

is one-to-one for

h^ (xy) = 0 m m (x, y) \

•

h^ : ((Sm )2, Sra)-» (S2m, eQ ) given by the for

x, y e Sm . Let

: ( (s “ )M, Sm)-> ((S 2n,)oo, e Q)

CHAPTER II.

18

be the combinatorial extension [9] of h^, commutative.

then the following diagram is

h

(2.6)

—

(Sm )00

>

(f* ).

---- >

(S2m)00 .

In general, the combinatorial extension mapping

h : (S™)^ (Sicn:1)oo of a

h r : ((Sm )^,)^_-j )-* (S^™, eQ ) is defined by the formula b(x1?...,xt ) = n h'(x0(1).---,x0(k))

where letters

o

is a monotone increasing map of k-letters {1, 2,...,t)

{1, 2,...,k}

and the order of the product J] in

(S^)^

into t~ is lexico­

graphic from the right (left). Define a generalized Hopf invariant

(2 .7 )

H =

h^

H

" n 1 : jt(EK -» sm+1 )-» n(EK — > s2m+1 ).

In particular, we have homomorphisms Proposition sider elements

by the formula

2.2:

K

Let

and

L

H : n^+1(Sm+1 ) -»

(S2m+1 ).

he finite cell complexes. Con-

a€n(EK-> 5ra+ 1 )),, P€ji(L-»K) P€k(L-»K)

and and

yye ^^(S

),

then

H(a » E p) = H(a) • E and

H(E7 ° oc) = E (y#y) ° H(a).

Proof: H(a ° E P ) = (n"1 ° i* ° 1^* ° i^1 ° a Q )(a « E P ) by (2 .7 ) = ^((i* ° ^

° C ’)(no«) 0 P)

^

= (fi~n o i* o h^* . i"1 o n0 )(a) o EP =

H(a)

o

EP

(1.12)

by (1 .1 2 )

.

H(E/ ° a) = (n”1 ° i # ° hr# ° i"1 » n0 )(Er » a) = n"1 ((i* o hp# « i'1 )(n E 7 ) ° siQa)

by (2 .7 ) by (1 .1 2 )

= n~1 (fiE(r # 7 ) ° (i» ° hp » • i,1 )(nQa ) ) by (2 .2 ), (2 .6 ) = n"1nE(r # 7 ) ° (n"1 0

0 «,)( 1

and elements

a€^(EnK-^ Sra+1 ), pejt(L-* K)

a o Enp = p o 7= 0.

satisfy the condition

19

and

7£jt(M->L)

Then

H{a, Enp, En7}n C (Ha, Fp' fi, En7}n . Proof: {a, Enp, En7)n

By

Proposition

of

7-

(-1 )n a 0 Up'y

is the composition

ota-si (En (K Up CL)-> Sm+1)

7

1.7, an arbitrary element of

of

a

and the n-suspension

By Proposition 2.2,

ejr(EnK ^ En (K Up CL))

This shows that

En7

of a coextension

H((-i)n (a o ER7 )) = (-l)nH(a) o En7-

be the class of the Inclusion, then

a 01 = a.

sion element and

of an extension

By Proposition 2.2,

H(a) is an extension of

H(a).

H(a) =H(a

1

is a suspen­ ° 1 ) = H(a) ° 1 .

Then, byuse ofProposition

1 .7, we have that

(-1 )nH(a 0 ER7 ) = (-1 )nH(cn) 0 En7

{H(a), Enp,

Thus the proof of the proposition is established.

We remark that the above discussions on the

Let 1

is contained in

(Sra) 00 and the results

on

generalized Hopf invariant are still true if we replace the sphere

Sra

by some suitable space, for example finite cell complex. Theorem 2 . k .(James [10], Toda [21]).

Let

p

be a prime and

Let h : (S111)^-^ (S^m )c,be a continuous mapping which maps and

epn] homeomorphically onto

h* :

((Sra)oo, Sm )

is even, then for

Spni - eQ.

If

m

(Sm )p_ 1

is odd and

m > 1•

to

eQ

p = 2, then

tt^( (S2m )to) are isomorphisms onto for all

i . If

m

h* : Tti ((Sm )oo, (Sm )p_1)-^ «i ((Spra)oo) are isomorphisms onto

i < (p+i)m-i

and isomorphisms of the p-primary components for all

i.

By use of Theorem 2 of [ 6 ] , we have that

Proof:

b* : ^((S” )^ are isomorphisms onto for

i < (p+i)m-i.

Next we recall the cohomological structure of

(S™ )oo, which is com­

puted from the cohomological structure of the k-fold products Sra

under the identifications to

u^

of Hkm((Sra)oo)

m

is odd, then

then if i

m

dual to the cell

i ^ u ^ = u 2k+1

and

= (

e^™,

if

of

Then we have that, for generators the following relations hold.

^ 2 ku2h = ( k+h ) ^2 (k+h)*

a.= Z

Is even.

: ^((S™)^,

^ ra)^..

S^x.-.xS™

m

is odd and

If

p = 2

m

and

is even, a

=

Then it is easily verified that the injection homomorphism a

)-» H^~ ((Sm )1,

a

)

is an isomorphism

onto for

i < pm,

If

the

CHAPTER II.

20

induced homomorphism into for all

i

h* : H^( (S^111)^, A ) —» ^ ( ( S 111)^,

by means of cup-products, where ^((S™)^, Now let of

a

)

a

)

G

under

is a subgroup of

is an isomorphism

^ ( ( S 111)^, a ) which is

i*.

be a set of the pairs {&, x )

X

& : I1-> (S^™^

(Sn )ro and

)

and that we have an isomorphism h V u s P 10)^ a) ® G * ^ ( ( S ™ ^ ,

isomorphic

a

is a path with

such that

i(i) = h(x).

x

Let

is a point p(i, x)

=

h(x) = j0(1 )- Then (X, p, (sP™)^) [13]-

is a fibre space, in the sense of Serre s t Consider the spectral sequence {Er ' } associated with this fibre

space,

then by the maintheorem of [1 3 ] we have an isomorphism E* ~ B* ® H*(F, a ),

where

B* = ^((sP™)^, a ). The pairs

imbedding of X.

(S™)^

(i, x) into

By the retraction,

of the constant paths X such that

(S™)^

h is equivalent to

form an

is a deformation retract

p.

F = p-1(e ). 0 We shall prove that the injection

i(I1) = h(x)

The subcomplex

(S111)

of .j is

contained in the fibre

morphisms

i* : H*(F, A) « H*( (Sm ) Let

to

i*.

Let

GQ

i1 : F C X be the injection.

be the subgroup of

is in the image of

sition

a

).

i*

andi*

H*(F, maps

)

a a

)

GQ

corresponds to

then we have

G0 ) = o

E*,

) ® G.

dr (B*

is equivalent

which corresponds to

G

for

i < pm.

by

i*.

for

fQ

tained in

i*

is

equivalent to

the compo-

for

G-

r > 2

Since

r > 2.

dr (B* ® 1) =

o

for

Consider the limit of

B*

H*(X,

a

)

r > 2, G0

« p*H*((Sp m ) ,

is isomorphic to E* and H*(X, a ) # dr-images in B ® GQ are trivial. Assume

is an element of minimum dimension in Then

that

B* ® GQ

This means that every

Gq.

GQ

isomorphically onto

then it is a graded module associated with Then it follows that

as modules. that

i* ° i*

: H*(X, A ) -> E0^* -» E°’* = 1 © H*(F, a ). It follows then

and its limit in E*

in

Then

is an isomorphism into

The injection homomorphism

dr (i ® G0 ) = o

a

induces iso­

1, a).

Thus i* : H^CX,A )-> H^(F,

H*( (Sm )p__1,

iQ : (Sm )p_1 C F

dp (l ® fQ ) is in

B*® GQ

H*(F,

a

)

and hence

which isnot con­ d (l ® fQ ) = o

21

GENERALIZED HOPF INVARIANT AND SECONDARY COMPOSITIONS

for

r > 2.

Since

1 ® fQ

is not a

d -image, then

But this contradicts to the isomorphism fQ

does not exist.

B* & GQ « E*.

Consequently we have proved that

i* : H*(F, A) ~ n ( ( S m )p _ v

1 ® f Q4 o

in

Thus such

E*.

an element

H*(F, A ) =

GQ

and

A).

Applying generalized J. H. C. Whitehead's theorem in [15], to the injection odd and

iQ, we obtain isomorphisms

p = 2

iQ^ : ir^( (Sra)

1)

*^(F)

and isomorphisms of the p-primary components if

if

m

m

is

is even.

Compairing the exact sequences of homotopy groups associated with the pairs

((S™)^, (Sm )

lence) of (S®^ statement for (Sra)p_1 ) into

into i*

.j) and

X,

(X, F)

by the injection (a homotopy equiva­

it follows from the five lemma that the above

is true for the injection homomorphism of

*.(X,F).

((S™ )oo,

Since the diagram

*1 ((Sn,)oo, (Srn)p_ 1 )

------ > «± (X, F)

P* ,.((Sp m ) J is commutative and since

p^

is an isomorphism onto for all

that the theorem has been proved, The mapping

i,

then we see

q. e. d.

h^j : ((S™)^, Sm ) -» ((S2111)^,

the condition of the above theorem.

e0 ),m

> 1,

i < 3m-i

and

satisfies

Thus

hm* : ni ((stD)oo' are

isomorphisms onto for all

i

if

m is

phisms onto of the 2-primary components

odd or if

for alli

if

Now consider the exact sequence for the pair (2.8)

» ^ ( S ® ) - * «((S“ ) J - .

Define a homomorphism (2-9) where

A

is even. ((S™)^, Sm ) :

* 1

Then

° p = dO^^a)

° d~]p)

°Ep)) = a(ht^Vft1 (aoE20))by (1 .1 2 )

C = A(a°E2p ). The element 2m-cell the cell

(Sm )2 - Sm

A (L2m +1 ) is the class onto

Sm ,which is the image of

(Sra - eQ ) x (Sm - eQ )

(Sra x Sra,Sra v Sm )-> ((Sm )2,Sm ). nition of Whitehead product that =

L2m+1 ^ ° 7 e ^* Proposition 2.6.

satisfy the condition

the attaching map of

Let

E(aop)

onto Sm v Sra

the

under the

the

attaching map of identification

:

Then it is verified directly from the defi­ A(t2m+1 ) = +•

a

g

*(K-» Sm ), P

= p o 7 = 0 . Then

Thus

g

nk (K)

_+ i Lm > Lm ] 0 7

and 7 g ir^(Sk )

23

GENERALIZED HOPF INVARIANT AND SECONDARY COMPOSITIONS

H{Ea,Ep,E7}1 = - A_1(a°p) o E27 . Proof: Let representatives of and let

and

p respectively.

b : (CSk ,Sk )-> (K U^CS^, K)

the injection

i

(S^,

of

Sm

b : (Sk ,eQ ) -> (K,xQ ) be

Consider the space

K U^CS

be the characteristic mapping. By (2 .3 ) and (2 .^),

into

i^ a o p = 0 . Thus the composition

implies that in

a

a : (K,xQ) -* (Sm ,e0) and

and we can extend the mapping

a ° b

Consider

E(aop) = 0

is null-homotopic

over

((Sm )x , Sm ).

a : (K UbCSk ) ^

Then there exists a mapping

a

]r

a'

such that the following diagram is

commutative: (CSk,Sk )

------ > (K UbCSk,K)

---5--- > ( ( S ^ S ® )

| P

| \

(5^+ \ e 0 )

where a o b Thus

p

and

p*

represents a' o p

are shrinking maps of (1 .1 7 ) given by use of a ° p,

and also

then

a ° b

represents an element of

a 1 represent an element of

Next consider a representative 7

of

7.

By (1.18), the composition

1 .7 , the composition

dary composition

----^ ---> ((S2tD )„>eQ ),

represents

a o c : Si+1-> (S™)^represents an

{3^ a, p, 7 }.

is a common element of

d~1(aop).

of a coextension

E 7 . By Proposition element of the secon­

{h^o a o c} = {a'° p

Thus the class

h^ti^a, p, 7 } and

Since

(crop )).

c : Sl+1-> K U^CS^

p 0 c

d^.

0 c)

hm^(^"1 (aop)) o E 7 . We have

HCEof,Ep,E7 ]1 = HCft”1 i^.a ,Ep,E 7 }1

by (2 .k)

= - H(ft~1 (i* ol, p, 7 } ) by Proposition 1 . 3 {i# a,p,r}

by (2 .7 )

(d-1(a°p)) o E27

by (2 .1 0 )

= ft"1 (hrn^(^”’1 (a«p )) o E 7 )

by (1 .1 2 )

=and

A - 1 (a°P) o E 2 7

Then it follows that common element

ft”1[a1 o p 0 c).

= ftj1

H{Ecr,Ep,E7 }.,

- A 1 (a«p) o E 2 7

and

have the

These two sets are cosets of the same sub­

group H(Ea ° Ejri+1(K) + jr^+ 1 (Sra+1 ) o E27 ) = HE(cz) o Eni+1(K) + H*. + 1(Sm+1 ) ° E27 = H* .+ 1 (Sm+1 ) o E27 Therefore we have the equalityH{Ea,Ep,E7 )1 = -

by Proposition 2.2 by (2 .1 1 ) . A

1(a op ) o e27

and

the

2k

CHAPTER II.

proposition is proved. Finally, it is well known that

H[ in > tn ] = + st^n-l

for even

n.

Then it follows from Proposition 2.5 Proposition 2 .7 . H(A (L2 n+l ^ = — 2 L2 n-l

for even

n.

(cf.(11 .1M).

CHAPTER III

Reduced Join and Stable Groups Consider the reduced join 3€JTp+h(Sq );

(Barratt-Hilton [5]•)

= (-1 )

a#e Proof:

^

^

• EP+kp = ( - l ) P V p . E^+V

It was known that

be a mapping given by morphism of degree (-1

«#tn = Ena.

(-1 )P S .

ijoc = (-1 )knEna.

Let

g (0p^s (x,y)) = 0s^r (y,x), The relation

)pnip+n 0 Ena o ("1 )^P+k^nip+k+n

L„#a

holds.

ap g : Sr+s -> Sr+s

then

= (a

IT

a

is a hemeo-

} o (a#/

XI

) o {a

iijJJ”r.K

By (2 .1 ),it follows that

By (1 .5 ), 0

a#£ = (a°Lp+k) # = = (_1)(p+k)hEqa o Ep+kp and

ae*p+k (Sp ) and

then

Proposition 3•1

=

a#£ejr.p+^+k+h(Sp+q) of

a # = U p °a) # (P0^ +h) = (ip#P) 0 (a#Lq+h) = (-DPVp-E^a. The difference

E^"1a o Ep+k~1p - (-l)khEp~1£ o E^+h_1Qj vanishes

under the suspension homomorphism sequence (2 .1 1 ).

q. e. d.

E,

and thus it is an image of

of the

A

The following result for this element was proved in [22:

Theorem (^.6)]. Proposition 3*2 k < Min. (p+q-3 .2p-^ ) and Eq-1a 0 EP+k-1p _ (-■, and

Let aejTp+k(Sp ) and

^€TCq+h ^ ^ ^

h < Min. (p+q-3,2q-^ ),

and assume that

then

e Eq+h-1a =

o E2^ 2H(a) o Ep+k"1H(e)

2 ([tp+qL_i ^ip+q_1J 0 E2q“2H(cr) o Ep+k“1H(p)) = 0. (Recently, this formula

is established without restrictions on By Proposition 2 .5 , we have

k

and

h [^].)

[ip+q_• ,>Lp+q-1] 0

H(p) e A (E2q-H(a) » Ep+k+1H(f3)) C *p+q+k+h_ ( S P+Q" 1 ). Proposition 3«2,

p+q+k+h < 3(p+q-i)-3« 25

2H(a) o Ep+k_1

From the assumption of

Then this is the case that

A

is one

26

CHAPTER III

valued.

Thus we have the following corollary. Corollary 3*3;

tio n

3 .2 ,

Eq_1a

.

Under the same assumption of the previous proposi-

Ep + k _ 1 p -

The reduced

( - l ) k h E p _ 'p

°

Eq + h -1 a

= A (E 2 q H (a )

o Ep+ k + 1H (3 ))

joinyields us to give some relations in secondary com

positions. Proposition

3 .^:(Proposition

i).

that ae

the condition

Assume

a#p = e#7 = 0 ,

E P + h + r p ^E p + h + q + k ^ }

b.6 of [25]-)

(^ (Sp ),

then the secondary compositions

( _ , j h k + k i + i h +1 { E P+Ep +q+k7jEq+k+r+ia) + (_, j ^ g P + q ^ g C L + r + J ^ EP+h+r+^gj

contains the zero element•

Proof: Let a,b spectively, and let respectively. H, H 1 : Ss+U

A^

and

Denote by s P ^ +r,

and

in

c

be representatives of be null-homotopies of

the identity of

s = p+h+q+k+r+i,

Sn .

a,p a # b

ta *

H

lq+r’ lp+h * P * V = {Eq+ra,

and

re­ b # c

Consider mappings

given by < t < 1

( a # i q + r )(1p +h# B2t-1 )(x)

for i

(A1-2t# ir )(1p+h+q+k^ o)(x)

for

o < t
K # Sn = EnK.

s2n+k

The restriction

1K # f |

EnK - S2n

(2 n+k+i)-cell attached by

is a

tion is homotopic to zero in have a homotopy which maps EnK

EnK.

Gs : En+kK -> EnK

s2n+k

such that

represents

to a point.

G 1 = F 0 E1^.

Ln # a = (-1)knEna. Enf

By the

to

S2n,

Since

then the restric­

homotopy extension theorem, we

between

GQ = 1R # f

and a mapping

Then there exists a mapping

G1

F : s2n+k+l

For the class of such a mapping

F, we have

the following Lemma 3*5: satisfy (3*1 )• Let Assume that

(i -(-i)k ) En+k-tla

where

of *n + k ( S n )

then for any coextension

there exists an element

a*

of

i : S2n -> EnK

F ° Enp.

01 # (-1)knLn *

such that

is the injection.

Denote by

AQ = l n # f

Gs : En+kK -» EnK

Gg

EmK = K # Sra. By Proposition 3 .1 ,

to

between

0^ : K X Sm -> EraK (m = n or n+k)

Then there exists a homotopy

t : Sn —> Sn

and

*2n+2k+i

°l!

tK # a = (En+kp )*(3^ a* + (-1 )nkEh7 ) in *(En+kK-* EnK ) holds,

tion which defines

from

K .

?eir2n+2k-h+i

Proof: We shall give precisely a homotopy and

and an integer h > 0

be the homotopy class of the identity of

k < 2n - 2,

the equality

a

Let an element

A1 = f #

t

,

where

is a mapping of degree

1^. # f

the identifica­ # a = (-1)nkEna =

1

Ag : (S2n+k,eQ )-> (S2n+k,eQ )

1

denotes the identity of

Sn

(-1)kn. Now define a homotopy

by the following formulas.

For a point

0n+k n (x,y)

of

S2n+k, we set

0n (df (X,2 S-1 ),

and then for a point

0n+k(d* (u,t ),v)

(u ,t),v ) ) 5

nk

Gs maps

s2n+k

of

t (y))

(d1(u, (2t-s)/(2-s)), f (v)) ( 0 (d1(u,(2t-s)/(2-s)) = I 1 Gs . 2 t ( V , n+k( f ( u ) , v ) )

e0- Since

En+kp

h s
52n+k+1 ^

Next consider the composition

It is calculated easily from the above formulas that

Enp ° F

is given by

( 0n+k+i,n(dn+k(u'2t“1 }'f(v)) ^ ^

°

i < t 1 1,

= \ eo

t < i,

( 0n+k+l;n+k (dn+k (x^ - U ) ^ T^ )} if 0 < t < i, where

0n+^-

= 0n n+ k ^ u ^ v ^‘

indicates the sum

ma^ consider that the above formula

H ] + (HQ - H2 ) of the three mappings

respect to the (n+k+l)-th coordinate

H 1, H2

with

such that by taking homotopy

{Enp 0 F) = (H1) + {HQ} - {Hg},

classes and

t,

H 1, HQ, H2

where

HQ

is the trivial mapping

are given by

H l(0n+k+l,n+k(dn+k(u't)’v » = ,n * l , n V (“ ’t)-f(,1) ’ H2 (0 n+k+i,n+k(dn+k(u't)’v)) = 0 n+k+1 ,n(dn+k(x'* Obviously (-1 )(n+k+ 1

tHQ) = o

and {H,) = (ln+k+1 # f) = tn+k+1 # “ =

En+k+1a = (-1 )kn En+k+,a.

It is verified directly that

(1 n+k+l # t) o 0 ■> (Ef # ln+k) for a botneomorphism ( - 1 )k

of degree

T ^ }} ■

which is given by

Hg =

a : s2n+k+1 _> g2 n+k+1

^(♦2 n+2 k+ 1 (x,,•••,xn+k,t,yi,...,yn))=

,l'2n+2k+1(xl ' - - - , V t'xn+l’--->xn+k’yi’---’3rn )-

It follows that

^n+k+1 * (-1)k n ^n ) ° (-l)kt2n+k+l ° (Eot * tn+k) = ( (Enp)*{P} = (-l)knd - (-1 )k )En+k+1a.

-

1

)

(H2 ) = a .

Thus

On the other hand, it follows from the definition of the coexten­ sion, that

(En_hp)*(7 ) = (1 - (-1 )k )En+k"h+1a. By (1 .6 ),

thus

(Enp)*Eh 7 =

((eH'^p )

) = (1- ( - 1 )k )En+k+ 'a,

and

( E ^ ^ U F ) - (-l)knEhr) = 0 . From the assumption

The pair

(EnK,S2n)is

k < 2n-2,

we have

(2n+k)-connected and

S2n

2n+2k+i is

(2n+k) +

(2n-l).

(2n-1)-connected.

Then it follows from Theorem 2 of [6] that the induced homomorphism (EnP>* : n2 n+2 k+i.(EnK’s2n)-' rt2 n+k+i (S2 n + k + 1 ) is an'Isomorphism onto. Consider the following commutative diagram of the exact row.

30

CHAPTER III

W

k

+ i(s2n) —

> - W k + i ^

—

'> W k . i ^ 3211)

'--^(Enp)*

I

(EV* /o2n+k+i 2n+2k+1 ^

^

Then

(Enp)*({F) - (-O^E^r) = o

an element

a*

implies that

%

{F} - (-1 )knEhr = 3* a*

for

of It2n+2k+i (S2n)* Consequently we have proved that

iK # a = (Gq) = (G1) = (En+kp)*{F) = (En+kp )* (i* a* + (-^)kn^ P 7 ),

and the

Lemma is proved. Theorem 3.6:

Let an element

h > o satisfy (3.1 )• a* e jr2n+2k+i 3 ° E^oc = 0,

Assume that

a

of *n+k(Sn ) and an integer

k £ 2n - 2,

such that, for any element the composition

En3 E^a*

then there exists an element 3

of

^t^™)

satisfying

is contained in the sum of cosets

(-1 )km+kt+t{Ema,En+k3,En+k+ta)n+k +(-1)kn+h+t+1 {En3,En+ta, (1-(-1)k)En+k+ta)h+t_. Further if then

(1 -(-1)k )En+k+1a = o,

of

3.

a 1 =E^a. Let

3 ° E^a = o,

By (1 .1 6 ), we identify

a relation

a : sn+k+t+l_^sn+k+t+1 of degree = W

there exists an extension EtK

p f : EtK =Sn+t Ua , CSn+k+t

as in(1 .1 7 ),then we have

3

k

is even,

En3 ° E ^ * e (-1 )km+kt+t{e % E n+k3,En+k+ta}n+k. Proof: Since

->

in particular if

(-1 )t

with

Sn+t UQ , CSn+k+t

-> sn+k+t+1

where

be a mapping defined

p f= o « E^pfor a given by

3 c it(E^K

homeomorphism

a U n+k+t+l (t 1* •••>^ k + t ' ^

+t+i(V * - - ' W ' t' W +i'-**'tn+k+t)- Consid^

the reduced join

# a € *(En+k+tK -4 Sm+n). First we have P # a = (p •

e\

)

# (in • a)

= (P # in ) • ( a K # It ) # a)

by (1 .5)

= E1^ • U K # a # ( - 1 )kti )

by Proposition 3-1

= E11? • ( - 1 )ktEt ((En+kp)*(i*a* + (- 1 )knE^ y )) = (-l)ktEnP o (En+k+tp)*(Eti*a* + (-l)knEh+tr) = ( - D kt(En+kp')*(En+ka)*(En 6 . e V =

( - 1 ) ^ k + ' ^t ( E n + k p ' ) * ( E n P

o Et a *

+

by Lemma 3-5 by (1 .6 )

+ (-l)knEnP . Eh+S ) ( - 1 )knEn P • Eh + t r ) .

By Proposition 1.7 , EnP

.

Et+ h 7

e

( - 1 )h + t ( E n B , E n + t a , ( 1 - ( - l ) k ) E n + t + k a ) h + t

.

51

REDUCED JOIN AND STABLE GROUPS

Next

P # a = (im ° P) # (a ° in+k) = (im # a) ° (5 # In+1

for

then

composition

for

then

{r 3

(cf. Chapter V). Let

the generator

generates

Elln = ^n+i^n ° V n

t2in ,T}n ,2in+1}

r\n

irn+1(Sn ) %

consists of a single element

of

and the secondary rjn ° nn+1 •

(Se®

[20] and Example 2 of Chapter VIII.) Now let

a = rt1

in Theorem 3-6.

It follows then

Ee ° ^I'm+k+ 1 € {rtm+i’Ep’rtm+k+i}i for an integer odd,

then

t

which depends on

r

but does not depend on

EP . „m+k+1 = r(EP . W + 1 ) E r (% + k + 2 (Sm + 1)).

is contained in the same coset as t.o = o.

p . If

Thus

r

is

EP . tnm+k+1

Then we have obtained the corollary

32

CHAPTER III.

for the case that

r

is odd.

Next let

r

be even and

r = 2 s.

By

Proposition 1 .2 , s ^ n

Also

It follows s

° \ 1+1) e stn ' {2Ln> V

t(rin o r)n+1) €

s

is

C (rtn>

rtn + 1] •

which consists of a single element.

s2 (T}n o Tin+1) = t(rin

that

is even, i.e., if

If

2ln + 1} ° SW

t] , rLn+i^

and thus s2 = t (mod 2 ).

o nn+1)

r = o (mod U), then

odd, i.e., r = 2 (mod U),

Ep o t>im+k+1 = EP « s2ilm+k+1 = 0.

then

E£ o ‘ nm+] Jt2 n (Sn+1 ) is generated by E_1 (it2n(Sn+1 ;2 )) of

is the sum of

n

^ n - i ^ 1^

^In- 1 ' Ln- 1 ^ = 0

and

[ln-i' In- 1 ] ^ °'

we may

Since the kernel of

a(*2n+1 (S2n+1 ))

n

pends on the sign of that

H[

l] = + 2 L2n-i

E(a) € jt2n(Sn+1;2)

sequence, we have that

n2n-i(Sn )

the infinite cyclic group

{a}

[in_1, in_1] 4 o. Ln-i^ = °*

Since H([in , tn l - 2a) = 0,

then

that [in , Ln ] e E *2n_2

such that

a.

de­

It follows

E rt2n-2

and since

E It2n_2 ^ n_1) into

^

anc^

It is easily seen that [in, in ] - 2a.

[in , tR ] - 2a e E *2n_2 (Sn~1).

Since

E : *2n_-,(Sn ) -> *2n ^ n ^ ^ 1^

2 ^ + (a) = it2n_ .j(Sn ;2 ) + {a} .

jr2n_1 = E~1^2n(Sn+1;2 ) = ^2n_ 1(Sn ;2) + {a).

+

From the exactness of the above

Consider the difference

E([Ln > Ln ] - 2a) = - 2E a € *2n(Sn+1;2), (Sn+1 ) is an isomorphism of

t

Since

in , in ] ), where the sign

^i-rect sum

generated by

E(-n2 n _ ] (Sn ;2) + {a} ) = it2n(Sn+1 ;2).

=

(Proposition 2.7)-

and H(a) = tgn-i'

^2 n - 1

E :

Then we have proved

is even and

Set a = p - t(+ [

chose

then

ir2n(Sn+1 ) is finite by (b.2), then there exists an integer (2t+i )E p e ir2n(Sn+1 ;2 ).

H(P) =

and the 2-primary component

*2 n _ } (Sn ;2 ).

is even and

Consider the case that

Ln-iJ

is the direct sum

+ [tn , iR ] = M i 2n+1),

E ir2n-2^n~1^ which coincides with

the lemma for thecase that

^

is generated by

and an infinite cyclic group generated

H(p) = t2n-i

fLn-i' Ln- 1 ^

A

it: follows that

E (*2 n_2

*2n-'\^2n ^

*2 n - i ^ n ^

Since the image of

then we have

If follows then

Consequently the lemma k.i has

proved. The following exact sequence is the main tool for calculating the 2-primary components of homotopy groups

/, i \

Proposition b.2: The following sequence is exact. n _E> „ n+i H 2n+i _A» n„ i _E 1 —n+i H — ... . ... _ n± 1+1 _ » * i+ 1 » ^ Proofs By Theorem 2 .b and by the exactness of the sequence (2 .9 ),

a subsequence of the sequence (2 .1 1 ) is an exact sequence of the 2-primary components if the groups in the subsequence are finite.

Then we have the

CHAPTER IV.

36

exactness of the subsequences of (4.4) in which the groups *2 n+i’ ^2n + 1

and *2 n-i do not aPPeap-

E : ffn (Sn ) ->irn+] (Sn+1 )

(^-1) and

are isomorphismsfor all

of the sequences -»

*n+i'

n,

thefact we have

that

the exactness

-> . Then the only problem is the exactness

of the sequence n E n+i H 2 n+i A n E n +1 *2 n * rt2 n + 1 * n2 n + 1 *2 n-i * *2 n

'

But, the proof of the exactness of this sequence was done essentially in the proof of the previous lemma 4.1. Let of

p

Then we have the exactness of (4.4).

be a prime, then denote by

the stable group

G^*

(G^;p)

the p-primary component

By the exactness of the sequence (4.4) or by

(3 .2 ),

we have (4.5).

Em n :

->

phisms

for n > k +2 . Lemma 4 .3 .

nents,

then so are

m > n,

n> 1.

Let a

and

If

a and 7

0 o 7.

o Enp and

then each element of {a , EnP, En 7 )

-> (Gk ;2 ) are isomor­

E°° :

are

in the p-primary

compo­

Further, if a o En 0 = 0 0 7

= 0,

is in the p-primary components.

Thislemma states for elements

of the homotopy groups of spheres,

but. the statement is true for general case as follows. n>

1 .If

and fora e jt(EnY -»

Z),

pS (P 0

0 € ir(X-» Y).

(4.6).

then

Let

7 )= 0

for

pra = 0 and

{a, En|3, En 7 )n

pr+s

= 0 . Let

7 e

pS7 = 0

n(EW -» X), Further, if

for positive integers then

r,s

pr (a o Enp ) = 0

a ° En 0 = 0

and

0o

7 = 0,

p s (0 0

7) =

and

= 0.

Proof: By (1 .7 ), p o ps 7

and

pr (a ° Enp ) = pra

® EnP = 0

1 be the classof the identity of

EW.

and

Then it

follows

from Proposition 1.4 and ( 1 . 7 ) that pr + s (a, En(3 , E11 7)n

- pr ( (a,

EnP, En7 )n . En+,ps t

= pr (a .

+ En (f3 , 7 , pS t))

)

= pr a o + En {p, 7 , P3 l ) = 0 4.5

Thus, (4.6) is proved.Lemma

Let

n>

1. Assume that

a » Enn.(Sk ) = a . Enit^ C n1? J

course, a particular case of (4.6).

p = 2 , we have the following

For the case (4 .7 ).

is, of

J

J

and

a £ ^k+n^™*2 ^

and

7 e jt^(S^;2),

then

«.(Sk ) • r = «^ • r C «? • Further, u

J

SUSPENSION SEQUENCE MOD 2

a ° Enp

assume that

37

= p o y = o, then the secondary composition

{a, EnP,

En7}n is a coset ofa ° En^k+1 + Jt™+n+1 o En+] y and it is a subset of m 7ri+n+1' It is well known that for n = 2, ^ or 8, there is a Hopf fibering h

:S2n_1

Sn

of afibre Sn“1

and that *±(Sn )

E + h* :7rj__1 (Sn~1 ) © ^ ( S 211-1 ) is an isomorphism onto for all a mapping whose Hopf

i

[l7]•

This is true if we replace

i . a h : Sn_1 x ft(S2n~1)

notes a loop-multiplication in

ft(Sn ).

a(Sn ),

the proof are left toreaders.

jt?_ ]

-4

de­

The details of

In our case, we have

k .^ .Let

a

be an element

© rr/1-

onto

itjfor

of

*gn-i

(p, y)

then the correspondence jt?” ]

where

This mapping induces the Isomorphisms

of cohomology groups and thus those of homology and homotopy.

isomorphisms of

by

invariant is _+ i. Because of the above isomorphisms

are Induced by a mapping

Proposition

h

all

i.

such that

Ep + a o y In particular

H(a)

gives E

is an isomorphism into for all We shall apply following Lemma b. 5.

Isomorphism into for all and for

p e

n = 2, k or 8.

Let i.

r 1 0 p=

E :

(Sn_1 )

rp for

an

it^(Sn ) is an

arbitrary integer r

1(Sn” 1 ).

Proof: The first part is obvious by the Hopf decomposition of jt^CS11).

Since

E(rt,n_1 ° p) = E(r p), by (2.1),

it follows then the second

part • It follows easily from the proof of Lemma 1+. 1, (^.8). 7-) n-1 E ,2n_2

Let n be even. If /'\ A 2n+i „ n-1 © A ^2n+1 - E*2n_2 © Z

A ^L2n-i^ = [Ln-i; Ln-1] = °' and by

n (k.

and

then

jt2*1/E2 *2~12 ~ Z2 ‘ An element E

9n\ ).

if and only if

H(a) €

,

) = [in-1; Ln-i 1 J °> then *211-1 = n+1 -,-,2 n-i „ n-1 T« ^2n = E *2n_2 E*2n_2 . If

e2 : *2n-2

* 2n

a of ^ n - 1

is an isomorphism into

ismapped into

E2ir2n-2

2 jt2^“ J.

We have n o „k mC «k^ •n nm Proof:

If

k 1 m

and

k 4 2m-1,

then

= jtk (Sm ;2)

and (^.9)

CHAPTER IV.

38

is true by (4 -7) k = 2m-1

and

m

If

element

integer

r

a

then

is odd, then

consequence of (4.7). an

k = m,

of

= *k (Sm ;2)

Now assume that

^m-i

such that

° jr™ = ir^ and (4-9) is true.

* Since

quence (4.4), we have that

by Lemma 4.1 and (4.9) is a

k = 2m-i

of ^

shall show that

= 0,

22t+1 (p o a)

m

and let 2 ^ then

is even.

Consider

there exists an

By the exactness of the se­

2a = * * U m , l ] + E y

let P be an arbitrary element

and

H[im , im ] = + 2t2m-i'

H(2a) = H(r[tm , tm ]).

If

for some

7

e ^2m-2 *

be the order of

p.

0 oa e n2m_ 1 ('Sn ;2) C ^m-1

We and

(4 .9 ) is proved for this case. By (2 . 1 ) and by the product [2 8 ],

bilinearity and

naturality of the

22t+1(p o a ) = (3 « r 2 Z t h m , im J + P ° 22tE y t ~t . „2t

Whitehead

CHAPTER V

Auxiliary Calculation of

for

7.

1£ k
1 .=

for

G 0 = Cl) ~ Z,

ofor i< n.

= 0

Gk = (Gk ;2)

1 = E°°in .

where

jt^+1 • By (5-i), jt2 = 0 . Then it follows from 2 5 2 H : ^ is an isomorphism onto. Let tj2 €

The Groups

Proposition k.k that

be the class such that

H(tj2 )

that

n

T)n = En"‘2n2

for

= iy

then

>

r\2

2andtj=

generates

« Z.

Denote

E°%2.

Proposition 2 .7 , H(a(l5 )) = + H[i2> l 2 ] = ± 2 1 . Thus H : 2 -> "5 is an isomorphism. c a2 E 5 4-> \ *5 = 0 of Consider theexact sequence

By _+ 2Tjy

Then

for

a(l^) =

since

E is onto and its kernel is generated by

and also by (^.5), jt^+1 = (nn ) « Z2

for

n> 3

2r\2 '

T^us

and (G^2) =

(^A).

=^ 3^ ^ (t)} ^ Zg .

Con­

for n > 3

and

sequently we have obtained Proposition 5 •1 •

jc| = Ctj2 } =» Z,

(G^ 2 )= {Tj) « Z2 . We have relations By (5-1 ), (5 -2 ). or

tt|_1 = 0

The composition

:

for

1 = U n ) 33 z2

H(t]2 ) = 1 ^

and

i > 3 . Itfollows

a -» t)2 0 a

A(t^)= + 2 -q thenby Proposition b .b

defines an isomorphism

i > 3• The following lemma will be useful in the following. 39

■z

^

2

40

CHAPTER V.

2a

Lemma 5-2. Assume that then for 2

E a

an arbitrary element

and

A(E2a)

20 = ^

= o for

0 of

o e a ° ti^+1

a

an element

{tj ,2 1 ^, E a}.,, 0

hold. Such a

of

Jt^(S^),

the relations

belongs to

H(0 ) =

^ rt^ + 2 m

= 0. Proof: ByLemma

4.5, 2t^

to the secondcomposition

{r)y2Eiy

»a =

E a}^,

2a = 0.

Apply Proposition 2.6

then

H((3) € a "1(t)2 o 2 l 5 )o E2a = a "1(2t]2 ) o E2a = +

° E2a = E2a.

20 e 2{t]5, 2i^y E a ] } = {t^, 2 1 ^, E a}] ° 2ii+2

Next

=

a, 2 1 ^}

o E {2

_C tj5 °

by Proposition 1.4

E a, 2Li+i^i

Proposition 1.3

^ t)3 o -(e a o T)i+1 ) = t)5 o e a o n1+1 by Corollary 3 •7 • r\^ ° -{2

The composition

E ar,

2Lj_+i-*i

For an element by (2 . 1 ).

of

7

tj^ 0 -{2 1 ^, Ecu, 2t'j_+i^i

consists of a single element.

20 = tj^0 Ea ° nj_+2 •

that

0 7 0 2Li+2 ) = %

*i + 2 (S )>

Obviously

40 = 0

and

A H 0 = 0.

of the above lemma. v 1e ii).

00

2

Therefore wehave

0 e *^+2‘ A (E2a)

=

q. e- d. The element

(5.3)*

0 2Er = 2 ^4 0 E 7 = 0

By Lemma 4-5. it follows that tj^ 0 7 0 2l-j_+2 = °> and thus the / 4\ t] 0 Jt1+2(S ) o 2l1+2 = 0 . This shows that the composition

subgroup

of

jt^(S5 )satisfies the condition

Consider an element

H(v *) = tj5 The Group

Denote that

2

V

and

2 1 ^,

v l of

2 v1 = ^

2^ 0

^

=

0

then we have

« t^.

*^+2. = 1n ° nn+l

^

and

= n ° n’

then

E,1n = Vt-1

E r)n = t)

Proposition 5-3-*£+2 = (r)2 } - Z2 Proof: By (5-2), TT^ Ji ^ A jt2

of the sub­

co set

o * 1 + 2 ( S k ) 0 2l±+2 + ^3 0 2l4 0 E Jri+2(Si+) = t)3 o Jt1+2(SU ) o 2l1+2.

group

and

is a

jr2

Jt^ -S itj -A Jr|

is an isomorphism into.

Proposition 5*1,

H :

->

for

n >

2.

*2 = {r)2 } ^ Z2 . Consider

Then

of (4.4). H(jt^) = 0

is onto.

the exact sequence

By Proposition 5-1, and

Then

E

is onto.

A(*g) = 0

and

A : jt^ -> By (5*3) and E

is an

(G2;2 )

AUXILIARY CALCULATION OF

isomorphism into.

*n + k (Sn ;2)

FOR

Therefore, we have proved that

*5

Proposition b.K,

*5 •

E

*6 = 0

then

is an isomorphism into. Thus

is onto.

->

By

is

an iso­

k

and for

= U . For n >

^+3*

The Group

Lemma 5.k.

There exists an element H(v^) =

l

rj and

14.

of

a = t)2 .

r^p

satisfies thecondition

(1 - ( - 1 ) 1 )E% 2 = 2 t]£ = 0 . Then there exists an element

a*

{Erar)2, E 5 p, E t + ^j]2 }^ for any

E2p 0 E^a* e (-l)ra

such that

2 Ev^ = E 2 v ’.

Apply Theorme 3 . 6 for

Proof:

of

n^(S^)

such

element

p e itt+2 (Sm )

By (2 . 1

),

p o EtTl2 = 0 .

such that

p = 2

Consider the case that = 2 E r\2 = 2 t|g}^ -

E

the proposition is proved by (^-5). iii).

that

is an iso­

Next consider the

E: n

U1

5

->

n = 3-

by (5*1 ),

morphism onto, and the proposition is proved for (G2 ;2 ),

2

E :

morphism onto and that the proposition is true for exact sequence

i < k < 7

= 0 . Thus

t = 1.

and

-{t^, 2ig,

E2 (2 1 ^ ) o Ea* = 2 Ea* is contained in

Now we show that

(5 - 0

for

n > 3

T]n+1)t

consists of two elements

En 3 v !

and

First consider the case

t > 1.

By (U.7 ), Proposition 5*3 and by

and

t £ n- 2

the secondary construction -En"^vl.

(5 -3 ),

we have that the above secondary composition is

+ n2 +2

0

nn+2

definition of 2 ik , % ),

Which

v',

is generated by nn

° nn + 1 ° nn+2 =

C ( - O n'5(V

2in+1, nn + 1 ) n _2 c (-i)n'3(v

11n ° E Itn+2

and

U n > 2 ^n + i ’ ^n+1 ^

2 Ea*

v1

to the generator

As 2 Ea* = + E 2 v 1, H(a*) = (2 s+1 )l^

E2 v 1

is

C

ve have that these secondary

E 2 v'

of

or then

s.

Then

are cosets of the same subgroup

is divisible by

for an integer

E11 ^v' e (En

2 in+1, nn+1 )t -

Then (5**0 is proved for the case

Now the element maps

2En~3 v'.By the

Then it: follows that

^ + Itn+2^Sn^ 0 ^n+2 * By

compositions coincide.

o

t > 1.

By (3-2), 7tn+3(Sn+1) = E *n+2(sn)* 2l n+ 1 } T)n+i^i

acoset of

Proposition 1 . 3 and by (1.15)> we have

(5 .U) is verified easily for the case

{ T\n , 2 in + 1 >

As

2.

t = 0.

-E2 v'. Since v'

H : itg

is not divisible by

-> 2.

Then it follows from (*+.8 ) that

H[i^, i^] = ( - 1 )u 2 t^,

we set

2

CHAPTER V.

a*

=

-

(-1 )u

s [ L k ,

l

^]

= -a* + (-1 )U (s+1 )[ l ^, L ^ ]

if

2Ea*

if

2Ea* = - E2v ’ .

secondary compositionE -E2p o E^ v ^, and

where

^+5^3

p is

contains

an element of

,

of Lemma 5.k, we have that the P t E p ° E or

For the element -Z.

E2v'

satisfies Lemma 5.k.

Then it is verified easily that the element Lemma 5*5*

=

rtt+2(Sm ) such

that

p 0 t)^.+2 = o

t > 0. Proof: In the proof of the previous lemma, we see that 2 t - E P ° E a* is inthe secondary composition ofthis lemma.

or

E[i^, l^] = 0, v^.

then E^v^

= E^a*

or E^v^ = - E^a*

O 4“ E p o E a* Since

bythe definition

of

Then the lemma is proved. We denote that

vn = E11-1^

k andv = E°°v^.

forn >

We also denote that in = nn 0 % + 1 ° ' W

for

n ^ 2 and

I5 =

1

" 1 •

By Lemma 5 •k and by (5 •3 )> we have (5-5)-

For

n > 5,

2vn = En"^v'

and

^

i^vn =

•

It follows from Proposition k.k and Lemma 5-^ (5-6). 5

The 7

*3.-1 ®

i

correspondence ^ QntQ *

Proposition 5.6.

(a, p) -» Ea +

it2

= (t)|) ^

4

- {v,) -

n!j = {vj

©

gives an isomorphism of

Zg, v (Ev' }■» Z © Z h ,

C 3 = tvn } “ Z8 (G^ ; 2 ) ={v} ^ ZQ .

and

° P

^

n>5, 2

Proof: By (5-2) and by Proposition 5-3, we have ■x

TC

XJ

^

^

p

~Z

JJ

= {r]2 o CL

{ri2 } « Z2 . Consider the exact sequence By Proposition 2.2

and by (5-3),

H(v' °

Then it follows from Proposition 5-3, that E

is an isomorphism into.

(5-3) that of

n|.

(^.8),

v r is of order

Then= E t)2 k.

Since

) = H(v') o tj6 = H :

is of order

-» 2.

jtg = {t)^} ^ Zg,

o

2

of (k.k). = r]2 .

is onto and thus It follows from v'

=

is a generator

Thus= {V ') » Z k - By (5-6), «lf -- {v^l © (Ev1) » Z © Z k . By c 2 x 2 ^5 *5 itg / E~ Z2 and E : itg -> jtg is an isomorphism into. It follows

AUXILIARY CALCULATION OF

from (5-5) that {yn } ~ Z 8

for

8

is of order n> 5

and

*n + k (Sn ;2)

FOR

1 < k < 7

and it generates

( G y 2) = {v} ^

43

jtq. By (4.5),

Zg.

Jtn+3 =

q.e. d.

In this proof we see that (5-7)-

H ( v ' . t)6 ) = 115

It is verified easily that the kernel of ated by

2 v^ - E v 1•

iv).

The Group

- Ev'). (= _+ [1 ^, l^]).

^+4*

By (5 -2 ) and Proposition 5 -6 , we have Lemma 5-7then

E2a e 2 l ^

If

E(t]2 0 oc) = 0 . Proof:

jr5 gis gener­

It follows from the exactness of the sequence (4.4) that A( 1 ^ ) = ± (2

(5-8)

4

E :

it| = {t]2 ° v'} ~ Z^.

rtj_+2 (S^)

0

for an element

E (t)2 0 v 1) = 0

In particular

E 2 (t)2 ° a) = °E2a €

We have

and o 2^

a e jt^(S^),

A(v^) = _+ (tj2

o v’)-

0 jt1 +2 (S5 ) =

0.

E(t)2 0 a) = 0 . By Lemma 5-4,

Then it follows from Lemma 4.5 that

E2 v ’ =

2

= 2 1 ^ 0 Ev^, then it follows that E(tj2 0 v 1) = 0 . By the exactness of 2 *5 the sequence (4.4), we have that itg = a ( txq ) = {a(v^)}. Thus A(v^) = _+ (n2 0 v ')*

q. e. d.

Since 0

_>

Ejt^ = (E(t)2 v ’)} = o,

jt^ from (4.4).

an Isomorphism onto and

then we have an exact sequence

By (5-7) and Proposition 5-3. we have that

nj =

0 ^5 } ~ z2 •

Now the

H

is

first two parts of the

following proposition is proved. Proposition 5 -8 .

2

itg

={tj2 v 1} ^ Z^ ,

*7

=

0 ^

^ Z2 '

n8

=

^v40 ^

® ^Ev’

=

{v50 T18} ^ Z2 '

*9 *n+4

= 0

Proof: The result for Consider the exact sequence

and (G^;2)

«g

^ Q

A :—> 7t^

=0

E

is onto.

,

.

=

By (5 -8 ) It

The kernel of

E

(by Proposition 2 .5 )

A(r)^)= a(l ^ ) ° = ( 2 -

it!j of (4.4).

K

is an isomorphism into.

follows by the exactness of the sequence that is generated by

~Z2 © Z2

jtq follows from (5 -6 ) and Proposition 5*1. ^

and Proposition 5 -6 , we have that

n> 6

for

0 ^

E v 1) 0 n 7

(by (5 .8 ))

o (2 T)y ) - Ev' o n 7 = Ev 1 o Tjrj .

CHAPTER V.

= Eits =

Then we have that

0

rio

is generated by

°

T)3 •

(5-9)-

Next consider the exact se11

.1 1 11

of (i+A).

Then

E

is onto and

6

Now we shall prove the formulas

= v' . rig,

>!n • vn+1

= 0

for

n > 5

and

V]

n > 6.

for

First consider the element 2 (t]^ o v^) = 2 t)^ ® = 0.

Thus

x v* ° ^

H (rj

'■3

= o,

t) x = 0 or

v^)= E(t)2 # x\2

1.

Applying

) 0 H(vu )

(r]^ 0 115 ) 0 L y

=±

Then this implies that

= 0 Then

o ^

2 (rj

then

n',

tj^ o

H, we have 2.2

Proposition 3 . 1

and Lemma 5.k

by Proposition 5 .3 ,

= H(x v f o n6 ) = xr)k x = 1

by (5-7).

and

° vs = E v 1

Next

2E(r)^ ° v^) =

by Proposition by

% and

Since

generates *15

v1 0

Since

for

° v^.

then it follows from Lemma k.5 that

for = tj^ 0

tiq = 0

hy (5-5)-

Thus

By Proposition 3 .1 ,

n > 5= 0

vn 0

and

= 0

for

n > 6.

Consequently

all the formulas of (5 *9 ) are established.

= 0.

By the last result,

Li0

By (U.5),

2) = 0

iQ+k = (GV

for

Then the proof of the

Proposition 5-8 is finished. 11 T1 1

In this proof, we see that

£ ^ “ 10

^ is trivial.

Since

(5 - 1 0 ).

is generated by

A(in )

v5 ° ^8 *

We have also in the proof A (T]q )

(5-11 )• v).

The Group

Proposition 5-9*

nn+5

*

p

*7 = *3 =

=

Ev'

is onto, i\q

>

since

E : nc

it follows then

=

AUXILIARY CALCULATION OF

Proof:

*n + k (Sn ;2)

FOR

1 < k < 7

The first statement follows from (5-2) and Proposition 5-8.

By Lemma 5-7,

Eit2 = (E(t}2 ° v') ° r^} = o.

exactness of (4.^) that the sequence

o -> itg

It follows from the

itg ^

is exact.

Lemma 5-7, Proposition 5-6 and by (5«5), we have that the kernel of generated by

=

bv^

.

By A

is

By Proposition 2.2 and by (5-3)>

H(v ' °

T]|)

= H(v’) o

= t]5 o t)| = T)^ .

It follows from the exactness of the above sequence the second statement of the proposition.

By (5*6), we have the statement of

.

Next consider the exact sequence 9 A i i E 5 H 9 A i + E 5 *9 * *10“ * *io“ > *8 *9

*11

of (U.1+).

By Proposition 5-8 and (5*11),

morphic to

I;.

itg.

Z2 .

Q

Then

It follows that

it:j0 « Z2 E : a

1+

->

(t]|) =

by Proposition 2.5 and (5-11)• *10

A(t]n ) =

5

hy (5 .1 0 ), 2 .7 ,

A

has to be mapped by

^

Q

is onto.

(tj9 )

o

A

0 T .i 2g 0

isomorphically into

i

and

n8 = E v 1 o T)y

by Proposition 2 . 5 ^

~ Z2

jr^

a -t-Vnno and thus

and (5*10).

from

It follows that

we have that

n

‘2 = 0 .

E * ^ = 0.

In

Then we have an

= (A(i

21

1.

By Proposition

)} = Z.

From the exactness of the sequence 7

XT’

By Proposition 5*8 and

is onto and its kernel is generated by

H(A( i13)) = + 2 t,, .

kernel iso-

Then we have that ^ * 9 = ^v 5 0 ^

0

exact sequence

has the

^ E(v5 o rtg) = v6 n° ^tj9 n° ti10 = 0

By (5-9), fact

a

E : itg ->

itj2 — »

= o

By (^.5 ), we have the last assertion of the proposi­

tion.

q. e- d. In this proof, we see that

(5 .1 2 ).

A :

->

are isomorphisms into for

n =

b,

5 and 6 .

Next we shall prove Lemma 5 -1 0 . a(i15) g {v6 , t]9, 2 l101 € Jtn (S6 ) / 2jt11(S6 ). Proof:

It is easy to see that the secondary composition is defined

and it is a coset of 2ir] 1 (S^).

vg ° jtn (S9 ) + itn (S^) ° 2in

(cf. (U.7 ), (5-9))-

2 . 6 , we have that

H{vg,

r\ ,

=

o

jt^

+ 2itn (S^)

Applying the relation (5-1 0 ) to Proposition 2

l^q)

contains

in

0 E(2l]0) = 2 l1 1 -

We have

C H A P T E R

also 1 0

V .

H(a (l 1^ )) = _+ 2tn - It follows from the exactness of the sequence —» JT1 1 (S )

( ) *5

Eit10(S ). ponent

)

of

(2

that

.8 )

A (L ^

2 1 ^q]

T] ^

+

5

Since

Eir10(S ) is finite and it has the vanishing 2-primary com­

Eit^0 = o,

then

are contained in vi).

JT^(S

C 2 * ^ ( 3 ^ )) .. En10(S5 2irl1(S6 r,0 (S5 ) ) C

Therefore

T h e r e f o r e

( ^ ) A(t^)

a

l

and

-a U ^ )

{v^, t)^, 22 Ll 1q) . 1o ) n

The Groups

^n+5 * nn + 6 ’ p 2

p

Denote that

f o r ^or

4

vn =: vn 0 vn+3 v n ° v n + 3

Proposition 5 .1 1 .

n > k a n d n — ^ and

v

v

0 v = vv 0 v‘

=

jt| = {tj2 •° v ^ Z2, V'’ 0° n|} „§} „ Z 2 , - u 2 0 0

=

* 9

, >

1+

* 1 0 '=

}

=

Z g

}

=

Z 2 ,

(v^

,

■“ Z 8

’

n

nn+6 ’ nn + 6 '

f o r ^2E

^ n 1*V

{ v n

(G6 r6 ;2) ;2) = =

nn >> 55 ,,

t v 2 } *^ Z2 .. z 2

( v 2 )

Proof: The first. assertion is i s proved p r o v e d by b y Proposition P r o p o s i t i o n 5-9 5 • and a s s e r t i o n B y L e m m a 5 - 1 ,Eitg E j t= | = (t E] ) By Lemma 5-7, (E( 2( t ]o2 v0 ')V « ®

*9 ^ *9 ^

=

%■

0

0 . = °-

W e have h a v e also a l s o We

= 0 E jt^ = inPropo­

E * 2

I t

*9 ^ **7 7 ^9 ^

w e h a v e t h a t we have that

A A

0 0

i s e x a c t . is exact.

r o p o s i t i o n 55*8 - 8 a n d P r o p o s i t i o n 5 * 9 , By PProposition and Proposition 5*9,

B y

a n i s o m o r p h i s m o n t o . is an isomorphism onto.

t f o l l o w s t h a t Then iit follows that

i s

T h e n ji

I t f o l l o w s f r o m ( 5 - 1 2 ) t h a t It follows from (5-12) that

H H

. = °°.

=

t r i v i a l a n d are trivial and

E

a r e

=» Z g . C o n s i d e r Consider the „ ,, , R ^ „ , R ^ = ' = ’

o m o m o r p h i s m s o n t o . are hhomomorphisms onto.

E

a r e

N e x t w e s h a l l p r o v e Next we shall prove

(5 . 1 3 )-

a (v9 ) = + 2{v\), B y

P r o p o s i t i o n

A ( v 9 )

v' 0 v' 0

e e

*9 = A a

2 . 5

A ( l 9 )

=

_+

0

(2 v 2

then

t h e n

0

=

and

a(ti1 3 ) =

0.

( 5 - 8 ),

a n d

=

~ 0 0} }

A(r|^)

=

-

E ( v

(2

+

’

0

-

E v 1 )

o

v 6 )) •

A(v^) == _+_+ 22 (v^) We have ( v ^ ) •• W e h a valso e a l s o

A ( v ^ )

( t) ( t n ) ) o o n n| | C 2 1,)) == A A(tn

= v5 •18 ” ’ 9 = "5 ‘ ’ 8 = “*(v5 • v8 ) = 4E(v2 )

b y Proposition P r o p o s i t i o n 2.5 2 . 5 by

by (5-10) by (5-5)

= E A ( + 2 V^ ) = 0 . a

(t}15) =

a

(l 13) ° T)n

6 {v6, t]9, 2t10) o nn

By

B y

p p

P r o p o s i t i o n 5 5*6 * 6 a and n d b by y ( (5 5 * *6 6 )) w e have that h a v e tj a t jt1 = {= v ^ Zg. } Proposition we th1Q = Q {v^} » . 2 n2n-i - i A An n EE n n+i + i HH 2 2n n-i - i AA nn « «„ „ . e x a c t s e q u e n c e s Jtn + Q > *n + £ -» » ^ + 7 * n + 7 * n + 5 exact sequences J tn+Q -> *n+£ *n+7 *n+7 *n+5

Since S i n c e

(5.2).

It follows f o l l o w s f from r o m t the h e e exactness x a c t n e s s o of f (( b ^ .. k ^ )) t that h a t t the h e s sequence e q u e n c e

sition s i t i o n 5-95 - 9 0 0

2 • }

by Proposition 2.5 by Lemma 5-10

t h e

*n+k(Sn ;2 ) FOR

AUXILIARY CALCULATION OF

= v6 ° (tj9, 2L1qj, t)i 0}

- (v6 o E 6 v').

and

A(n 1 ) = o,

By (5-5),

47

by Proposition 1.4.

° {r^, 2 i 1 0 ,tj1C)}

By (5-4), we have that

1 < k < 7

consists of

o E^v’

v5 o E 6 v» = 2 v| = E2A (v9 ) = 0 . Thus

and (5*13) is proved.

Now apply (5*13) to the above exact sequences, then it follows that n = 5 , 6 and 7 -

the proposition is true for By Proposition 4.4, Thus the

is isomorphic to

by

n = 8 . For n > 9

proposition is proved for

E.

and for the stable

Consequently the Proposition 5 . 1 1

group, the proposition is proved by (4.5). has esteblished.

q. e. d.

Lemma 5 .1 2 .

{r]n , vn+1, ^+^5

consists of a single element

p vn

for

n > 6. Proof: By (5-9). this secondary composition is defined and it is a ooset

of ,n . *n+6(Sn+1) ♦ «n+5 (Sn ) . nn+5- By (3 .2 ),

T)n o *n+6 (Sn+1) = T)n

(sn ).Then by (4.7), and by Proposition 5-9, ° ^ + 5 = * ^+5 ° ^ + 5 '

Similarly n > 6

and generated by

A(i^) o ti^

if

*n+6 (Sn+1)=

and

grouP

E «n+5 = 0.

°

trivial if

n = 6 . As is seen in the proof of

(5 .1 3 ),

a(l^) © t)1 1 = a(t)13) = 0 . In any case, the secondary composition p consists of a single element, which is xvn for an integer x = 0 (1 .1 5 )

and Proposition 1.3,

shows that the integer the case

p =

contained in

and

and

x

vn + 2 ,

E{r)n , vn+1, rin+lt} C - h n+1>

does not depend on

n.

(nm+2, E 5vm ,

C

(w

x = 1 . Consequently the lemma

nn+5)-

Now apply Lemma 5 . 5

m > 6 . Then it follows that

for

E 2 vm ® Em+ 1 v^ = v2 +2

, vm+5> nm+6). Thus

v^ +2 =

is proved.

Remark. The Lemma 5 . 1 2 is true for n = 5■ vii).

The Groups

*^+7 -

By (5 .2 ) and Proposition 5-11,

*2 = 0 .

y

It follows from (4.4) that the sequence 5 A 2 E 3 nio *8 *9 By Proposition 5-9 and Proposition 5* 1 1 , we see that n

is exact.

isomorphism onto.

2 *9

It follows then

3

H

jt^ 0 = 0 .

By (5 -6 ) and by Proposition 5 -8 ,

= 0.

A

or

is an

is

This

CHAPTER V.

d e r tth h e e x a c t sequ sequen Next c o n s iid en ce o f (4 .^ 4 ))..

o = jt^ 0

S

ir^2

jr^Q

i o n 5 «ii t i io o nn 55- 66, , we have t h a t By ( 5 - 1 3 ), P r o p o s iitt io -11 and and PPrrooppoossi it we have

i s a homomorphism o f d edgere g ree e 2 2and and i it tss kkeerrnneel l i iss ggeenneerraate t edd Thus we have an isom isomorphism orphism Lemma 5 - 1 3 -

am

*5

H :

9

-» ^*^2

a'",

[{v v^,

such t h a t

g e ng e rnaetreast e s *12*12 ~ ~ ^ 2’ (°

AA

b y 4 v Q == ti ~« by ti~« y

7y

** Z2*

The s e co c o n d a rryy cco o m p o ssitio i t io n

o f a s i n g l e e le m e n tt,, d een no byy o ted b eelem le m een ntt

jt^2 ^

vqJ^

Ql q q,

co n sists

H ( a ,,M) M) = 4 v^ =

rj^

.

The

^ £ 3) ^ £ 3) Q

P roof:

The s e co c o n d a rryy co c o m p o ssitio i t io n i s a c o s e t of. o f.

* 9 (S5 ) o vv^. 9•

By ( 3 - 2 ) ,

Ejr1 l (S^) =

o o^ ^ = =0.0. ByBy ( 3(-32-)2, ) , PPrrooppoossiit t iio o nn 55-8 - 8 and byby ( 5 -(95 ),- 9 ), ^ ( S ^ )

v 9 = jr9 °

( 4 -. 7 ) and P r o p o s iitt io ion 5 5--8 8,,

v^» it1 2 (S ) +

Q 8 li n 0 ,,

(CVv c 5 ,

J

8 iq,

(v ^ ,

vq}^

V n )).t v Q

o

By P r o p o s iitt io i o n 2 ..6 6, ( 5** 8) By (5 8 ),, k (v k

Thus k n* ?^ 22 ^

o 8Lrj).

Then

H(ct,m ) =

kvy

~

00

o

u

By By ( 1( 1. 1. 15 5)>)>

C

({ V c ), v^

^

8 tt og ,

V v qo )} . .

o

o

elem en t i s a l s o c o n s i s t s o f th t h e same s i n g l e ele m ent

en t By (^ - 7 ), tth h e elem ent

±

0

v 9 = ^v5 = ^v5 0 0 ^8 ^8 00v v9^ 9^= =° ‘ °*^^ ^o ^ol l ol ovvs s th theenn tthhaatt th t heese co s en c odna d rya ry com­ com­

p o s i t i o n c o n s i s t s o f a as isni gnlgel eeelelemmeennt.t .

Then

° °jt1jt1 2 (S8 2 (S8 ) )= =

a ’ "i ctm

a ‘ 1 ( vv

i s in in

H(a,n) = H{v^, 8 l q , v q ) 1 = A- 1 ^

A (l^i9 (lu 9 ) = +

a m. crm

o ) 0 v^. ® 8 1 ^)

E 4 v ' )) = + Sv^ = - E v ’ ) = + (Sv^ - EUv’

o Ql ^) o

« «

c o n tta a iin ns

_ + ((b4 l^^)) » _+

=

v^

ved isom orphism and th e lemma i s p ro ve d by tth h e isomorphism

Z 2 *

e *

4 vv^ k ^

±

= t]^. r\^.

H : it^2 -»

d *

Next we p ro v e ( 55 -- i i+)-

The sseq eq u en 0 ces -» 0-»

The sequences

and

*^2*^2 S 55S 5 S -»

-> -o0 ->

SSS

jt^

-> -> -> 000 -> -> o00 -»

are a r e ee xxa a cctt.. By tUIJC7 h e “e Axcai Uc tLilC? n e so so o U 1f

JDj

A (jtj^ A ( i t ^ ) = A ( « ^ ) = A(j t j ^)) = 0. A (it11 ^)= aA((jj t ^ ) = 0. and t h i s shows

.

. .

-

11 k

a ( j t ^ )y — =

ZA ^ 7l ^

The l a s t two fo f o rm u la s o f ( 5 - 1 3 ) show t h a t

ion 2 . 5 By P r o p o s iitt io

^(*^5 ^( *^5 )= 0.

__

( ^ . ), -L i tk i-LS s su t t0oU p roU vV et? tLXicL h a tU UJLf 1f iXcUiXe^n IIL

4 /j \^ • ^

and ( 5 - 1 3 ),

i o n 2 . 55 ,, By P r o p o s iitt io

a(t^ =

a ( t j 1 5 ) o tj12 t)12 = 0

)> ( 5 - 1 0) and by ( 5 - 9 ),

]8 0 0.. A ( v n ) = A ( tl n ) o v 9 = v 5 o T]8 0 v9 = 0 T h iiss p rro o v e s w iith t h P r o p o s iitt iio o n 5-6 5 - 6 th a t

A(nJ^) = o. 0.

C o n sse e q u e n ttly ly th the e xa c tn e ss

o f ( 5. 14) i s e s ta b lis h e d . Lemma 5 . 1 4 .

There e x i s t e le m e n ts

ctqq O

e

8

a* e

7

and

AUXILIARY CALCULATION OF

€ «6 13

such that

H(ag)

*n+k(Sn ;2)

HCa" )

2a f = EcrM ,

= t^ ,

2a" = Ea'".

Proof: Apply Theorem 3-6 for element

< k < 7

2Eag = E 2 an ,

=i

H(a ’) = t)13, and

1

FOR

a = v^.

By Proposition 5*11, the

2 E^~5 v^ ° E 7 “3 v^ = 2 v2 = 0

satisfies the condition

of (3 .1 )

g h = 3 . Then there exists an element

for

eV

. e V

for any

e ( - D m (vm + ^

p = 8 1 ^.

Now let E^ 8 i5

satisfying P o v t+i+ = 0 . 81^ °

Then

= 8 v^ = 0 by (2 . 1

8 Ea* =

)and

oEa* e -{v9, 8 ^ 2 , v12)? - {8 ^, v9, 2 v12)^. {v^, ® L-|2 ' v 1 2 ^ 7

The secondary composition

n1 2 ((S9 » v v1? E7^ ^ „ +t It1 s9 ) ) o 1 5 = vv9 9 • 0 E 7 jr^5 o v 1 5 = 00

E 7 ^9 (S 5 )

5 .8 , andI thus it consists of a single element.

i s a cose‘t

vt2 ^ 7 consists

of- E a 1" =

E

By Proposition 1 .2 , {8 i9, = {2 l9,

v

Since oddorder.

Thus

Thus we have that

vn

0

By Proposition 1-3 and Lemma ~iv^,

follows then

a1".

9 , 2 v12)^ C {2i9, hi 9 . v 9 . 2t12,v^)^

0 , v ] 2 )k = 2 L9 o E^tc1 2 (S5 ) + *1 3 (S9 )

= 2 E^* 1 2 (S5 ).

°f

by ((I^t - 7 ) and by Proposition

) 3 C {v9, 8 1 E^crm e E^lv^, 8tg, vg}^ 8t]2, v 12^7*

5 .13 *

8 11 2

) such that

E 7U, vt + n }7 + ( - D t { E S )vt+g )2vt + n )t+3

p e jtt + ^ ( S ra)

element

a* of

o v 1 5 = 2 E 4 jt1 2 (S5 ) + ir9 3 o

8 Ea* - E^am

iP]2 = jt1 2 (S^;2 ) « Zg,

=

e 2 E^it1 2 (S5 ).

then2 E^* 1 2 (S^)

there exists an odd integer

x

is a finite group

of

such that

8 x Ea* = x E k o"' = E k a'" .

Now, from (5*1^) and Lemma 5*13> it follows that the groups and

7

have k and 8 elements respectively and that

isomorphism into. Also it follows from (^.8 ) that ii. 5 9 E : tt^ 2 -> it^ is an isomorphism into. Then

8 xEa* = E^a,u 4 0 and 16x

is of order 1 6 and itgenerates

E^aM = ^xEa*.

the groups

7

* Zg

0

5

: «rj2

H(ct") = t^ 2 -

7

is an

has 1 6 elements and

Ea* = 0 . Thisshows that

xEa*

Z ^ . There

6

ou e

E a 1 = 2 xEa*

Then it follows that

a1

and

ar e 6

7

and

« Z^.

such that and

Obviously

aM

2

generate respectively

2 a 1 = Ean

and

From the exactness of the sequences of (5-1*0, it follows that and

-»

which isisomorphic to

exist uniquely elements and

E

2 ctm = Ea,n .

H(ct') =

50

CHAPTER V.

Next, the element (4.8),

xa*

is not in the image of

has an odd Hopf invariant.

exists an integer aq

xEa*

by setting

y

such that

E2 . Then, by

HfAft^)) = + 2 1 ^,

Since

H(xa*) = H(yA (t^)) + t

aQ = xa* - y A ( i ^ ) .

.

Then we have that

there

Then we define

H(cjg) =

and

E ctq = xEa*. Consequently we see that the Lemma 5.14 has established. We denote that

an = En~^ag

n > 8

for

and

q. e. d.

a = E°°crQ.

By (4.5), we have that for n > 8 and (G^;2 ) are isomorQ phic to and they are generated by an and a respectively. n 8 8 Now the groups * n +7 are computed except n ^ . The group is com­ puted by the following (5*15). which follows directly from Proposition 4.4. (5*15) 7

The correspondence

^

(a, 3)

Ea + ag ° p

gives isomorphisms of

8 .

15 onto

The results on

are listed as follows. 2 ^ 4 0 = *11 = 0 *

Proposition 5*15-

(a"') « Z2 < 3

- to") * = (a' ) * Z8 ,

15

= tas ) © (Ea') *= Z © Zo

“S +7 = ^ n 1 ~ Z16

for

9,

(G? ;2) = (a) *> Z , 6 .

Consider the exact sequence

17 A

8

->

2 ag - Ea 1,since

is generated by the element

E

9

-> jtljg

the above

. The kernel of result.

E

Then it

follows from the above exact sequence (5 - 1 6 ).

&(

= +

It follows from the beginning the definition of Lemma such

that e S

where

x

of the proofof Lemma 5 * 1 4 and

from

ag,

5 .1 6 .

t > 0

Let

p o vt+i 4

{tj5 , 2v6 ,

and

o < t < n-2 .

By (5*11+),

E :

7

it'^

composition

o Ej^2 + ^ Q o V]0

r\^ o Ejt^ 2

€

c (nn > 2vn+1, vn+4}t

Next consider the secondary

Proposition 5*15 and Lemma 5 -lb,

e3

eR = (-1 )n~^

2vn+1, vn+li}n_2

v 9 )1 which is a coset of ^

6

2tn+ 1’ vn+ 1}t

•

by

is generated by

(4 .7 ).

By

tj^ ° Ea".

is an isomorphism into. It follows that

E (2 1 6 o cr"-Ea,M) = 2 Ea" - E 2 a"= 0 implies

2 l6 ° a" = E a '". Then

tj5 °

Ea 1 " =

t 0 2 i£ o a" = 0 ando Ejt^ 2 = 0 . By Proposition 5*9, and (5-9) 5 2 *1 0 ° V 1 0 = ^v 5 0 ’18 ° V 1 0 -* = °* Therefore the secondary composition [r\^,2v^, is a coset of the trivial subgroup

0,

and it consists of a single

. By Proposition 1 .2 ,

element which has to be

It follows that

{V v6 ’ 2 v9 }1 c fV 2 v6 ’ v9V (tj^, vg, 2 v^}1 consists of e^.

It follows that

{r)^,2 tg, v^ ) 1

consists of

£5 €

2 v>9^1

By Proposition 1 .2 ,

2 L6 , v2 )1 C (t,5 , 2Vg, v 9 }, . 2

^^ 5 '

v6 '

Now we have that (6 . 1 ) is true for n > 6

.

^ ^ 5 * v6 '

n = 5*

Similarly, 2 ^ 12 ^ 1

Then the proof for the case

is similar to the first part of this proof.

q. e. d.

Next we have (6.2 )

a'" ° v 12 ^ n 2 .

e

7

m o d 4 (v 5 o Ea' ) .

Proof: Consider the secondary composition ° E^jt^ 2 + jt^ ° v|

which is a coset of

5-8

and (5-9),

by (4.7).

0 E 5 n] 2 = (v^ o E 5 a ,n) = (4 (

and Lemma 5-1^

* 9 0 v| = {v^ 0 t)q 0 v|) = 0 .

tion is a coset of

£ {v5 , 8tg,

By Proposition 5-15

° E a f)).

By Proposition

Then the secondary composi­

By Proposition 1 . 2 and by Lemma 5*13,

{^(v^ ° E a ’)).

O'" O V 12

2

{v^, 8 ig, Vg)^

V g ) 3 . V 12 C fv5 , 8 lg,

Vg)? .

By (6 .1 ) and Proposition 1 .2 , 2

ti5 ° e?

2

€ T)5 0 {t)7 , C. C ^ , 2tg,

2

2 1q , Vg)3 v | ) 5 = (tv5 ,

2tg,

v 2 )3

SOME ELEMENTS GIVEN B Y SECONDARY COMPOSITIONS

a1" o V] 2

Therefore the elements k(v^ o Ea 1).

r\2 °

and

are in the same coset of

Then (6 .2 ) is verified.

ii ).

The elements

53

q. e. d.

vn .

Consider a secondary composition

{vg,

° 7-

if

Proof: By Proposition 2 . 6 and by (5 *1 0 ),

V11 = By

L 11

° v ll 6 a " 1 ( v5 ° V

It follows that

° V 11 = H t v 6 ’

V

v l o 11 •

A (v 1 ^ )) = H(+ A( t1 ^ ) 0 v 1 1 ) = 2 1 ^

Proposition 2.7; H(+

oVl 1 = 2 v } 1 .

mod 2 vn -

Hfwg) =

By Proposition 1.^, (^-7) and by Proposition 5*9, 8v6

€ {v6 ,

t)g , v 1 0 ),

• 8 l llt =

v 6 . E {r)g, v 9 , 8

C v 6 ° En15(s8) = v 6 °

i

12)

= °-

It is easy to see that the secondary composition

{v^,

2 vio^

is a coset of (2 A(v13)). By Proposition 1 . 2 and Lemma 5 *1 0 , 276 = v6 ° 2 1 k

€ (v6, r, , v10) o 2tu C tvg, t, , 2v,0)

and A( v 13) = A(t13) o vn € (v6, n9, 2L10) . vn C (v6, T)g, 2v10) If follows that b v 6 = 2A(v1

A (v1^ ) = 2 v 6 mod 2 A(v]5).

) and 2^ 6 =

= A(tj15) o t]2 2 = AH(a») Since for

n > 7-

or o tj22 = 0

E A = 0,

then

.

Then it is verified easily that

= - a(v]5), since Sv^ = 4a(v]5) =

a(t^)

. 2 v^, = E(+ A(v^))

By Proposition 1 .3 , and (1.15)>

= 0 and

2 vn = 2 v = 0

54

CHAPTER VI. vn =

for n

(-

i

W

6 V6

€

(-1 )nE n ' 6 { v 6 ,

r y

v , ^

^ ^vn 9 ’W l ' vn+4^n-2 ^ ^vn' ’W l ' vn+4^t 0 < t < n-2. By (4.7) and by Proposition 5 .9 , the last

> 7 and

0 vn + ^ + vn 0 *^+8 = °*

secondary composition is a coset of

^

fol­

lows that the secondary composition consists of a single element which has to be

vn .

q. e. d.

Lemma 6 .3 : For SSSES

= vn . vn + 3

"vn « Tin + 0 = r)n ° T n+] = v3 ,

n > 6 , we have

. v,n + 6

.

Proof: By Proposition 1.4, * n 0 "n+8 € £vn ’

and

In °Vn+ 1 e

%

vn +k ] ° ’W S

1n + 3 '

° (vn+ l’ V +V

* C,W 3 ’ vn +k ’

= vn

vn +5 } = £V

vn+ l’ W

ln + 71

’ ° vn+6 '

It follows from Lemma 5 . 1 2 that these secondary compositions consist of a single element

v^ . Then we have the equalities of the lemma. 0 a 10 = v 9 +

Lemma 6.4:

n > 10,

and for

r)n ° an+1

'n+7 Proof: that where

t]^ 0

x

^5 0 v 6 = °*

By (5-9),

^-j q G

~x ^v

^1 2 ^

7 +

v 10 '

"vn . The second secondary composition is a coset of y

I9 0 e 5 * ? 2 + 'li ° Z v ]k = 11I9 » E 5 a'") + 0 =

(8(t)9

Proposition 5*15, Proposition 5*9 and Lemma 5 .1 4. by (6 . 1 ).

single element which is

an ° nn+? - nn • * n+1 = vn + ¥n

iii).

The elements

• a10)} = 0 ,

by (4-7),

Then it consists of a

Therefore we have that

- X V 9 + x s 9 = v 9 + e? . By Proposition 3-1,

Then

>

2 v 1 3 ^5

By Lemma 6 .2 , the first secondary composition

is an odd integer.

consists of

v 13

Then it follows from Lemma 5 - 1 6

n2 # ° 8 =

0 ° a!i = ° 1 0 ° n,7-

n > 10.

for

® a1Q =

q. e. d.

n .

We shall give an element

^

of

such that

H ( ^ ) = a 1" by

means of secondary compositions. First consider a secondary composition {tj3, Ev', which is a coset of ^

4i ?)1 C

0 E *3 + jt-q(S^) «>

k8 (S3 )

4iq,

jtg(S3 ) is finite and its 2 -primary component by Proposition 5 .8 , then ponent of

7tg(S3 ).

jtq(S3 ) o

by (4.7). ttq

Since the group

is isomorphic to

Zg,

lq = 4jtq(S3 ) coincides with the odd com­

Thus the above secondary composition contains an element

SOME ELEMENTS GIVEN BY SECONDARY COMPOSITIONS 2 itg

of itg.Since

= o,

55

it follows that

O e 2{t]3 , Ev ’, ^7}-, = {t^, Ev

, ^7}-,

C {t|^^ Ev 1, 8 1,^)i ,

°

q

by Proposition 1 .2 .

If follows from Proposition 1 .9 that there exists anextension p

e jt(EK

K =S

-> S^ ) of v' CS

for

r\^ ° Ep = EpQ*o = o,

such that

\ = 8 i^

and

p Q : K -» S

where

is a shrinking map given as

in (1 .1 7 ). "3

Next consider a cell complex

M = SJUpCEK

and its subcomplex

L = S^Uv ,CS^. Let i : L -> M and i! : S^ -» K be the injections and let 2 7 p :M ->E K and p 1 : L -» S be shrinking maps given as in (1 . 1 7) • Since of t)^ •

° Ep = o,

Obviously,

prove (6 .3 )

(Ei)*a e tt(EL -> S^ ) is an extension of

Proof:

(Ei')*a' = xv^

H(a)

any mapping of

S^

a mapping which

maps

since

.

S^ )

¥e shall

= Ep*a* for some a ’ € *(E2 K-> S5 ),H(Ei*a) = Ei*H(a) =

H(a)

(Ep*)*((Ei1 )*a') and

a',

a e *(EM-^

then there exists extension

for an odd integer

is represented by a mapping of

into

S^is inessential,

then

EM

H(a)

into

S^. Since

is represented by

eQ. It follows then H(a) 1) . O is a mapping which shrinks S = ES of

Ep

x.

S^ to

= Ep*a' EM

for some

to a point

eQ.

By Proposition 2 .2 , H(Ei*a) = H(a ° {Ei}) = H(a) o {Ei} = Ei*H(a) = Ei* Ep*a* = E(p ° i)* a ’ = E(i* « p* )* a = (Ep1)*((Ei')* a' ) • (Ei1)* a*

is an element of

jtg(S^)

and a !

is itsextension.

The existnece of an extension of

(Ei1)* a ! implies that

8 (Ei')* a ’ = o.

(Ei1)* a ’

Let

Thus the element

x be an integer such that

{r)^, Ev ', vy }1

a of

tj^

and a coextension

Applying the homomorphism V2

x,

only.

It

= (Ei)* a ° Ey ' for the extension 7 ’ e it1 Q(L)

of v^ •

H to the last equation,

= H(e^) = H((Ei)* a o E 7 1)

by

= H(Ei* a) 0

by

E7

jtg = {v^} .

consider the secondary com­

which consists of the elements

follows from Proposition 1.7 that (Ei)*

Is an element of

8 ig =

xv^ = (Ei' )* a*.

In order to estimate the integer position

(Ei1 )* a' °

* = (Ep1 )*xv^ 0 E 7 1

Lemma 6 .1 , Proposition 2 .2 ,

56

CHAPTER VI.

= XVj ° Ep' ^Er ' = xv j 0 E(p| 7 1 ) = xv^ 0 E 2 v5 = xv2 Since

p

by (1 .1 8 ).

is an element of order

2,

then we have that

odd.Consequently (6 .3 ) is proved.

*30 = 0

is a subset of

is

q. e. d. € ^(K) ofv^>

Next consider a coextension 7 p 0 E7

from Proposition 1 . 7 that

x

is an element of

then

{v*

it follows

8 ig, v^-\>

by (4.7) and Proposition 5 .1 5 . Thus

which

p o E7 = 0

and the secondary composition {ti3, EP, E 2 7 }] is defined.

G 7t1 2 (S3 )/(Ti3 0 E*n (S3 ) + *(E3K -* S3 ) » E37 ) « Ejtn (S3 ) = rj^ °Ejt3 1 • We shall prove

By (4.7), jt(E3K -4 S3 )

(6.4)

o E3 7 = 0 .

*(E4K -> S4 ) o E 4 7 = { }

Proof: Consider an element tension of an element 0E 3 7 ) e {X1

that

an element of

X'

of

jtq(S3 ).

8 iq, vq^3 *

\

of

*(E3K

\

-»S3 ).

is an ex­

It follows from Proposition 1 .7 ,

Since

x 1 0 8 1 8 = 8 x' = 0 ,

* 8 = {v* 0 t^} (Proposition 5*9).

y = 0 or 1 . By Proposition

.

then

X1

Thus \ x = yv ’ 0

is for

1 .7 , (4.7) and by Propositions 5-8 and 5.11, we

have (yv* 0 t]6 , 8l8 , v8 )3

2 C (yv1, tj6 o Q l q , v0 }3

= {yv1, 0 , v8 ) 3 = yv* o e 3 k| + Jt3 o v9 = 0. Therefore we have that established.

X

0 E3 7 = 0

and the first statement of

(6.4) is

The second statement is proved similarly, by using

that

it Q 0

vi o = {vi ] ° v 1 0 = (v4 !e- dNow we choose an element of this secondary composition and denote it by

u3

e {1^, Ep, E 2 7 )] € We denote also

that

These elements

|i

2 (S3 )/(1^

nn = En " 3

o E*^) . for

n > 3

2 nn = 2 ^ = 0

for

n > 3

e fV 2 tn + 1 ’ ^ °'")n-4 + (vn } ^ n Proof: It follows from Proposition 1 . 7 that ^ a

of

t)3

.

have the following properties.

Lemma 6 .5 : H(n3 ) = cr,n,

extension

n = E°° ^3

and

and

k'

= a »Er

satisfying (6 .3 ) and for a coextension

7 e jtn (M)

E 7 . Then H(h3 ) = H(a 0 E7) = H(a) ° E7 = Ep*a' • E 7 = a' » Ep*E?

by Proposition 2 .2 , by (6 .3 ),

for

an of

SOME ELEMENTS GIVEN B Y SECONDARY COMPOSITIONS

= a ’ 0 E(p* 7 ) = a 1 ° E 3 7 By (6 .3 ), (Ei*)* a* = xv^. sion of

xv^.

by (1 -1 8 ).

This means that

7 is a coextension of

Since

tion 1 .7 ,

a 1 o E3 7

57

v

II

OJ

and

v l 1 ^5

,

2 V g ,

vn )5

C (v5, 4vg, Vn )5

2Cv 5, 2 t8’ vl1 }5 _ < V 8t 8 > V 1 1 -*5 ° 2 l 1 5 C (v5, 2Vq , 2 v 1]}^ C It follows that

{v^, 2vq, v11 }^

of

2a = E28* + x o 1" ° v^2

and an integer

0 E a 1 = 2(E2e’ + 2y a

C (E2v '

v9 53

x.

By (6.2),

° E a 1) for an integer

for an element

of + 4y

a !" V]2 = 1^ o y.

a

Thus we have that for any

{v^, 2Vg, vn ^ 5^ E32b 6 1 = 22 ( cx + x E £ 1 + 2xy (a E2e

(6.6)

0 Ea 1) .

Then Lemma 6.6 is proved.

v).

q. e. d.

The elements

£n -

Consider the secondary composition 0 810 = 8v^ =0. Lemma

2Vq , v ^}^.

k.5

that

E(8tg 0 a') = 8Ea1 = 0

81^ 0 a' = 0.

Lemma

5 . 1^ ,

81 g, E a ,}1

by (2 .1 ).

C

It follows from

Then the above secondary composition is de­

5

Choose an element of this secondary composition and denote it by e {v^,

B l q ,

•Ea'}1 €

^ / ( v ^ o E*^).

We denote also that £n= E11"5 ?5 Lemma 6.7:

.

0 EnJ^ + ° E2ct!, by (^.7 )* By Proposi2 0 E a ' = {v^ ° t)q 0 2a^} = {tj^ 0 2 r]g ° a^) = 0.

fined and it is a coset of tion 5-8 and

[v^,

for

n >

H ( ^ ) = 8a^

and

5

and

5 = E00^ . 2 ° ^ mod

7 ° 2E*1 ‘^ .

60

CHAPTER V I .

8^

= 0

=

n > 5

for

provided if

^Eir7^ = 0 .

Proof: By Proposition 2 .b, H( 5 5 ) € H(v5, By (5-8), H { v 8 ig, Ea1)., Since

8 l8 ,

_+

Ea, ) 1 =

a

“ 1 (v 4

• 8 l ) ° E 2 o f.

= 4 ( 2 v ^ - v*) = 8 v ^ =

contains

H{v^, 8 tg, Ect, } 1

o 8i^.

° E 2 a' = _+ 4(2a^) = 8 a^,

+_

Next, we see that the secondary composition

°

vK ° Eir'

and the secondary composition {2 v^, ^ig, E^a' 7 ° Ejt^ = ° 2 Ejr.j^, by a similarway for {v 8 tg,

5

7

We have by Proposition 1 .2 , 2£5

and

{v , 8 ig, E 2 a " } 1

7

is acoset of 2 v^ E a 1 }-, .

) = 0,

H ( ^ ) = 8a ^ .

then we have proved that

is a coset of

by Lemma 5 . 1 3

H(v^ 0 Eitj^) = HE(v^ °

is a coset of

Thus

( 2 v 5 , 4 tg,

€ {v^,8 tg, Ea')10 2 1 ^

e

V '},

C {v

C iv^,

8i g,

2Ea1)1

- Cv5 , 8ig,

E2a"}1

, 8 i g , E 2o " ) 1

.

It follows that 2 ^

and

e {2v^, 4 l q , E 2 a " } ]

+ v 5 0 EirJ^

e {2 v 5 ,

0 2 l ]6 + 2 ( v ^

E 2 a n )1

C

(2v^,

=

{2 v 5 , l^tg, E 3 a * " } 1 .

°

Eir^ )

ktQ, 2 E 2 a " ) 1 + 2 v,_ o Eir7,-

By Lemma 6 .5 , Proposition 1 . 2 and by (5*9), 15

0

e t)2 ° Ctj7, 2 lq, E 3 a , n ) 3 + Ct]2 ° v3} C {T)3 , 2tg, E 3 a " ,} 1 + {0} = { 2 v 5 0 2 Lq , 2 tg, E 3 a 1 " } 1 C { 2 v 5 , 4 Lg, E 3 a ’" } 1 .

It follows that = n| 0 ^7 If for

n > 5•

‘t-Eit'j = 0 ,

then

8^

= 2^

m°d v 5 ° 2 En'[5 . n7 = 0 ,

and thus

8 t;n = 85

q. e. d.

CHAPTER VII

2-Primary Components of

i).

*n + k (Sn ) for

The groups

*£+8 and n£+9

The results for

«n+8 and *n+9

Theorem T-i:

= °>

rtn+8 = {en } " Z2 4

8 < k < 13 •

.

are s'ta"t:ed as follows.

n = 3, 4, 5,

® (£6 } - z 8 ® z2'

= (V

= (o' » r)llt) © Cv7 ) + {e^J = Z2 © Z2 © Z2 , "?6

= ^a8 ° ’l15 1 ® fE a ' ° 1 15 J ®

,t?7

= ( a 9 °’116 J

“ Z2

® {e9 } =“z2 ® Z2 ®

,tn +8 = ^ n 3 ® ‘S 1 “ Z2 ® Z2 ’ (G0;2) = (V) © {e} Theorem 7 .2 : n ?2

® Z2 ® Z2’

Z2 ' 10,

~ Z2© Z2 .

ir^ = (t)2 o e^) == Z2 ,

= ^ 3 ^ ® ^ 3 ° e4 3 “ Z2 ® Z2 ’

* n +9

= {vn 5® {tJn ) ®

n ?6

= ta' 01l4}

” 17

n^

® Z2

®

° en+1 } * Z2 ® Z2 ® z2 for tv7 5 ® (|J7 } ®

= (o8 0 I 1 5 1 ® (Eo' °

tT>7 ° e 8 } “ Z 2

n =

4 ,5 ,6 ,

® Z2 ® Z2 ®

Z2 ’

® (v8 ] ® £“ 8 } ® C,>8 ° 69) = Zg © Zg © Zg © Zg © Zg,

*?8

*]°

' Co9 ° ^l6 J ® fv|’® S ] ® N °£ 1 0 5 ~ Z 2 ® - tA(i21)) © tv3 0) © {Hl0} © {tj10 . e,,} - Z ©

*n+9

= tvn 5 ® ‘^n5 ® {^n * En+ 1 1 " Z2

© Z2 ® Z2 ’

(G9;2 )

= £V3 } © (u) © (r, ” £} - Z2 © Z2

© Zg .

First we have

H(Ej) = v2

Proposition 5 .1 1 .

Z2 © Z2 © Z2,

^

by (5.2) and Proposition 5.15.

n > 11

In the

0= n2Q— >

exact sequence of (4 .4 ),

n2 0 = 0

Z2 ® Z2 ® Z 2 ’

by Lemma 6.1

and this image generates

It follows then

= {e3) « Z2,

and thus

1 = [y\2 ° e3) « Z2 61

by (5*2).

by

62

CHAPTER VII. By (5 -6 ) and Proposition 5-9,

= (e^) ~ Zg .

By Propositions 5 - 8 and 5*9* we have an exact sequence 9 4 E 5 9 0 = *t4 -* *12^ * 1 3 * *13 = ° from

(4.4).

of (4.4).

Then

is an isomorphism and =

E

Z2

{e^) «

Next consider the exact sequence 3 A 2 E 3 H 5 2 _ n 13 > *1 I- **1 2 “"* * 1 2 * * 1 0 By Lemma 5 -2 , A * ^3 = {A(s^)} = {a(E2 s3 )} = o

It follows that

E

is an isomorphism into.

Lemma 6 .5 ,we have

that

H

.

since

2 e^ = o.

By Proposition 5*15 andby

is onto and that

* 3 2 = {p.^} 0

[t)3 o

} « Z2 ®

Z2 .

By (5 • 6 ) and Proposition 5•11 , 43

= Cv4 > ® W

3 ® K

0 s5 ) - Z2 © Z2 © Z2

By Proposition 2 .2 , Proposition 3*1 and by Lemma 5-1^, HCv^ o aQ ) = Efv^ # v^) o H(erg) = E(vq o V]1) o l1 5 = v| . Then in the exact sequence J

H 9 A 4 E 5 _9 15 15 13 14 14 H is onto by Proposition 5 .1 1 . It follows that E Q By Proposition 5 .9 , = 0 . Then E is onto and

is anisomorphism into.

= Cv|} © (n5 ) © {n5 - E g ) - Z2 © Z2 © Z2

By Propositions 5 - 8 and 5 .9 , it follows from (4.4) the following two exact sequences: 0

0

^

0

,

«’3-s «f„a

.

From the first sequence, we have the result for 7 *2 .

By Lemma 6 . 2

andProposition 5 -6 , we have

has 1 6 elements and it

is

generated by

have

"vg and

is onto and

e^. Since

8 v^ = 0 ,

ir^ we

= (v6 ) © (s6 ) - Z8 © Z2 .

By Proposition 5 -8 , n

J3 0 - jt1 7 of (4.4)

that H

in Theorem

By Lemma 5 •i

>

6

jrJ3 = 0 . Then we have the exact sequence E

7 H

13 A 6 E *15 *16

and Proposition 2 .2 ,

7H 13 *i6 ~^

* 1 4 51 1*5

2 -PRIMARY

COMPONENTS OF

Since a 1 o 2 t)^

= o,

8

«n+k(Sn) FOR

generates

= Zg

< k < 13 2 (o' ■>

and since

Ajri6

in Theorem 7*1 is proved. = H(a 1 ^2^_).

The kernel of

A :

2vg. for

is generated

Then it follows from the exactness of the above

2 (af * > t]2^) = 0

sequence and from

_

i-s generated by

= {v^} © {e^) « Z2 ® Z2 . Then the statement

It follows that

E

.

]o

By Lemma 6 .2 , and Proposition 5 *6 ,

where

=

then we have that *j5 = (o' . n,t) © E # f 4 » Z2 ©

by

63

that

* ? 6 = [0' ° ’I 14] ® E n % > is an isomorphism into. Thus we have the statement for

7 n ^

in

Theorem 7 *2 . The results for results fo: for

7

7

and

and ^ ^

are verified directly from the

and from Propositions 5-1 and 5*3. by use of

(5-15).

of (4.4).

Next consider the exact sequence 17 A 8 E 9 H 17 A 8 E 9 H 17 A 8 *19 * *1 7~* n iQ~* *1 8 “* *1 6 ”* *1 7 ^ *1 7 ^ * 1 5 By Proposition 5*15 and (5 - 1 6 ), we have that A :

an isomorphism into.

8

9

E : *■,£-> * ^7

Then it follows that

7 ->

is onto.

is It is

verified easily from(5 *1 6 ) and Proposition 2 . 5 that A(t)17) = Ea 1 o^ Thus the first two

A

and

a(t^7 ) = E a ’ o ^

of the above sequence are isomorphisms

. Into.

It

follows from the exactness of the sequence5that „9

E : and by checking the rela­

isomorphisms tion

E(t^ 0 a") = E ( a ,n ° r\]2^ = ^ ^ 4 0 a 'n) = 0

from Lemma 5 . 1

Next we shall prove (7 .6 ).

The secondary compositions

consists of

e^

and

5 2 ?ltai£iS en 2 ^2 Ln' vn' ^n+6 ^t^

(v1, vg, tj^)

and

{Ev1, v^, ^o^i

respectively. The following secondary compositions :

{ 2 v n ,v ^ ,

w^ere

nn + 6 ) t .

n > 5

and

t

t v n ,2 v n + 3 ,, n + 6 ) t ,

Cv2 ,

2 t n + 6 , r,n + 6 ) t ,

are appropriate integers such that

the secondary compositions are defined. Proof: First consider E v 1 o Eir^ 1 +

{Ev', v

w

h

i

c

h

is a coset of

o rj^ ^ = 0

by (4.7), Proposition 5*9 and Proposition 5*15* k Thus this secondary composition consists of a single element of « 1 2 = {8 4 }

« Zg,

which

E00 xe^ = xe

0

t) =

{a 0

is

x = 0 or

xe^for

is an element of r\]

=

Cv

+

e} .

= E°°{Tin , vn+i> 2 vn+4^

that

By i) of (3 -9 ), contains

x= 1

and that

E°°en =

mod

: *^2— » (Gg;2 ),

then

which is a coset of xe e

belongs to by (6 .1 ).




Thus

v + e

{Ev’, v^, rj^ Q}1

consists of a

single

e^.By Proposition 1 .3 ,

element

E {v ',

Since

Consider E°°

< 2 v, v, r\ >

xe = e This implies

1 .

E

vg ,

n9 J

C

(E v ',

v? ,

m 0Jt

•

: jtn (S3 ) ^ * 1 2 (S4 ) is an isomorphism into by Lemma 4.5, then it

follows that

{v1, v^, r\y)consists of a single element

E~ 1 e^ = e^ .

The remaining part of (7 -6 ) is proved by similar methods to (6 . 1 ).

ii).

The groups

n£+10

and

*£+i T

Apply Lemma 5 - 2 for the element ment

|i 1 e (tj3, 2 1 ^, ni f ) 1

such that

"3 € n^2,

H(n') = E2 n3 =

then we have an ele­ and

CHAPTER VII.

66

2 n'

= t)3 o ^

and

E( 2 u') =

4.5

that

oni3. 0

2 (j.T =

0

Proposition 3 .1 , r, 2 # n3 = ^

By

0 ug = E(r)| ° ^ ).

» r)llt = 0

» u6 = n5

-

t)u

It follows from Lemma

(j.^• We have obtained H(fi’) = 1^

(7*7)*

and

2 n' = t]2 °

Then the results for the groups ^ + 1 0

anc^

*1 1 +1 1

ape stated

as follows. = tn2 0 Ei*) ® U 2 0 ^3} - z2 © z2 ,

Theorem 7 •3 » " h

=

{e1} © {?i3 0

} s*■ z k © Zg,

{v^_ 0 ffr} © {Ee '} © (1)4 ° u5 ) *= Zg ' © z k © z2 , = for n = 5,6 and {v 0 a _ ) 0 { ti "n+l0 ' n n+3 'n * ^n+1} " Z8 ® Z2 8 ) ~ Zg © Zg © Z2 , "18 “ {a8 0 v i 5 ] ® f v8 0 " n 1 ® ll8 0 ^9 l a 9 . vl6) © (n9 • 1^10) = z 8 © z 2 j = 10 *20 *-ff10 0 V 17^ ® C, • n,, } - Z k © Z2 ' 11 *21 = {CT11 0 V181 ® £ n1 1 ° ^ - Z2 ® Z2 ' n for n > 12, tln ° Mn+1] ~ Z2’ *n+10 ii

0

CVI

0

(tl 0 n) = Zg . p n1 ^ = (n„ 0 e 1 } © ir\2 0

Theorem 'J.k: "14 =

^5

=

U'} © {e3 ° v, ,} © {v1 0 s6 )

} « z h © z2 , Z h © Z2 © Z2 ,

tv4 0 a ' ° n,^) © (v4 0 V^} © {Vlf » e } ® {Em’} © {s4 } « Z2 © z2 © z2 © z^_ © z2

© {Ev1 0 {£ } © { 0

«?66 n1 ^ t?6 ) ©

22

Vg ) © {v5 ° e8 ) * Z8 ® Z2 ® Z2 ' ° Vl4) : » Z8 ® h

(Vg

12

>

« Zg © Z2, for n = 7,8 and 9 , " {?n} © t“n • vn+8) 10,11 and for n > 13 , *n n+ 11 = ten ) = Z8 for n = ^ 12 ( A U 25 )) © (?]2) « Z © Zg , C n

23

(G-| ^;2 ) = t£) - Z8 . First we have and by Theorem 7*2.

n^2 = (r)2 0

by (5

Next we have

(7*8)H(v1 o e 6 ) = ti5 ° e6 For,

© {Tj2 0 e^} ^ Z2 © Z2

and

H(s3 ° v^ ) = v3 .

H( v ’ o e^) = H(v’) 0 ££ = ^

0 e£

and

H ( £ 3 0 vj1 ) =

H(e^ ) 0 v11 = v2 0 v11 = v3 , by Proposition 2.2, (5 -3 ) and Lemma 6.1. The images of

H

in (7 -7 ) and (7-8) generate

7.2. Thus in the exact sequence

by Theorem

2 -PRIMARY

COMPONENTS OF

3 H 5 A 2 71114- *14 “ * *12

is an isomorphsim into. "3

is onto.

Then we have that

E

O

0111:0 and that the group

b 1, t}2 o

67

By Lemma 6.6, and Theorem 7*1 we have that

R

H : — * *13 by

< k < 13

E 3 H 5 * *13 * *13

H : f t ^ — » jt^

of (4 .4 ), the homomorphism

8

*n+k(Sn) FOR

= 2e1 and

ti3 0 n^-

is of order 8 and generated

Thus

={£’} © {r]3 o

0 Z2 .

By (5-6) and Proposition 5 *15 , ^ 14 = tv4

°

® {Es ’} ©

°

} ~ Zq © Z^ © Z^ •

Consider the composition of homomorphisms 9

where

= {a^} % Z ^

7

and

^ = {a '} ~ Zq

H ° A :

Q

^

7

,

by Proposition 5 *15 * By

Lemma 5 *14 , Proposition 2.5 and by Proposition 2 .7 , (H - A)(1kj 9 ) = H(A(E3a")) = H(A( i 9 ) ° Ea") = H(a(l9 )) o Ea" = + S l 7 « Ea" = + 2Ea" = 4a' . Since

H ° A

H o A

is onto and that the image of

order 8 or

is a homomorphism, this relation Implies that the homomorphism 9

:

A

-»

4

is a cyclic group of

16. Then it follows from the result of the group A(a9 ) = x (

for an odd integer

x

that

o a ') + yEs1 + zt^ o ^

and some integers

y

and

z.

It is verified that

= En?3 ® ■ Then it follows from the exactness of the sequence .9

A

16

*

4 E 5 H 14 * *15

9 15

that the sequence

is exact. of

By Lemma

5 n. s u c h 15

6.6,E2s I is divisible by 2. Let abe an element 2 that 2a = E s ’. Then it follows form the exactness of the

last sequence that

H(a) = v|

and « Z Q © Z2 .

= {a} ® {t)^ 0 Furthermore, any element of

R

and

a

of

H(a) = v|

satisfies the above decomposition

O

2a = _+ E s 1. In particular, we may take

(7-9)-

H( v 5

o Qq

o Qq

a, since

) = v|

which has been proved already in the proof of Theorem 7*2. °

as

°

Thus

« Z q © Zg

and we have '( 7 - 1 0 )

2 ( v ? ° os ) = + E 2e ’

and

4 ( v n » an+3) = V

° en+2

£22.

n > 5 •

68

CHAPTER VII. We h a v e

( 5 *2 ) ,

by *13

C on sid er

^2

=

th e

°

exact

*5 15 of

(4 . 4 ),

th e

in

classes

Thus

th e

follo w s

^

of

n f,

v'

®

of

E

is

2A ( v ^

^2

2

0 ^ 4 } % Z4 ® Z2

E

3

13

w h i c h we k n o w t h a t

kern el

Le m m a 5 • 1 >

e'5 0

H

5

14

H

is

and

e

g en erated

14

o n to

and

ir^ /E jt2 ^

is

g en erated

B y L em m a 5 - 2 ,

«* v ^ .

by

a(v^

0 ag )

0 ag ) = a (+ E 2 e 1) = A (i ^ ) °

(+

.

by

0

iig)

= 0.

By P r o p o s i t i o n

2 .5

and

s 1 ) = 2 ( t )2

A(rj^

o e 1).

It

th at

(7-11)

a(v ^

© a g ) = _+ ( t ) 2 E n 2 2 = { t )2

and It follo w s

from

is

easy

to

»

resu lt

for

jc^

in

Z2

= 0.

V]])

Then

it

( 7 *7 ) th a t

of

o Vl 1} ~ Zk © Z2 © Z2 .

7 * 1*- f o l l o w s f r o m

Theorem

.

= 2 (8^°

0 eg)

^

0 li^

T]2

«

2 (v*

= t)2

mod

\±^)

0

see th a t

2 n'

th er e l a t i o n

o £ f)

= (m.

^

.

2-PRIMARY COMPONENTS OP

x2

Theorem 7-7:

*n + k (Sn )

FOR

8 < k < 13

= (r)2 • v' » ug) ©

75

■> v 1 ° rjg » e^]

z2 ©z 2 n l6 = (v1 0 %

II

A CD

-* U1

”17 ’

^n n+1 3

Cv2 .

n

CO OJ

II « - -=t

r~ OJ

*11-

Ug) * Zg © Zg © Zg

°8) ® {v4 0 *>7 0 ^8}

tv5 0 a8 0 V15 1 ® fv5 ° ^8 ■

(vn ° an+3 ° vn+10 3 “

Z2 © Z 2 for

Z2

(Og ° v 1 5 } ® (v8 ° a11 ° v 1 8

*n+, 3 = {on

4 1 -

° ' V ~ Z2 ’

(0' »

•

vn+7]

*

Z2

£2E

'125)==

Z2

Z,

i (e 0 T)24) © (E6- 0 n24) « Z2 (Ee 0

Z2 ®

n

® (011 0 V 1 8 } *

W

] “

n = 6 and 7 ,

®

Z2

’

Z2

"27 = (A^ 2 9 )5 ^ Z ’ *n+13 = (G i 3;2 ) = 0

for

n>

15*

First, by (5-2), Theorem 7*4 and by (7*12), we have *14 = ^ 2 ° |a'■*© = ° © By Lemma 5*7,

E( ti2 o v ' °

^2 e3 V1 1 ^ © ^ 2 v* e6^ (t)2 o v » o v 6 ) @ Ct]2 o vf o s6 ) « ) = E( t;2 ° v 1 o e^) = 0.

© Z2 0 Z2 .

By Proposition 2.5

and Lemma 5 «2, A(2^) = A(_+ E2n *) = A(i^) o (_+ n 1) = 2 (tj2 ° ia1)• Thus A (^) = ± U 2 ° n* ) mod Ct]2 0 v’° 7 6 ) + {t]2 ° v * o e6 ) and Then

= 0

E( ti2 ° n *) = 0.

and we have from the exactness of (4.4) that the following

sequences are exact. 2 3 H 5 A o — * *i5 > *15 * *13 A 2 E > it3 H > n5 and tt1 rj > jt2 1^ 1g 1g > t,^ »0 * j *14 *1 6 * 16 ri5 By (7*11 ) and Theorem 7.4, the image ofo ag under A is an 2 5 element of order 4 in As = {v^ o ag} © { 0 = Zq © Zg, *5 2 Theorem 7*3, the kernel of A :— * *1 3 ^as at most ^ elements. By

Proposition 2 .2 , (5*3) and (7-10 ), we have H(v 1 o n6 ) =

o ng

and

H(vT o t)6 o 6r^) = ^

o

These elements generate the kernel of the homomorphism

= ^(v^ o aQ ) . A

from the exactness of the above upper sequence that * =

(v< . ,6) e lv' • 16 • e?) = Z2 © Z2

and it follows

by

76

CHAPTER VII.

By B y ((5*2), 5 - 2 ), ^ 2 ° v *' 0° ^ 6 * ® {t12 ^2 ° 0 v? v ' 0 ^6 15 0 • 8E 7^ fl ^“ Z2 z 2 0® Z2 z 2 *• Anl5 = fr)2 We have, by Lemma 5*7, {E(t]2 ) 0 |i^} ii^} ©© CE (T)2 oo vV 11 )

Ejt2 ^ E A*j -=

(E(t)2 0 V v '1) ) o o (E(t)2 °

0 Eg) 6g) = = 0 0 . . °

It follows from the exactness of the above lower sequence that c p (7 *23 ) *• (7-23 A :— > Jt15 Is o n tIso , o n t o , O R and that the group jr-g is isomorphic to the kernel of A : n'g which is a homomorphism onto. then the kernel of this

A

As

xijg

»« Zg © Z2 © Z2 ,

has just two elements

o0

and

p -> ir^

by Theorem 7*4, 4^.

By Propo­

sition 2.2, 2 .2 , (5*3) and b y (7* 1*0, H(v1 •

Tig • u ?I ) =

j

° Hy\ =

j

'

It follows that *16 "?6 = {( v ' *0 ^6 16 ° 0 ^T3 = “ Z2 • k k The results for the groups and

are obtained from and are obtained from

(5 •6 ), Theorem 7•2 and Theorem 7•3• O k Consider the homomorphisms A : jt^q — >

Q

k

and

A :.

By using Proposition 2.5, 2 .5 , we have , 2 A(a9 ■ * ^ 1 6

A(v|)

x )

0 a'

=

- A(v|)

0 v 13

= aH(v^ 0 ag) 0 v 13 = 0

M u 9 ) = A ( t g ) • n7 A U 9 1'

S 1 0 }

“

(i 9 )

a

by (7 .. 1 6 ) ,

+ E v 1 « nT 0 e g

»

Ev' 0 t)7

by

by (5 .8 ),

0 11^ •

e8

•9 ),

(7 .

= Ev'

0 ^

by (5.- 8 ),

0 e8

A (CJ9 «' V16 } = A (Og ) 0 V 14 =

X

(vk

°

a'

by (7 .. 1 6 )

0 v l 4 ) + E e 1 0 v lif

= x'(v® . a 1 Q ) + E ( s 1 0 v 1 2 ) a

where

x

(t19

and

e

'

(i1 0 )

=

A(tg) 0 r)7

0

x ’ are odd integers.

n8

= Ev'

by (7 -1 9 ),

0 T)7 0

by (5.■8),

Then from the exactness of (4.4),

E n ^ = {v*} © {v^ ° nQ } © {v5 o t)q 0 e 9 ) « Z2 0 Z2 © Z2 and

Ejt^ = iv^ ° t]q 0 1^ ) « Z2 . By conserning the structure of the groups

that

A :

morphism into. (7.24).

has the kernel

{v^}

and

A :

9

and

9

we see is an iso­

Then it follows from the exactness of the sequence that E : jt^ q — > Jt^9

is onto

and that the following two sequences are exact:

2-PRIMARY COMPONENTS OF

n _*> E tt> rt, g

*n + k (Sn )

*5 i t

H —

FOR

By (7-13)^

A :

->

77

kit^

i9 t1^ A

0 — > ^*^7 — > *?8

8 < k < 13

—*0 is an isomorphism into, then it fol­

lows that A l = E *^6 = {v5 ] +

{v5 ° ^8} + (v5 ° ^8 ° e9} ~Z2 © Z2 © Z2 ‘ ByProposition 2,2 and (7*9); H ( v 5 ° a8 0 V 1 5 ) =H ( v 5 ° a8 ) ° v 15 = v 9

By (7 -1 0 ), =

{E2 v » »

'

2(v5 ° cjg 0 v15) = + E2£' o v15 CE2it3g

T)rj ° n8 ) = Ea{t)9 o h10) = o . Then it follows from the last exact sequence that n?8 = ( v 5 ° °8 ° v 1 5 ] ® ( v 5 0 18 ° V

- Z2 © Z2 .

Consider the exact sequence n A 5 E 6 H 11 A 5 * *18 * *18 * *16 * *1 9 ”"* * 1 7 By (7-17) and Proposition 5-15, the kernel of generated by

2 al 1

A :

+ H(A(a13)) by Proposition 2 . 2

which is

jt!jg—> jt^g and 2.7*

By Proposition 2 .5 , (5 *1 0 ) and by Lemma 6 .3 , we have 3 _ 11 > == v5 ° ^8 ° v9 = V5 ° V8 = v5 a (e 11) =

and

Then the result

jt^q

= {A(a13)} «

follows from the exact

sequence and from the relation (7-25).

8a (q 13) = v6 This is provedas follows. 8 a (a 1 ^ ) =a(i13) 0 8 an g

v 6 ° ^9

{vg,

by Proposition 2 . 5 by Lemma 5 . 1

r)9, 2 t10)o 8 an

0.

=v 6 ° ^ 9 € vg ° - E ^ { ti5 , 2tg , Ea'

= {vg, t)9, 2i10)^ 0 E 6 a ,M C {vg, t)9, 2 l10) 0 8an By (4.7),

{vg,

v6 ' "n

ti9, 2t10) 0 8 a n 0 8on

= (v6 0 4

by Proposition 1.4 by (1 -15) and Lemma 5 •1^ • is a coset of ° “n 1 = 0 •

Then we have the equality (7-25)* Next consider the exact sequence

J

1

20

_A

* *18

JL J

* *1 9

H ^11 A 19

5

17

’

is

78

CHAPTER VII.

The last homomorphism isomorphism into. E

is onto.

A

is clarified as above, we see that it is an

It follows from the exactness of the above sequence that

By Proposition 2 .5 , (5*1 0 ), (5-9), (7-10) and Proposition 5 .1 1 ,

we have A(v^ )

0

=

18

v9

°

=

0

M n n ) - v5 • 18 ° ^9 and

A ^11

°

e12)

v5 0

=

1111

°

the kernel of

A

3

11 *2 0

:

18

£ 12

The homomorphism t13

11 ~ 2 and that is generated by „311 and

"1 9 = (v6 ° 0 0 O v 16

Then we have that (7 .2 6 ).

0 8

4

and

A(a^)

jt.jg 1 «

41

-

jt2 j

are generators of

is an isomorphism onto since ^ and respectively.

It follows from the exactness of (4.4) that the sequences *7 0 *13 A 6 19 19 17 and 6 _E^ J .13 A 0 * "19 * 2 0 21 are exact. A

onto

2v£ ° 4 °*

and we have

2 v13

By (7*18), the generator

«11 ^^

« Z2

is mapped by

4Thus °* the Thushomomorphism the homomorphism A is Aan is isomorphism an isomorphism into = =o.o.

7

it' q

By Theorem 7-1, the group « Z2 ®© Zg and

of

e -|3 * Then the result

is generated by

jt^q Jt20 = {v^ (v^ ° a1 ctiqQ ° v 17^ }) «~ Z2

into

v’ 13

of Theorem Theorem 7*7 7*7 is is

a consequence of (7-27).

= A(£ 13} = 0 * This is proved as follows. A ( s 13; 13; ° (-en t ;) 13) = A ( - e 13; 13) == AA( Ui 13) (-en) ) e {v6, {v6, t)9, t)9, 22ll11QQ)) oo(-en —

f 1.

^

o »

1

o

( —a

by LemmaLemma 5-10 5-10 ^

■ (v6 ’ V 2 t1 0 ]6 •(’e ii) = v6 o E 6 {t]3, 2 1 ^, 8^}

q 1 n p p

TT1

=

„9

slnce e 6 "5 = *?i by Proposition 1.4

C v6 o E6 * ^

by

(4.7)

= {v6 o n9 o n 1Q} + {v6 o E 6e')

by

Theorem 7*3

= {EA(m_1 1 )} + {v 6 - 2 v 9

by

(7-1 0 )

=

a (v"13

Proposition2.5 2.5 bybyProposition

0

+

[2v\

o

a 12 } =

0

+ e 13 ) = A (r)13 o a 1^ )

o a 12)

.

by Lemma

6.4

= a(t1i 3 ) ® a ]2

by Proposition 2 . 5

= A(H(a1 ) ) o a ]2 = 0

by Lemma

5-14.

2-PRIMARY COMPONENTS OF

*n+k(Sn )

FOR

8 < k < 13

a(s13) = a(v]3 + £i^) " A ^V1 3 ^ = 0

Thus

79

and (7*27) is proved.

By (5*15), Proposition 5*9 and Proposition 5-11, we have = «

4

*21 = {o8 ° v1 5 ] ® Cv8 ° °11 ° ''IS1 * Z2 ® Z2 • 8 0 = jt2 Q — E>9 — » H« j 17of

In the exact sequence by Proposition 5-8.

=

Thus we have

Consider the exact sequence 8 E 9 H 17 A *22

*22

of (^.^), where

*2 3 *2 3

= 0

~

and

8

E

9

H

17

*22

^v?7^ ~ Z2

*23 =

21

o.

*21

*

(h-.k),

*22 ^

'

P^positi011 5*9 and

Proposition 5 .1 1 . By Proposition 2 . 5 and (7-1 9 ), we have for an odd integer x

that 1 7 )

A ( v 1 y )

0

2o

v lQ

q

o

v 15

-

x v q

0

a n

o

v l(

~ "8 u U11 "18 * It follows from theexactness exactness of of the the above abovesequence sequence that that rj

9 _ f 2 -I *22 9 ° vl6 ~

2

and that 8

(7 -2 8 ). (7*28).

9

EE :: it22 -» n|3 ^23 "22

and 5-9-

= tvl9 J

is a homomorphism onto. n2 ? = "23

Z8 and

= 0 by Proposition 5-6, 5-8

It follows from the exactness of (^.4) that 9 1o , 2 i r? E : tt22 ~ ^ 23 ^aio 0 V 1 7 ^^ Z2

and the sequence n is exact.

21

JL

J ° A 22

22

A

J

20

By (7*22), the kernel of the homomorphism

2v19 = + H(A(v21)).

It follows that

Obviously the homomorphism nel isgenerated

by

A

A :

*2 2

^v21 = ri^i*f°ll°vs from the

is generated by .

*22 = (A(v21)} *

on^°

an(^ -^s ker-

exactness of

that the following two sequences are exact:

p1

21 *23

OJ OJ

n*

*2 3 —2 1 A . 10 25 *2 3

II

11

0 -

11 H r 3 *2b ^2 £ 1,} '

by Proposition 5 .8 . By Lemma 7*5 and Proposition 2 .2 , we ^ 2 1 and t)21 see that the elements Q l and 0 l ° r)2^ are mapped onto tj2

where

=

o

respectively and they are of order 11

2 .. It follows that -

- -

CHAPTER VII.

80 and

41

-

(S'

.

• v^g) ~ Z 2 © Z2

r,2 3 } © ( « „

Next consider the exact sequence 23 A

11 E "24

*26 of

( 4 . U).

12 H *25

B y P r o p o s i t i o n 2.5 and

23 A *25

11 E *2 3

(7-21 ) w e

have

A ( v 2 3 ) =A ( t 2 3 ) » v 2] TThus h u s we w e hhave a v e tthat hat ((7-29)7 - 29)-

t h e kkernel e r n e l oof f the

E jt^

= a,,

12 H

23

*2k

o

n2k

v^g .

== ((E01 E01 ° °ti^24^ ^ ) Z 2% Z2 a n d anci t h a t^ a t

A : jtg| -> *11

is ggenerated e n e r a t e d bby y

22 vv 23 2 3 .*

B y L e m m a 7*5 a n d P r o p o s i t i o n 2 .2 , w e s e e t h a t t h e e l e m e n t s and

G o T)2lf

are mapped by

H

onto the generators of

r e s p e c t i v e l y and t h e y are of o r d e r

2.

and

T h e n it f o l l o w s f r o m t h e e x a c t n e s s

of the above s e quence that =

and

{9} ® { E 0 ' ) » Z 2 © Z 2

= le o l2jf) ©

[E6'

B y use of Pr opos i t i o n

E 0 ’€ E an ^23^3 *

The secondary composition jr2 ^

o « £ of the proof

are isomorphisms into.

of Theorem 7-4 that

. and

Also we see easily in the last part

A : ir^

-* *2 ^

is an isomorphism into.

It follows from the exactness of the sequence (4.4) that nJJ = (E0) » Z2 and

= (E9 ° Next consider the stable element

« Z2 . E°°0 e G 1 2 - E ° ° 0

of the secondary composition < a,

v,

t) >

e

G 12 /( a 0

+ Gn

° r\) .

is a n element

2-PRIMARY COMPONENTS OF

*n + k (Sn )

FOR

8 < k < 15

81

By (^.7), Proposition 5-8 and Theorem 7*^, a ° G5 + G 11 ° n = 0 + U

< a,

Thus

°

o ni8) = o .

v, tj > consists of the element

E°°0

only.

By i) of (3*9),

< a, v, r\ > = < tj, v, a > . Since

T|n ° v12 =°

by (5*9) and since

(7*19)

and (7-2 0), then the secondary composition

defined and its image under an

element of

E°°

is

vl 1 o a li^ = 2a1 1 o v l8 =

< tj, v , a >.

that

a >

by

v12, a1^}1

is

By (4.7), there exists

= {01} belongs to this secondary composition.

2 11 2 E «23 = {E 0 1} = o, we have that < r), v,

0

contains

Since

0. Thus we conclude

E°°0 = 0. Since

it follows that

E°° : *2g

(G12;2)

is an isomorphism onto by (4.5), then

E20 = 0.

From the exactness of the sequence (4.4), we have that 27

" *25 =

is onto'

(l2^) = E0

and

* 2 6 “ *S+ 1 2 = (G !2 ;2) - °«

fOT

a

n *

By Proposition 2.5, we have ^(^2 7 ^ = ^(i2-^) ° ^25 =

° ^25

Then it follows from the exactness of the sequence (4.4) >■' that — and the sequence is exact.

Thekernel

of

0 A

14 H 27 A 13 *27 *27 *25 is generated by 2t2 7 1^_ 27 —

have

Finally by (4.5) andby the exact

E -n2g =

= —H(A(t2^))*

Thus

we

fA/ \■> „ iA(l29 /J ~ Z . sequence (4.4), it

"n+l3 = (0 13;2 ) ■ 0

for

is computedeasily that

n ^ 1 5.

We see in the above discussion that (7*30). jt^+1

A(t2y) = E0,

a (t]2^)

are isomorphisms into for

= E 0 1 and the homomorphisms n = 1 2 , 1 3 and 1 4.

A :

->

PART 2 .

CHAPTER VIII

Squaring Operations Let Sq1 : Hn (X, Z2 ) -» Hn+1(X, Zg ) be Steenrod's squaring operation.

In his paper [18], Steenrod gives a homo­

morphism of

in terms of the functional squaring

operation.

nn+j__i ^ n ) into

Z2

The homomorphism will be denoted by H2 : *n+i-1

H2

Z2'

may be definedas follows.

andconsider a cell

complex

uniquely determined by

a

Let

1 > 1‘ a

be an element of

Kq = SnUa CSn+1 1

ofChapter I

up to homotopy type. H2 (a ) i 0

if and only if the squaring operation

irn+-?-i(^n

which is

Then

(mod.2 ) Sq^ : Hn (Ka > Z2 )

Hn+^(K , Z ) is

an Isomorphism onto. For the case

i = i,

H2

may be defined for the elements of

2jtn (Sn ) and we have a homomorphism H2 : 2Kn (Sn ) -* Z2 . Since sequenceo -> Z2

Sq1

is the Bockstein homomorphism associated with the exact

Z^ -» Z2 -> o

of the coefficient

groups, then it is easily

verified that (8.1)

H2 (2rin ) = r

(mod.2)

for an integer

r .

It follows from properties of the squaring operation that [18] (8.2)

i). ii).

H2 (a) = 0

if

ac jrn+i_ ( S n ) and

H2 ° E = H2 .

It is known [1 ] that Proposition 8.1. *n+k(Sn ) H2 (a) i 0

i > n .

Z2

Let

H2

is trivial unless

k = 1,

is onto if and only if

(mod.2)

if and only if

(3 or 7 respectively),

n > k.

a = r\n , 82

i = o, 1, 3, 7.

Let

(vn

or

n > k+i, an

then

Hg :

then

respectively)

SQUARING OPERATIONS

mod

2jtn+k(Sn ).

a = tj2, (v^

Let

or

aQ

n = k+1,

83

H2 (a) 4 0

then

respectively) mod

(mod.2 ) If and only if

2jt2k+1 (Sk+1 ) + Ejt2k(Sk ).

Proof.

Consider projective plane of complex numbers, quaternions k+i 2k+i or Cayley numbers. This is a cell complex of the type =S uacs Sqk+1 4

such that

°*

Thus

map, in fact, the Hopf class. morphism onto for for

n > k+1.

n = k+i.

4 0

H2 (a)

We have that

for the class

aof

H2 : Jtn+k(Sn ) -> Z2

Itfollows from ii) of (8.2) that

It follows from i) of (8.2) that

the attaching is a homo­

H2 isonto

H2 is .trivial

for n
k+i . Then the group

By Propositions 5.1, 5*6 and 5 *15 ,

nn , (vn

generates the 2-primary component of Jtn+k(Sn ). the kernel of

H2

if and only if

is

2*n+1 > Sn+1Upo. £, £ CSn+i. H2 (E(3) thenSq1Sq1 Sn+1Upof CSn+i. follows (E3) = H2 (p) (f3) 4 4 0, 0, then / o/ o in inSn+1Upof CSn+i. It It follows by by the the induced homomorphism that

H*(Sn+1Upof CSn+i, Z2 ) -» H*(X U en+1 Uf CSn+i, Z2 )

Sq1 un+1 = un+.+1.

q.e.d.

We shall see in the following examples how the lemma is applied, in which Adem’s relations in iterated squaring operations are essentially useful. Example 1 . Since k = i, 3> I,

for by

2

-» z2

then the elements

tj ,

vn

ls onto fQI> n > k+1 and

crn

and

are not divisible

and thus they are not zero. Example 2.

Since

H2 :

{2i n , T)n , 2 iR+1} 4 °*

The secondary composition

*n +2 ^ n ^ = ^ n ° ^n+i ^ ~ Z2'

tion consists of a single element

it follows that the secondary composi­ rjn ° tj ^ . (2Ln (2Ln>>

To prove this we assume that that

2ln+i^ 2ln+l^

it(K by Proposition Proposition 1-7, 1-7, there there exists exists an extension a aee it(K a coextension

3 e jt(Sn+2 -> K)

is zero, where

K = Sn U en+2

and thus

Sq2 / 0 in

K.

of

2in+1

has

Let

tj

->Sn Sn)) of of 2t -> 2tn

a o p

such that the composition

U en+1

which is a cell complex of the

form

SnSn UU en+1 en+3.Since en+1 UU en+3.

then

Sq1 Sq1/ /0 0 in in SnSnU Uen+1. en+1. ItItisiseasy easyto to see see that that Sq2Sq] Sq2Sq]// 00in in

has has

Let3 g 3 it(Sn+^ g it(Sn+^ p. -» L) L)be be a coextension of p.

2i 2i

as the the attaching attaching as

exists since since ((33 exists

class, class, L. L.

® pp aa ®

Construct a acell cell complex complex M M= =L LUgUgCSn+^ Sn UU en+1 en+1 UU en+^ en+^ UU en+\ en+\ CSn+^= = Sn use of the above lemma we have that

and n and

as the class of the attaching map

L = SnCK S

contains 0. 0. ThenThen contains

== o. o.

then by by then

1 2 1 Sq Sq Sq ^^ 0 in M. M. On On the the other

hand, we have an Adem’s relation Sq2Sq2 Since there there is no Sq2Sq2 == Sq1Sq2Sq1. Sq1Sq2Sq1. Since cell of dimension

n+2,

Sq2Sq2 = 0

Thus we conclude that

in

M,

but this is a contradiction.

nn > 2Ln+i"* ^ °*

Example 3- The compositions

2

2

2

an+i and and an an are aren°t n°t divisible 00an+i

thus they notare trivial (Adem).(Adem) We shall show for a2, a2, the other by 22 andand thusare they not trivial . Weshall showfor the other cases are proved similarly.

Assume that

2

an = 2a

for some element

a

of

Itn+ 7h (Sn )* Then we may construct a cell complex K = Sn U en+® U en+1^ U en+1^

SQUARING OPERATIONS

such that that

en+B

en+1^

and

en+1^

are attached by

an

un , un+8

represented by

and

Sn, en+®

en+1^

respectively.

Sq8Sq8 un = Sq8 Now we have a relation

a

respectively, and

be cohomology classes mod 2 which are

un+1£

and

and

a n + 7 ~ 2 Ln+i4 *

is attached by a coextension of

Let

85

Then we have that

= un+lg .

Sq^Sq^ = Sq^Sq^ + Sqll*Sq2 + Sq 1 ^Sq 1

of

Adem. Obviously, Sq1 un = Sq2 un = SqNi^ 0 . Thus Sq^Sq^t^ = 0 , but this is a 2 contradiction. Therefore we have proved that an cannot be divisible by 2 . Example 4. W

1' {V

vn+l*)’ tvn’ an + 3 ’ vn+10)

’W

[r\n ,

The elements of the secondary composition

vn+1,

vn+T, an+1Q) are_not

divisible by 2 and thus they are not trivial. We shall show this for the last secondary composition.

For the other

secondary compositions, the proofs are similar. Assume that there exists an element 2a

is contained in p 0 7

position 7

of

an+10,

K v sn+18

of an extension

Let

F : K v Sn + x £ > Sn

Let

G : Sn+1^-> K v sn+1^

tion

F 0 G

Since

K

and

represent

P ° 7

Adem’s relation, Sq8S q S q 8 =

x

Sn ) of

an

Ug CSn+1°

2a

is

that a com­

and a coextension for

5 = vn+7- Let

sn+18

with the base point in common.

P

K

on

and represent

7 on

equals to

2 a,

Sn+l8. 7 +

then it follows that the composi­

By a similar way to Example 2, we con­

L = Sn Up C(K v Sn+lB) = Sn U en+8 U en + 1 2 U en + 1 9

struct a cell complex and a coextension

* n +lQ(Sn ) such 1o

be a mapping which represents the sum

is homotopic to zero.

L U en+20. Then

*(K->

g

K = Sn+TU en+11 = Sn+T

be the union of

(- 2 Ln+i8 ^*

p

of

Proposition 1 .7 ,

{an , vn+^, an+10-*'

where

a

g

jt(Sn+1^-> L)

of

(G).

Let

it is verified that Sq^Sq^Sq^ / 0

M = L in

M.

CSn+1^ = By use of

we have a relation Sq4Sql6 + Sq16Sq^ + (Sq18 + Sq1S q 1*)Sq2+ Sq^Sq^Sq1 .

Since there is no cell of dimension

(n+i ),(n+2),(n+4) or (n+1 6 ), in

right side of the above equation vanishes in Thus we have proved that any element of

M,

M, the

but this is a contradiction.

{an , vn+r^, 0n+io^

cannot be

divisible by 2 . Next we shall consider about the operation Lemma 8 .3 . Let n > 1 6 , en+8 U en+1^

then there

such that, for a generator

u

Sq1^.

exists a cell complex of

K = SnU

Hn (K,Zp)^q1^u 4 0

and

86

CHAPTER VIII.

that the attaching map of en+1^

en+^

Sq

e 1^.

By Cartan’s formula,

Ug =u 1^

for the classes

and the attaching map of

and

Uq

M # M

of two copies of

be a shrinking map which defines

complex of the form and

S1^ U e2^ U e U

S1^ U

mapped by o>*

onto

Sq1^ ^ o

in

M # M.

e^2

16

by

such that

The cohomology mod 2 of

Ug x Ug

and

S

and

in the product M

Then

S^ # M.

and let M # M

M x M-

M # M

that

an

represents a coextension of 2an+y* 8 16 Proof. Let M = S U e be given by attaching

Then

E^M

represents

e^2

are

Then it follows

ande 2 ^ represent

g

E Oq = a £

ig # ag

and

respectively.

By Proposition 3*1,

l8 ^ a8 =

g

E Oq = a^. Thus there

of the identityof degree 1.

exists anextension o1 6 U ,, e22k-» qi6 m f« : S S U 16

S

such that

e2

Identifying each point of

2k

e^

2k

2k

is mapped onto by mapping2 4 e2 with its image under f, we obtain

a cell complex K = S16 U e24 U e 32 . We shall prove that this complex n = 1 6 . There is

for

tification. have that

e2^

tion of M # M

f.

still holds.

is attached to

sider the class of the of the orientation Let

e^2 . Let

and

respectively.

S1^

thecondition ofthe lemma

p'

e^2

in the above iden­

S1^ U e2 ^ = E^M,

Since

by a representative of

attaching map of

of

p

satisfies

no changement for S1^ and

Sq1^ j- 0

Thus

K

°-]£'

e^2, which depends on

F : M# M -> K

then we Next con­

the choice

be the above identifica­

be mappings which shrink

Then there exists a mapping

F'

S 1^

of

K

and

such that the following

diagram is commutative. F

M # M

K

P' o f The mapping S2i+

6nto

F' S2^

maps

V s f ) U e 32_ ^ ___ > op

eJ

by degree 1.

S2ltu e 32

homeomorphically onto Let

^2

eJ

.

and maps

2k

S1

L = (S2l+ v S2l+) U e32. By shrinking

and S2^

SQUARING OPERATIONS

to a point, we get from 00

which get

eJ

L

the reduced join 16

is attached by

S1^ # M = S2i+

U e 32,

E

in which

e 32

shrinking maps preserve the orientations.

in

M #S 1^

in

S2\

of the attaching map of

e

represents

S2

on

and consider the homomorphism p

S2i+,

and

classes of F ‘,

we

:

By use of (1 .1 8 ), we have that

e 32

P0*(?) = 20 2 k

+ 7 * By changing the orientation of

we have proved that

such that the

the attaching maps see that the class choice of

K satisfies the lemma for

7

and only

p

e32,

(SZ k )

is a co­

e 32

in

K

if it is necessary,

n = 16.

n > 1 6 is proved by considering

The case that

the

Theorem 2 of [6 ].

extension of 2 a2 3 ‘ Then it follows that the attaching map of represents

by

(S16 U e 2 k ) -> n

*

:*3 1 (S1^U e 2 ^ , s1^) ^ ^3 1 (S2l+) by

p Q. pQ_^

S2^

p Q : S1^ U e 2 ^ -> S2l+ be the restriction of

e 32. Let

induced by

in 24 to a point we

2 0 2k by a suitable

orientation of S1^ U e2\

-30 UeJ ,

Now, we see that the attaching

M.Then, by the mapping ^2

24

= ^2

is attached to

Lrepresents the sum of the

andS1^ #

16

Remark that these discussions are allowed

under suitable choice of orientations in

ofe 32

M#S

CTg = a2 4 ‘shrinking

l t6 # CTg = a2 i^ (Proposition 3 .1 ).

map

87

En" 1 ^K-

q .e .d . Theorem 8 .4 . contains an element

a

o 7 V where

a = 2a1 +

n+k. 7 i € «n+1g(S ) and ii). element

a

Z ei

and

consists of a

single element.

By the definitions of

we have £ = E°°^ e E°°{v 5 , 8 i 8 , E a M C < E°°v 5 , E°°8 Lq, E°°a1 > 8 1, 2a >

by

C < v, 1 6 1 , a >

by

= < v

Lemma 5 . 1 4 (3-5)

•

Thus we have obtained (9*1)*

< v, 8 1 , 2 a >

< v , 16 1, a >

and

consist of a single element

(;.

By i) of (3-9), (9 .2 )


=


consists of a single element

(; .

By ii) of (3 .9 ), we have a relation -< mod.

v,

8 i,

2a

> + < 81,

2a,

+ 2 a ° G^ + 8 1 o G ]1 =

v o Gq

8Gn -Since

coincides with the odd component of
=

2
+
mod

89

8G ^

= ^G^.

8l> = 0 then

8G11

(;

90

CHAPTER IX.

Then it follows that < v,

Ql

,

C

2 a>

C

Since

2G11/8G11

By i) of (3-9),

< v,

< 1 6 l , o, v >,

where


+

2G1]•

< Q l , 2 o, v >

Ql

2 o , v,

(9-3)-




8G1]

for an odd integer

x.

< Ql,

v

>D

thesecondary compositions are cosets of the same

sub­

2

o

v >. By (3*5),

,

< Ql, 2 a, v > = < 1 6 i, o, v >.

8G ] 1 = 16G 11.Thus 1 6 1 > = < 1 6 l,

>+

then we have that the

contains x£

2a, Ql > = < Ql,

< v , o,

Ql,

2 a, v > +

8

is generated by the class of 2£,

secondary composition

group

l,


•

=


= x t; + 8G11

=


v, 2 a , 8 l >

=
,

1

are the same coset ofthe

,

i,

t}3 ,

T) °

£ '

Tl > ^

subgroup

n >


,

T)2 , £ >, 2

21 ,


, < 2 L ,

< 2 1, T], T] o £ >, ,

o,




L > ,

2

?


=

r\,v 3 > = 2 G 1 1 •

< 2i,

First remark that a 0 G, = a 0 (GUJ2) = 0

, by (7. 1 0 ) and

8 ° G2 = {e v ) = 0 ,

by (7- 1 8 ),

T1 • G10 = h 2 0 n) = [^} C 2G11; 2 ^ 2 n • g 9 = U 2 0 8 ) + {T) 0 V } + {t,2 0 n) = 0 + 0 + {4^] C 2Gn and

v

3

° G 2

= { v 3

° t]2 } =

)

,

0

Then the secondary compositions

20^.

in (9»^) are cosets of

Then, by use of (3*5),

and

T),

2

=




e

= < 2 l, ri3 ,

a

>,


,




=

2

= < t), t) o

< r),

e

,

2i >

By i) of (3*9),



=


2 t, T\2 ,

tj,

08 ,

and < V 3,

2l> 2l>

=




e>

,

T) >

,

.

LEMMAS FOR GENERATORS OF Since

0,

(G^;2) =

v2 ° < v,

then

2 l > C v2 o G^

t\)

91

*n+1 1(Sn ;2)

= v2 °(G^; 2

) = 0 .

By(3•5 ), < v3 , r\,

21 >3 v2 o < v,t), 2 l >

< v3,T], 2 l > = 2G-j

Thus

1• Finally, we

= 0.

have

< ^ 0 6 , r], 2 1 > =


+ 2G11

=


+ < v3,

=


2 1 >

n,

by (3*8)

T),

6.4 and Lemma 6 . 3

by Lemma

= < a, T)3, 2 l > • Consequently we see that (9*4) is proved. Lemma 9* 1* < o>

n3,2 i> = < 2 1 , t]3,

a > = (; +

t),2 l > = < 2 1 , r],

t] o e> = ^

=5 +

2Gn

< Tj,T] ° £, 2 i> = < 2 1 , T] o£, rj > = ^ + 2G^

,

, ,

< r\2 , e, 2 i > = < 2i, s, T)2 > = £ + 2Gn Proof. By (9*4), it is sufficient to prove that < r\ ° £, We shall show

tj ,

21 > ^ 0

thatthe assumption

2 G ]1 .

mod

e

0 mod

2G 1 1

leads

us

to a contradiction, then the lemma is proved. Let < Ti »

tains

nbe sufficiently large,

e, T), 2 t> E 0 0.

2G n ,

n(K1

a]

*n+i 1 (^1 )

2Ln+l0'

€

the attaching map of L 1 = K 1 U en + 1 2

of

p 1 . Since

of

a 1 . Let

a1 o

->

Sn ) of

¥here

tj

° en+1

Assume

has

H2 ^n+9^ ^ °'

t]n+^

then en + 1 2

be constructed by attaching

that con_

is the composition of

and a coextension

K i = Sn+9 U en+1 1

en+1 1 . Since

= 0,

n > 13*

then < T,n • en + 1 , nn + 9 > 2 in+10 >

By Proposition1 .7 , the trivial element0

an extension

Let

mod

for example

P1

€

as the class of

Sq2 / 0 in

K 1.

by a representative

then there exists an extension

a e it(L1 -» Sn )

f 1 : L 1 -» Sn be a representative of

a.

By Lemma 8 .1 , we see that

Sq3 = S q ^ q 2 4 0 in Similarly, from the relation struct a cell complex

< v3,

L1 • tj, 2 L > = 0 of (9*4) we con­

L 2 = Sn+ 9 U en + 1 1 U en + 1 2

such that

92

CHAPTER DC.

Sq3 = Sq 1 Sq2 i o

in

L,

and a mapping

such that tative

of

f2 * ^2 ~ ^ represents v3. Let

f-g | Sn+9

:Sn + 9 -4 Snbe a represen­

f ^ f2

nn « Then the mappings

and

f^

define

a mapping

f : L 1 v L 2 v Sn + 9 -> Sn , where

L 1 v L 2 v Sn+9

point in common. ponent of

is the union of t\

Since

JTn+ 9 ^sn^

L 1, Lg

°en+l'vn

and

^y Theorem 7 *2 ,

then

Next, we introduce a result of [23]* (n+9 )-skeleton is

Sn and

having the base 2 -primary com­

M'n gener>a;te

f* : W L i V L2 V Sn+9) ^ W is a homomorphism onto the 2 -primary component of

that its

Sn+9

and

3^ irn+9 ^ n )*

Let =0

*j_(K9 )

be a CW-complex such

for i > n+9*

Then

Propo­

sition 4.9 of [2 3 ] states that (9-5) • of

Hn+ 1 0 (K9, Z2 ) % Z2

Hn+ 1 0 (K^, Z2 ) such that Consider a space

in

+ Z2 + Z2 . There are generators

S

h^, i^

and

Sq2 h^ = Sq3 i^ = 0 .

of paths in

K n which start in 7

Kn 7

and end

Sn . By associating to each path the starting point, we have a fibering p f : S -> K 9

in the sense of Serre tract

and

p'

[13], such that

E

has

is equivalent to the injectionof

pect to the retraction.

eQ.

Sn

as its deformation re­ into

with

res­

Y = p ,-1 (e0 ) of this fibering consists

The fibre

of the paths starting at

Sn

It is verified easily from the homotopy exact

sequence of the fibering that the injection homomorphism i* :

i > n+8

is an isomorphism onto for is an

(Y) -> «i (S)

(n+8 )-connective fibre

jt^(Y) = 0

and

space over

Sn .

for

i
Sn and this is equivalent to the injection

i :Y C S

by the retraction.

Thus

p. : n±(Y) -» ni (Sn ) is an isomorphism onto for

i > n+8 .

By Proposition 5, Chapter III of [13b we have the following exact sequence of cohomology groups associated with the fibering

p1 : S

K^.

LEMMAS FOR GENERATORS OF

H1 (S, Z„) -i-> H ^ Y , Z0 ) i < 2n +

where

= o

for

then we have that

Since

i > n,

E

z

E ' IT Tli+1 (k 9, Zg)

for

93

* h1l(Sn »2)

H1+1 (S, Z2 )

is a suspension homomorphism. z

Since

is an isomorphism for

H (S, Zg )

n < i < 2n+8.

commutes with the squaring operations, then it follows from (9*5)

that (9*6).

Hn+9(Y, Z2 ) ^ Z2 + Z2 + Z2

isomorphic to

Z2 + Z2

=.n+9 S y

Y

HIi+^(Y, Zg „2/) contains a subgroup

which vanishes under the operation

Now consider the mapping complex

and

L

f.

Since the

,1c. 2 Sq3 = Sq'Sq^

(n+8)-skeleton of the cell

is trivial, there exists a mapping

P : L 1 v L2 v

such that p o F = f.

Consider the following commutative diagram. n+9

W Y)

n+9

o a ]^ = 0 . Then the secondary composition

vn

< 8 1 , v, a >

is defined and it is mapped onto

is a coset of

< 2 L, 4 v, a > = < 2 1 ,

Moreover this

8G11 + a 0

, a >.

= 8G]1 . By (3-5),

by

E°°.

< 8 1 , v, 0 >

Then it follows from Lemma 9•1

that

g < 8 it v, a > for an odd integer E(xvt

and

E ( E 2 a'"

The composition

x. 0

.

^

10^

a l4 )

Thus, 1

x^ 1 0

{

g

=

Vg

X 5 n

G

=

-

€

E3{v^,

8tg,

C

- [vq,

8 tn ,

8 t 1

1, v11,

0

Vq

{8tl1,

cti

^

v n '

-{vq, 8 1 ^, v^) o

q

v 8 0 nl5 (sn) 0 al5

o

vn )

by Lemma 0

= v8 '

"15

° al5

E : jr2 1 (S^)

^ (k'7) by Proposition 5 *8 . -» *2 2 (S8 ) is an

isomorphism into, by Lemma 4.5, then it follows that

Since

2 a ,M=

0 and since

x

1 -3.

is acoset of

E(xv^ o ^io) = E(E2 ct,m o ct^). Since

xv? .

1 .4

5*13

by Proposition

=0

Thus

» , UJ

by Proposition

Cv8, 8 l n , vn } 0 a15 Vq )

We have

5 , 0 = E2 o " ' . o , u .

is odd,then

we have

E2 By Lemma 5-2, and its proof that (7*10), we have that

C(V

2t10'

^

2 ^1 0 , £io^i

H{t^, 2 1 ^, e]+)1 = e^.

It is seen in Theorem 7*3

impliess a = e s ’ mod 0 By use of > 6 _ 1 _ r p 6 2 v^ 0 a 1 2 = ^ E e 1 = E e 1 is contained in

H(a) =

{tj^, 2 t10, e 1 0 )-j • Therefore the secondary composition contains contair

tj^

2^0

a12 + 2v^ ° a12 = 4v^ ° a]2 = tj| ° en

proved. 95

{tj^, 2 i10, "^10^

- 0.

Then (10.1 ) is

96

CHAPTER X.

By Proposition 1 .7 , (10.2)

0 = a o E p

for an extension ~Vy,

where

2i ^. Next

a € n(EK -> S9 ) of

and a coextension

K = S9UCS9is given by attaching the

base

P e ^(K)

S9

of

of

CS9 by

we have

(1 0 .3 )

v? o Ea = 0 . By Proposition 5*9,

2 n1 2 (ST ) =

2 l 11)1

t1 2 (S7 )=Eit, 1

isa subset of

1.9 that

jt12(S7 )

has no 2-primary component, then

(S^), (in fact

*12(S^)

*1 2 (S7 ) = 0 ).

Thus

o 2i12«Then it follows

(v?, n10’

from Proposition

° Ea = (E p)* 0 = 0. By virtue of the relations (1 0 .2 ) and (1 0 .3 ), we can define a

secondary composition (v?, Ea, E 2 P ) 1 C n2 1 (S7 ). Choose an element

k

^

from this secondary composition and denote

that = E11"^ Krj for n > 7 and k = E°° . — 2 Lemma 10.1 . 2 = v7 ° v1(5- mod mod. Vrj 0 £l0- rjg o Krj € {vg, 2Li2 ^ 7 " 7 Proof. By Proposition 1 .2 , 2k7 e (v7, Ea, E23), « 2 1 ^

C (v?, Ea, E2 O

. 2 1 ,9 )), •

By Proposition 1 .8 , ^ 0 2 ^18 e for the homomorphism nihilates 2 1 ^ °

i*

j^gtS9 ).

v9> 2^17^

induced by the injection

Q

i : S 7 C K,

It follows then from Corollary 3 . 7

which an­

and Lemma 6 . 3

that E(p o 2 il8) = Ei# V 1Q . ril8 = Ei, v3 0 . By Proposition 1 .2 , 2 Krj € {v7, Ea, E^(p o 2 1 ^ ) } ^

and

Vrj °

{vy, E 2i*Ea,

€ {v^, ii10, v11)1 0

C {v^, tj10,

= {v^, tj10, v^1)1 .

By (4.7), Proposition 5-9 and by Theorem 7*4, the secondary compo­ sition

{v^, t)10, v31)1

is a coset of

v7 ° Eir20 + 111^2 ° v?2 tv7 ° ^1 0 ^ _ p It follows that 2 /c7 = o mod. ° £1 0 *

2-PRIMARY COMPONENTS OP

*n + k (Sn )

FOR

k = 1U and 15

97

Next by Proposition 1.4, ° {v^, Ea, E 2 f3} = (fig, v^, Ea} o -E3P

° Let

7

be an element of

7 o -E3 p.

{t^, v^, Ea} C it(E3K -» S^ ) such that

The restriction of

which coincides with

v|

y

S12

on

by Lemma 5-12.

is an element of y

Thus

^

0 ^

=

{tj6, v^, t)1q},

is an extension of

.

It follows from Proposition 1 .7 that ng • K? = 7 • -E3P

€ fv|, 2 i ]2, v 12)3 .

q.e.d.

Next consider the following secondary composition

Choose an element

{e3’ 2tH' V 1 1 1 6 > 7^ from this secondary composition and denote

that 7n = En ~ 3 *8 ^

for

Lemma 1 0 .2 . ^(¥3 ) = 27n = 27 = 0

for

n > 3

and

0 ag 0

7 = E°°

. 0 rjg ° ^

mod.

and

n > 3•

Proof. By Proposition 2 .3 , Lemma 6 .1 , Proposition 2 . 6 and by (5.13), H(73 ) € H{b 3, 2L11,

}g

C (H(e^), 2t]i; vii^6 = and

2L11' v11^6*

H(H(73 )) e H{v2, 2 tn , v2 1 }1 = A - 1 (v2 o 2 1 10) ° v ^2 = - v 9 ° v ^2 = v 9 * It was seen in the proof of Theorem 7-7 that °ag

ators

v

Since

HE = 0 ,

° v1 ^

and

E(v^ °

n'g

0 i~ig) and that H(v^ o

H(73 ) = v5 0 a8 0 vi5

2 ^3 ^ ^83 * 2 ^1 1

Since

ag

o v ^ ) = v3 .

we have that raod*

v5 0 ^7 0 ^8 *

Next, by Proposition 1.4,(5 .9 ) and Corollary

and

has two gener­

0 = e

^1 1 ^ 1 0

o

= {e^ o 2a^} = {26^ « o ] 1} = 0, and

ii)• The Groups

° 2 L1 8 — £ 3 ° ^2 ^1 1 * ^ 1 1 *2 *"1 7 ^

e s3 0 t2 *--]^ vn ' 2l17^ * ]1 is a coset of 0 j^q o 2 1 n

O o {2tn , v^, 2t^}

273 = 0

then it follows that

27R = 27 = 0

for

n > 3-

rr^ +1 ^ .

We shall prove Theorem 1 0 .3 .

3*7,

jt2g = [r\2 ° v 1 ° r\^ 1^ } « Zg,

q.e.d.

CHAPTER X.

* 1 8 = (e4 ° v?2} ® ” 19

= Cv5

0?8 ] ©

*20

={ct"

0 a l 3 5®

°v7 ° V 1 5 J®

tv4 °

Cv5 0

~8 * V 16] ~ Z2® Z2

^6

°

v% ]

~

zh

®

Z2

"22 =

(o|) ©

(Ea1 oa15) © (Kg) ^ Z,6

"|3 =

(CT|) ©

t*9) = zi6 ® ZU’

^ n ’ +

{* n ] * Z1 6 © Z2

"26

= (al2}+

*28

=

*29

= (a15]©

*n+ lir

’

'

f«7) *> Zg © Z k ,

= {a ' o a^) ©

*2+1*’

Z2 © Z8® Z2 ’

U 12)

©

© Zg© Z^

n = 1 0,

for

,

11

13,

,

Ca(v2 5 )} “ Z 16I® Z2 © Z4 ' *■ Z8 © Z 2 >

(an }©

(k15 ]

* zk ©

Z2 ’

U n ] ” Z 2 © Z2

{*} = Z2 © Z2 , 2 a„ = a « a and a2 = a o a. n n n+7 --First = {t}2 o v * o r\^ ° ^

n > 16

fo£

,

( G llt;2) = {a2 } ©

where -----

zg, by Theorem7-7 and by (5 -2 ).

)«

By Lemma 5*7 and by Proposition 2 .5 , we have (10.4)

A(

o r]q o n ) = ti2 ov1 o r\£ o

(jl

the exactness of the sequence (U.4) that 5 morphically into the kernel of A : -» Theorem 7*6

and Theorem 7*7, the groups

elements respectively.

Thus the kernel

and H: 2

*5

irj^ ofA

E*2g = 0 .It follows ->

maps

from

iso-

which is onto by (7 *2 3 )* 2

and is

have

8

and

By

4

isomorphic to Z . By

Proposition 2 . 2 and Lemma 6 .1 , H(e3 ° v2}] ) = H(e3 ) ° and

^ 0

by Theorem 7 .6 . ” 17

By

(5 *6 )

*18 =

tvU °

and ©

(vU °

,

Consequently we have obtained that = ( e 3 0 v1 1 } - z 2 Theorem1 .b, ^7 ° V 15 ) ©

K

•

0 v?2 ) ^ Z8 © Z 2 © Z2

By the exactness of (4.4) and by (7*24),

The group Theorem 7 .k.

E : « 1 q/a«2 0 ~ ^ 9 • Q jt2Q « Zg + Z2 is generated by

^

and

_

o v^,

by

2 -PRIMARY COMPONENTS OP

*n + k (Sn )

FOR

k = U

and 15

We have a(79

o viy)=a(v9 ) o v 15 = A(£9 ) =A( 1 = +

)

by Proposition 2 . 5

o ^

( 2 v i+

o

^

-

E v 1

o

-

t by

•

v 22

2vk «

and *?9 =

by Proposition 2 . 5 and (7-13)

v22

eh o

by (5-8).

o

(c

p

0

21 J ,

'•^

Q

then uuoii

2 ois generated

Thus

(v 5 • £ 9 ) ©

(v 5 vQ • v l 6 ) = Z 2 © Z 2

and (1 0 .5 )

the kernel of

A : jt| 0 -> ir^

( H 9) = 0 n 1 • ,) •

is

Next we prove

Lemma

10. 4. H{cr,n, H{ cr",

ai 2^1= ^ 9 = 1 6 Ll 3 , °]k^l

is a coset I 1 3 0 E,t2 ? + A l

° al5

=tTli3 0 V1U] + (ri1 3 » ” tvi3 * +^ 1 3

by (4.7),

Theorem7-1 and

= ^ 1 3 -* i6 l i4' al4}1 '

of

Lemma 6.4.

0^ 1

By Lemma 6 . 5

elU} + £nf 3

• o15)

r

and Proposition 1 .2 ,

m--|3 e ^ 1 3 ' 2 L ] kf E a * ^9 +^v 1 3 ^ = (ti1 3, 2^^ 1 3 * 1 ^ 14* al4 ^1 Thus we conclude that

H{a', ] 6 l ^ >

a-j^^

is a coset containing

^1 3 * We have H{ a ", 1 6 l1 3,

ct1 3 } 1C CH(a"), 16 l 1 3, = ir}2u , 1 6 1 ]3 , a 1 ^ }1

by Proposition 2.3 by Lemma 5.14

and

100

CHAPTER X.

and

12 G

111:

C [r^,

i 6 l 13,

by Proposition 1 .2 .

a 13)]

Since (r)1 1, ] 6 l ] 3> 0 i3 ^1 is a coset of 11 12 T14 *2 0 ai4 = ^11 '1 1 °145 '1 3] + ^ 1 1 e13J + {v11 by (4.7), Theorem 7•i, Lemma 6 .3 , (5*9), (7• 1 0 ) and (7 -2 0 ), then H(or ", i6 l13, a 1 3 }1

>11

12

Similarly, H{af”,

16 i12,

a12)l C (H (a'"), l6i]2, a]2)} ^ 9,

bt>9 = r,| ° nn

and

a 12 ^1

€ {t||, i 6 ^ 2 ’ CT1 2 ! 1 ' where

is a

,6 ii2> ° 1 2 } 1

coset of E*

11 19

13

Thus H{a'\

I6 i12, ct12 }

1

q.e.d. Now consider homomorphisms A : * 21 1 1->5

1 A : * 22

and

6 *20

By use of Proposition 2 .5 , we have a(ti1

II

CO

( a.

>

a

n 12) = A H{oM, A(a,,

= =

V5

° v8

° v 16

V5

° 78 ° V16

V5

° v8 0

+

° e£

by (7.17)

'16

by (7.13)

1

v i6 ’

*7 2

-

AH( ct1)

•

e 12

=

A (^ 1 3

v5

+

•

1

U)

13

by Lemma 10.4,

0

V 16

= AH( ct ’}

II

>

on on •-

a (n

*

)1 =

0

-p-

=

5

161

: A(H( cr', l6tU > °lU} 1 +

by Lemma 6.3 by Lemma 5.14,

„3

It follows from Theorem 7 .2 , Theorem 7.3 and from the exactness of the sequence (4.4) that (1 0 .6 ) ~

Z2

the kernel of and

that

E

:

By Lemma 9.2,

A : it 6

jt2 q

7

*21

E(v^ ° ^ )

*?9

is

H,,

= E ( E a ,n 0 a ^ )

follows from the last assertion of (1 0 .6 ) that (,°'7)

v6 °

’ 2o" * 0 1 3 '

» n }2),

En^

is an isomorphism i n t o . = E(2a"

o a^).

It

2-PRIMARY COMPONENTS OF

*n+k;(Sn )

FOR

k = U

1 01

and 15

It follows from (7 .2 6 ) and from the exactness of the sequence (k.k) the following sequence is exact. 0 Etti9 We have

0 e 12}

* ^v 11^ + ^ 1 1

* n20

°*

H( a" 0 a, 3 ) = H( a") 0 a13 2 = 1,1 * a13

by Proposition by Lemma 5 . 1 U by Lemma 6. 4

" 111 * ~1 2 + ^11 ° £ 12

by Lemma 6. 3

= V11 + *11 ° £ 12 2 H ( ^ . v2u) = H(76) 0 v 14 3 = v 11 0 v 2 14 = V 11

by Proposition by Lemma 6.2 .

we have the result = (a” 0 a13) 0 {v6 • vfu) - Z k © Z2

4

.

It follows from ( 1 0 .6 ) and the exactness of the sequence (*+.*0, the sequence H 13 n _. 6 E l 0 20 -- * 21 -- * 21 is exact.

^ Z 2 + Z 2,

Since

Considerthe

elements

and

2 a 1 0 a1^ = E(a" °

orders

° cr^, then

^)

2Krj = E(7^ °v^) + 2xE( a" o

and where

o'

x = 0 or 1. 8

and

4

has at most ^ elements.

*2 1 ^ * 2 0

then

This shows that respectively.

a

)

by

Lemma 5.14

by

Lemma 10.1 and (10.7),

a' ° 0 ^ and

are elements of

Thus we have

* 2 i = £0 ’ 0 CTi1+5 © ^ 7 } % z 8 © z4 * By (5.15) and Proposition 5.15, Q * 2 2 = ^a3 0 ^ i (J) 1 o ©^

558 ^ 1 5 ©^ Z8 ^© ^4

*

It follows from (7.28) and the exactness of the sequence (k.k) that « 1^ 3 .

E :

By Proposition 2 . 5 and by (5 .1 6 ), a(ctiT)= a U 17) o a ]5 = + ( 2a q - E a ' )

= + (2Qq - Ea'

°

° ct15)

Thus we have "23 = f

© CKg) = Z, 6 © Z^

and ( 1° .8)

the kernel of

a : ir^ -» * 22

is

{8a1T} ^ Z2

In the exact sequence 25

J L *9 J L 23

10 H

24

19

^ 24

1 02

CHAPTER X.

of (4.4),

= o

by Proposition 5.9.

is generated by a(v^),

Thus

E

= fv^} « Z2

since

Proposition 2 . 5 and (7.2 2 ), 2

_

A(v,g)

is onto and its kernel

= A ( v 19)

» v 20

= v9

by Proposition 5 .1 1 . By 2

O v 1T

.

Then we have that ”2° ■ where

2 k 10 = 0

2 k io = 8ctio*

an isomorphism into. ( 1 0 .9 ).

E :

+

is

It follows from the exactness of (4.4) that

-> *2°

is onto.

It follows from Proposition 5 .8 , Proposition 5-9 and from the exactness of (4.4) that

E :

->

is an isomorphism onto and that the

sequence 11 E 12 H 23 A 11 *2 5 — " " 2 6 ^ "26-^ *Sk

0 is exact.

Thus *25 ' {0?1} + U 11J ~ Z 16 ® Z2 E n 25 = ^CT?2 ^ + ^*1 2 *

and

By (7.29), the kernel of which is _+ a( v 2^)

by Proposition

A

Z1 6 ® Z2

:

-> * 2 ^

2.5 and

is generated by

Proposition 2.7.

2v 2^,

Then the result

* 2 6 = la% ] + U 12) ® f ^ v25» = Z 1 6 ® Z2 ® Z k is obtained by use of the first relation of the following

( 1 0 .1 0 ).

a

=

(

A(

4a ( v 2 5 )

^ 7 ) ‘ 8o?3 A( ^29^ =

and

= 0

* ,

A( l3 1 S.3

^ ■ 2° h ■ for the case oc =

then we

have, by Lemma 5 .1 ^, A(ti35) = a(EU t,®, 0 E 1\

3) = a(EU H( = { ^K 1 *28 = (0 ^) © and By (U.3 ),

5 } - Zu © z2

>

"30 = {°?6 ) © U 16] =» z2 © Z 2 n *n+i 4 = {°n} © U n} :- z2 © z2

for

:

(GiU;2 ) = Co2} © U) - Z 2 © z ’2 * Consequently Theorem 10.3 is proved. We have from ( 1 0 .1 0 ) and from the exactness of the sequence (4.4) that tt-t13 - tn!5) - z2, = (2i13) = Z and that (1 0 .1 1 ) 28 H"3i n+1 are homomorphisms onto for n = 13 and n = 14. 7Tn+16 iii) . The .groups «^+15 for n < 126 7 Choose elements p1^ e: *20 , p ’" G 21* P e it22

: itn „ n+ 1 5

and

as follows. P P

IV M1

e e

p"

€

P'

€

{a,,f > 2 l 1 2 ’ 8CT12)1 {crn * 0 , 3 ).1 > U 13 '

>

{a ’ > Q L ^ k ’

*

( °9 ’

1

'

16 L16 ^ CT16 } 1

By Lemma 10.4, we have (10.12).

H(pIV) = n9 °

= K 9, H( p" ') = r,n

= n 15? , H(p") = n,3

—>

104

CHAPTER X.

o mod (v^) + {tj1^ o e11+} Theorem 1 0 .5 .

and H(p') = Bor^. 2 2 = {n2 °- e 3 0 vi^ ~ z2>

*n+ 1 5 = ^ n } ~ Z2

n = 3 }- Z2 © Z 2

and

n20

=tp^5 © S

*1,

- tp"'} ©fig} « zu© z a

it|2

= (o") © U ' » 7 U ) © [a' »ellt)

4

,

,

© Ci?} =»Z8 © Z 2 © Z 2 © Z 2

= ^°8 0 v 15^ ©^ (a8 0 ei5} © fEo"}© (Ea 'o

*23 © (Ig)

(Eg1 °

, >

= Zg © Z2 © Zg © Zg © Z2 © Zg ,

*2U

= (p’} © (cf9 o 7,g} © (ag « el6) © [ 1 ^

~ Z, g © Zg © Zg © Z2

*25

» tEp' l © ( * 1 0 0 ^T7) ® ( e i 0)

n”

- (E2 p') © ( e n ) « Z , g © Z 2

,

^ 7

= (E3 p ') © ( ¥ , 2) - Z , g © Z 2

.

“Zi6 © Z 2 © Z 2

,

First we have from (5.2) and Theorem 10.3 * 1 7 = * n2 By (7 -1 2 ) and Lemma 5.7,

Thus

E(n2 ° 2 E jt^ =0.

° vn^

that

0 e 3 ° v 1 1 ^88 Z 2

= E ^T12

*

° v ’ 0 v6 ° v 14^

It follows from the

,

= 0 *

exactness of the sequence

(4.4) that the following sequences are exact. ( 1 0 . 1 3 ).

- A* * 2 7

n

0

By (10.4), Cv^ °

o v15)

most two elements.

a (v 5

— > 0

__ ,

,

„3H 5 A,

* *1 8

* o.

° t]g o ug)

18

.2

*

16

Since

'

«^g = (v? ° ig ° ug) ©

Z2 © z2, by Theorem 7.7, then the image By Lemma 1 0 .2 , H(¥^) ^ 0.

Hn^Q has at

Thus we have

*?8 - (?3} “ Z2 • It follows from (5.6) and Theorem 7.6 that ~ Z2

•

Consider the exact sequence ^9 ___ , ^ __E 5 21 19 20 of (4.4). jr21 = 0 by Theorem 7.6. Then (10.5) Thus

and (1 0 .1 2 ), the kernel of 4 q

A

9 A 4 20 18 E is an isomorphism into. By IV 2 is generated by H( p ) = ^ ° ^ n .

has 4 elements and it is generated by

and

p^".

Then

2-PRIMARY COMPONENTS OF

*n + k (Sn )

FOR

k = 1U and 15

105

the result it\Q =

Cp17) + {e5) - z2 © z2

follows from the first relation of the following lemma. Lemma

1 0 .6 .

= 0 ,

and

r13 a" ° t *21 14 a ' ° *22

mod

’ £ E pW

2 p" 2 p"

= E p" 1

mod

2 p*

= E 2 p"

mod

' >

16

*

24

°9 °

By Proposition 1.4,

Proof. a OJ

= a ' 0 E{ 2 t1 1 , 8ai1 > 2 I18 ^ 21 1 3 ’ 8ctT2 ) 1 0 2 t 20 By Corollary 3.7, a"’ 0 E{2ln , 8 a, ,, 2 L1g} contains a” 1

0 8 a^

2

° I 1 9 = 8a'” 0 °12> ° 71, 9 = ° is a coset Of

•

0 "' 0 EC2tn , 8a, ,, 2 l1 q]

arM ’ then it follows that

0

E*19 °

2 l 20

=

2

an 1

0

E^ 9 = 0

>

2 p ^ = o.

Next we have 2p"’ €{cj", ^t13, ^®13)1 0 2 i21 C (a", E p 37^

eE { 0 M | ,

2t12,

U 13,8a13)1

by Proposition

1.2,

by Proposition

1.3,

by proposition

1.2.

8 a 12) 1

C {Ean ', 2 l 12, 8al2)1 = {2a", 2i

8 ^ ^

C ^a > 1+11 3 > 8ai3^ 1

The secondary composition CcrTT, ■ ,3,8a 13} , is a coset of i t r , 1 2 6 o " p 1 2 o 6 a o E ji20 + *u o 8(rlU = ^^+16

are isomorphisms into for 11

It is sufficient to prove that of (4.4).

*22

is generated by

A(tn ) = A(cn ) 0 Q e v ^ ° T t 20 = 0 Thus

a *22

= 0.

byProposition 2 .5 ,

*23is

t 1,

a *22

13

= A*23 =0,

n = 5

6.

by the exactness

since Theorem 7.4.

o t]8 °

and

Then

by Proposition 2 . 5 and (5.10) by Theorem 7 .6 .

generated by

tji3°

i-t^,

since

Theorem 7.3.

and by Lemma 5.14,

A (1 1 3 o n1l+) = A(n13) o m12

= aH(ct’) 0 iji12 = 0

.

1 06

CHAPTER X. 1^

Thus

= 0

and

Since

(10.14) is proved.

2tt23 =

^y Theorem 7 *1 , then

o' o E2 itJ| = 0 . It follows from (10.1U) that

4?

) = 2 a'

E(a" a" ° rr2131

13 E* 21

Then, by

Lemma 10 .6 , we have (10 .15).

Ep

IV

We have an exact sequence 6 H , E 0 “20 ' n2 211 * 7 ^11 1''11 w ^ 12J from the exactness of (4 .4) and from (10 .6 ) and (10 .14) ,5

0

Then it follows,

by use of (10.15) and (10.12), that n®, = tp"') © te6) - Zu © Z2 We have also an exact sequence

r13 22

where

_13 7 22 22 ° ei4^ % Z 2 © Z2 © Z 2

6 E I21‘

0

fv1 3) ® {m-i3) ® ^ 1 3

by Theorem 7.2.

We have H( a '

A) = H( a ') 14' = I 13 '14 :li+) = H(cr') >

H( cr'

H( pn) = n 13

and

Thus*22 Obviously

by Proposition 2.2,

14 3

= ^3

'14

mod {H( a 1 o 7 1k)} + (H(ar o elU)}

is generated by

2 (crT o v^)

E*^ ,

a' ° 7 ^,

by (10 .12).

a T ° e 1^

and

p".

= 2 (a 1 0 e^) = 0 . By Lemma 10.6 and Theorem 7 .1 ,

2 P" = Ep”

for some integers

by Lemma 5.14 and Lemma 6 .3 ,

'13

a and b.

'114. + b a ' 0 eU Applying the homomorphism

H to this equation,

we have that >]3 + bT1] 3 o E ] k = H( Ep” ’ + a a

It follows that

a = b = 0

(10.16).

ba 1

14

= 2H(p") e 2 rr13

0

£14>

.

(mod.2 ) and Ep”

= 2p

By the exactness of the above sequence, we have the result *22 = ^p' ^ ©

^CT’ 0 v 11+} © ^a ’ 0 e-]i4. ^ © f £:Y^ - ZQ © Z 2 © Z2 © Z? . Q *23 in Theorem

By (5 .15 ) and by Theorem 7 .1 , we have the result of 10.5. Next consider the homomorphism a

rr1? *25

23

By Proposition

2 .5 and by (5.16),

A( V 1 rj)

=

A(

I

)

A ( e 17) = A ( t 17)

(2 a p - Ea 1) » v1c = Ea' 15 e 1 5 = ± (2c7q - E a ’) o e = Ea' '15

15 '15

2 -PRIMARY COMPONENTS OF

Since

7 ]^ and

*n+k(Sn) FOR

generate

~ Z2 © Z2,

then It follows from the exactness of (^.M (10.17).

k = U

E : *2l+ ->

and 15

107

by Theorem 7.1,

that

is onto

and that E *23 = (E2 pM) © (cr^ o v l6) © {cr^ o

« ZQ © Z2 © Z 2 © Z 2

We have an exact sequence 0 from (10 .8 ).

^8 ai7 ^ -- * 0

> Ett2-3 — >

The element

p’

satisfies

H(p') = 8a

and

2 P ’ = E 2 p"

° 7 ,A + ( 0 e,,}. by (10 .11), Lemma 10.6 and Theorem 7.1. 9 lb 9 lb7 Then we conclude that

mod {

4k

= {p,) © to9 ° 7 165 © {a9 ° £16} © {I9} “ Z1 6 © Z2 © Z2 © Z2 • Remark■ It can be proved that 2p' = E2p" by use of the methods

in the next chapter. Lemma 10.7. ----7 n ° crn+g = 0

and

Proof.

£ n 0 a n+o q = 0 for n > 6 .

£^ o a 1 1

for an integer

tion,

We have for some integer

HH(?3) = H(v5 ° ag o

Q 3

Apply

an °

n+ ( = 0

_ = {£^} » Z2< 5

9

H 0 H : jt^q -» rcjg -> jt^g

for ---

Thus

n — > 11

£^ °

to this rela­

y

+ yE(v^ ° n7 o Mg))

= H(v^ o ffg) o v 1^ = v2

n — > 3*,

is an element of

_ = x£^

x.

for ---

+ 0

by Lemma 10 .2 , by Proposition 2.2

» v15 = V3

by (7 .9 )

H(H(e3)o o 1 1) = HH(£3) ° a11

by Proposition 2.2

and HH(e3 0 a^) = = Since = x HH(i”3)

H(v 2) 0a ]1 = HE(v 2) ° a 11 = 0 by Lemma 6 .1 . Q is an element of order 2 , then the relation

implies that

£n 0 an+g = En "3 (£3 0 a^)

x = 0 (mod.2) and =0

for

e3 ° an

=0.

HH(e3 0 cr^)

Obviously

n > 3 . By Proposition 3.1,

°n 0 en+7 - En'11(CT8 * e3) = en 0 °n+8 = 0 Wext a16 € tv8 , n,,, v12), 0 al6 = v8 ° E( t>10’ v 11< “H 1 C vg 0 = (vg 0 EA(v21)) = 0

.

by Lemma 6.2, by Proposition 1.4 by Theorem 7 . 6

108

CHAPTER X.

Since = 7q 0 a ] £ cr^) = 0

E 2 : * 21 -> * 23

=0

for

into, thenE 2 (7^

is an isomorphism 7^ ° crli+ = 0 . Thus

implies that

°a^) 0

7n o 6.

q.e.d.

As a corollary, we have (1 0 .1 8 ).

A(a19) = a9 o 7 1 6 + a9 . el6, ct1q

o 7 1? = *1Q oe1T

and

an 0 7n+T = ° ^ 11 * For, by use of Lemma 6.4, Proposition 2 . 5 and (7-1), A(a,9) = A(tlg) • a17 = (a9 « n16 + = ff9 0 ^16 0 a 1 7 + v 9 0 a 1 7 + e9

° a1 7

= a9 o (71 6 + elg) = crg » 7 1g + ag . e,6 . Thus

o1Q ov1? = o 1 0 o e 1 7

and then

+ EA(a19) = * 2 5

21 *26 = 0

but

*27

2

= (v21) » Z2

E(a10 o 7 1T) = an

has a non-trivial kernel.

by Proposi­

0 7 iQ = 0

by (10.18).

It follows from the exactness of the

above sequence that *26 “ (E2p,) © {?n } “ Z 1 6 © Z2 and that (10.20).

a

: *27 -2 5

is an isomorphism into and

a ( v 21) = o 1Q

o

=

a10 ° e17 * Since

1 23

-27

2^ = *28 = 0

Proposition 5.8 and Proposition 5.9, then 11

E : *25 "*

we have from the exactness of the sequence (4.4) that an isomorphism onto and that = {E3p M © {¥12} - z l6© z 2

.

Consequently all the assertions of Theorem 1 0 . 5 are proved.

12

*27

2-PRIMARY COMPONENTS OP

iv) . The groups

^

*n + k (Sn )

for

FOR

k = U

1 09

and 15

n > 1 3.

We shall continue the computation of the groups *^+1 5 * 12 1^ Lemma 1 0 .8 . i) . E : «27 -> it2g is an isomorphism into and H : k’I / Eitg| *= Zg. ii) . E11”13 : itgg ->

are isomorphisms for

n>

13

and

n JL 1 6 .

iii) . E : jt^ -»is an isomorphism into and

= E^q ©

(a ( l 33)} = «30 © Z . Proof.

i) is proved by the exactness of the 1o o sequence (4.4) , and by the results Hir^ = ir\2 ^] « Z2 of ( 1 0 .1 1 ) and 25 *29 = 0

The assertion of

ProPosition 5 .8 . Next we prove

(1 0 .2 1 ).

a(v27) = 0 , A(Tjg9) = 0

and

A(n31) = 0 •

For, by use of Proposition ,2 .5 , a(v27) =a( L^rj) 0v 25 =

by (7 .3 0 )

Ee o v2^

= E({cr^2, v 1 9 > ^2 2 ^ 0 v24^ = - E(a 1 2

o{v19, tj22, v23))

by Proposition 1.4

= - E(a 1 2

o 7 ^)

by Lemma 6 . 2

=0

by (1 0 .1 8 ),

A(n29) =a(ti29) on2Q =

and

a

= A k ° '"'as = 0 (ii31) =A(t31) o iig9 =

k a ^ k ° t! 28

by (1 0 .1 0 )

< 2o ®5 ° i}29

by (10.10)

’ = °1 5 0 2ti29 = ° where we have to remark that the composition

(a *1T J7 *3 2

o

,

0

,

> p_.3 1 *31 ji

^

.33 = Q it33 32

By Proposition 2.7, we may replace by

A tt33 . Then the

lemma is proved by these exact sequences and by (4.5). Next we prove

q.e.d.

110

CHAPTER X.

Lemma 1 0 .9 . There exists an elementof 2 p 1 ^ = E^p?

Proof. < a, 1 6 1 , cr > by

7

e < a, 2a,

and that

p' e {a^,

Since

1 6 1 ^,

follows that

Gq 0 a = 0

o a. (GQ;2 )

a = a

E°°p ’. Apply (3 .1 0 ) to

is generated

e °a = 7 0 a = 0 .

< a, 1 6 1 ,

andthat

E°°p ' e < a, 1 6 1 , a >

then

Gg °a = (Ggj2 )

is a coset of

such that

>•

e (Theorem 7.1).By Lemma 1 0 .7 ,

and

element

ir^g

a > consists

p = 161,

and

It

of a single

then

we have that

E°°p' € < a, 2 a, 1 6 L > . By (3.5), 2 < a, 2a,

Ql >

< a, 2a, 1 6 1 >

Since

= < a, 2a,

81 >

is a coset of

we have that there exist elements

°2i C < a, 2a,

161 >

.

Gg ° a + G 1 ^ ° 161 = 16G.,^, then

a e < a, 2a, 8 1 >

or € G 1 ^

and

such

that E°°p1 - 2a = 16a1 . Set

p = oc + 8 a',

1 0 .8 ,

E°° : * 2 8

p e < a, 2a, 8 1 >

then (G-]^2)

is an isomorphism into.

Then it follows that there exists an element 2 p1 ^ =

E^p*

and

E°°p1 3 =

p^

2 P^

By Lemma

Obviously, of

p e < a, 2a, 8 1 > .

Since the order of is 3 2 . It follows from

2 P = E°°p1 .

and

xr^g

p e (0^;

such that

q.e.d.

= E p' is 16,

then the order of

p^

i) of Lemma 1 0 . 8 that

*28 =

% Z 3 2 © Z2

We have also that ( 10.22) .

H( p13) = Uv25 = t,35 .

Denote that Pn = En_ 1 3p 1 3

for

n > 13

and

p = E°°p13 ,

then it follows from Lemma 1 0 .8 the following theorem. = (p16) © (F1g) © {a( l ^ ) } » Z32 © Z2 ©

Theorem 1 0 .1 0 . nn+i5 = and

© f£nJ

Z 3 2 © Z2

n > 13

Z ,

n ^ 16,

and

(G,5;2) = (p) © (?) = Z32 © Z2 . Next we shall prove the following relations.

(10.23) .

T)n ° «n+1 = Fn

By Lemma 10.1, _ the definition of e^, of (3.9),

for

n > 6

and

«n » 9.

i| . * e E”{vg, 2i12, vlg) C - < v 2, 2 l , v >. _ 00 2 2 s e E { 2t11, v 1l] C < e , 2 l, >. By

= - < v2, 2 l , e >.

v

The stable secondary

By i)

2 -PRIMARY COMPONENTS OF

compositions cosets of

0,

Lemma 10.7.

, < v

since

G^ ° v

= G^ o v = G^ ° e = 0

r\ 0

k

= e .

morphsim into, it follows from 0 3*1'

K7 = ®6’

*n ° ’W

Obviously, =

are

by Theorem 7.6

and

and by Lemma 6.4,

< y 2 , 2 l , v > + < v^, 2l, 6 > = < y 2 ,

Thus we conclude that

ill

2 < v , 2 1 , r\ o a >

and

, 2i, e >

k = I4 and 15

FOR

_

2

By (3.5), (3.8)

*n + k (Sn )

Since

E°°(r|^

2l, rj > o a € G8 . a = 0 .

E°°

0 Kj) = t) o

nn o «n+1 = en

• 6. for

n > 9 .

CHAPTER XI.

Relative J-homomorphisms. The homomorphism J :

SO(n))

of G. W. Whitehead wasdefined as follows (of. [26]). be a representative of

an element

a of

Let

jr^(SO(n)).

f : S1 ->

SO(n)

Define a mapping

F : S1 x S11'1 -* Sn'’ by the formula

P(x,y) = f(x)(y), x e S^, y e Sn_1. Let G(F) : S1 *

be the Hopf construction of with

Sn+1, G(F)

F.

Sn_1 -> Sn

By a suitable homeomorphismof

S1

* Sn_1

represents an element J(a)

«1+n(Sn) ■

e

The Hopf construction G(h) : A * B -> EC of a mapping

h :A x B C

a join of

and

A

A x I1 x B

B.

We consider that

G(h)

A * B

by identifying with the relations

(a, 1, b) = (a1, 1, b) each point of

is defined as follows, where

A * B

a, a ’ e A

for every

by a symbol

(a,t,b)

A * B

denotes

is obtained from the produd : (a, 0, b) = (a, o, b T) and and

b,b' e B. We represent

with the above relations.

Then

is defined by the formula G(h) (a, t, b) = dc(h(a, b) , t) ,

where

d^ :C x I1

EC

is

For two mappings

a shrinking map which defines f :A A ’

and

f *g : A *B isdefined by

the formula (f

* g)(a, t, b)

g : B-» B*,

EC. their join

A’ * Br = (f(a), t,

g(b)).

Consider a mapping p :A * B which is defined by the formula EA x B

EA # B

EA # B

p(a, t, b) =

(d^(a, t) , b) , where

is a shrinking map which defines 112

EA # B.

:

It is verified

113

RELATIVE J -HOMOMORPHISMS

easily that

p

shrinks the subset

A * bQ U aQ * B

homeomorphically elsewhere.

If

the subset

is a contractible subcomplex of

thus

p

A * b0 U a0 * B

A

and

is a homotopy equivalence.

B

to a point and maps

are finite cell complexes, then

EnSO(n)= ESO(n) #

the homotopy equivalence of the

case that A = SO(n)

A =

Sn+^

p :

and

and

B = Sn_1 ,

* Sn~1

Sn+i

homotopic to a homeomorphism

and

In particular, we denote by

pn : SO(n) * S11'1

case that

A * B,

the join

Sn~1 and B = Sn_1.

S^" * Sn ~1

is a mapping of degree

In the

is homeomorphic to _+ 1 . Thus

p

is

pQ .

In general, we have the following commutative diagram. A * B (11.1)

f* 8

A' * B ’

P

v EA # B

|p

v >. EA' # B'

Ef # S

Now let r*n : SO(n) x Sn' U Sn_1 be the action of

G(P)

of

J(a)

SO(n)

as the

rotations of Sn ~1 .

Then therepresentative

satisfies the formula G(F) = G(rn) - (f * in _i} ,

where and

in_1 Sn+i

is the identity of

such thatpQ

preserves

is the class of G(rn) ° (f of the diagram (11.1)

Sn_1. By taking orientations of the orientations, we have that

* in_-,) 0 P0 *

S1 * Sn~1 J(a)

It follows from the commutativity

that G(rn) 0(f * ^n-1^ ° ^0

is homotopic to the

composition GCrn) • qn • Ef # V ,

where

qn

is a homotopy-inverse of

= G(rn ) » qn . Enf ,

pn . We denote that

Gn = G(rn) » qn : EnS0 (n) -> Sn .

We remark that homotopy. (11.2)

GR

is independent of the choice of the inverse

Then the homomorphism

J

up to

is defined by the following formula.

J = Gn# » En : «i (S0 (n)) -» nn+1(EnS0 (n)) -4

Next consider the natural injection

i

* n + 1 (Sn ).

of

which is given by considering that each rotation of of

qn

S0(n-1) SO(n-i)

into

S0(n),

is a rotation

S0(n) leaving the last coordinate fixed. Lemma 1 1 .1 . The restriction

GR | EnS0(n-i)

is homotopic to

m

CHAPTER XI.

Proof. We use the notations d^(A x [o, i]). We identify

A

C+(A) = d^(A x [i, 1])

with

C+(A) n C_(A)

a -> d^(a, J) . Then the restriction of

rR

on

and

C_(A) =

by the correspondence

S0(n-1) x Sn ~1 satisfies

the conditions rn | SO(n-i) x Sn~2 = and

rn (S0(n-i) x C+(Sn'2)) C C+(Sn '2)

rn (S0(n-l) x C_(Sn~2)) C C_(Sn"2) . It follows from the definition of the Hopf construction that

G(rn)| SO(n-l) * Sn‘2= G(rn_,), and

G(rn )(S0(n-i) * C+(Sn ~2)) C EC+(Sn'2)

G(rn)(S0(n-i) * C_(Sn~2)) C EC_(Sn~2). We have also similar properties on

pn | SO(n-i) * Sn '2= pn_,, and

pn :

pn(S0(n-i) * C+(Sn"2)) C C+(En_1S0(n-i))

pn(S0(n-1) * C_(Sn'2)) C C_(En'1S0(n-i)) . We show that there are homotopy inverses

qn

Q.n_-| of Pn_-|,

and

respectively, such that qn | En_1S0(n-l) = qn_,, and

qn(C_En~1S0(n-i)) C SO(n-i) * C_(Sn'2). Let

eQ

qn (C+En_1S0(n-1)) C SO(n-l) * C+(Sn '2)

U be the closures of a regularneighbourhood of

U e0 * Sn~1 in S0(n) * Sn_1

U n SO(n - 1) * Sn' 2 , n _p

C_S

such that

suitable simplicial decomposition of Sn"2,

SO(n-i) * C+Sn-2

S0(n) * eQ U eQ * Sn_1 retract of

U,

then

and

U

U_

we have that the images

This is possible if we take a

S0(n) * Sn_1

SO(n-i) * C_Sn~2

are contractible to a point.

such that

SO(n-i) *

are subcomplexes.

Since

UQ ,

U+

Further, applying the identification

V = Pn (U), VQ = Pn (UQ), V + = Pn (U+)

and

are contractible to a point.

Themapping

pn

maps the outside of

outside of

V. Then we obtain a mapping

EnS0(n) - V

and by extending over

exists and it

topy inverse of

V

qn

into

U

homeomorphically ontothe

by setting U.

Since

is unique up to homotopy.

qn = p~1

U

on

is contractible,

Further,

qn

is a homo­

pR .

Now, we can choose the extension of C V

=

U_ = U n S0 ( n - 1 ) *

is contractible to a point. Similarly,

and

suchqR

and

UQ

is contractible to a point and it is a deformation

Pn ,

V_ = Pn (U_)

U and its intersections

U+ = U n SO(n-i) * C+Sn~2

are all contractible to a point.

S0(n) *

qn (V+) C U+

3X1(1 qn (V-^ C V ->

since

qR Uo>

over V

such thatQ.n (V0)

U+ 311(1

U-

are

RELATIVE J-HOMOMORPHI SMS

contractible.

Set

En-1SO(n-i) - V Q inverses

qn

q

= qn I En_1SO(n-i),

and thus

and

qn-1

qn_-j

and

qn-1,

then

qn-1 = p"^

is a homotopy inverse of

on

Pn_-j • Then the

satisfy the required condition.

Consider mappings qn

11 5

Gn

and

Gn_1 which are defined by use of these

then the following conditions are satisfied.

Gn I En'1SO(n-i) = G n_,,

Gn (C+En~1SO(n-l)) C E C +(Sn'2)

and

Gn(C_En_1S0(n-1)) C EC_(Sn '2). a : Sn -> Sn

Let

and

p : Sn -» Sn

be defined by the formulas

«(*n(t,,..., tn_s, tn_1; tn)) » ^(t,,

tn_2, tn , tn_,) a

p(d (x, t)) = d (x, 1-t). Then we have that each other,

p ° KG ^

homeomorphically onto

= - E^n _ithat EC+(Sn ~2)

a

and

maps

and

C+Sn_1

EC_(Sn-2)

p

are homotopic to

and

C_Sn_1

respectively.

It is

easily verified that a ° EGn-1 satisfies the same conditions as n_2 Since EC+(S ) is contractible to a point, the restrictions of a o EGn_-,

of

on

C+En_1 S0(n-1)

En_1SO(n-l).

is homotopic to

and

a ° EGn_1

C_ . Then

is homotopic to

- EGn-1. Thus we have obtained the lemma. Corollary 11.2.

G . GR

and

are homotopic to each other fixing the points

A similar statement is true for cr o EGn_1

and

Gn I EnSO(n-l) p ° EGn_1 =

q.e.d.

The diagram

jt^( SO(n-1))

~J.

nn+i-i(sn"1)

1*

I

v

E

^

S0( n))

J

> «n+i(Sn)

is commutative. This is a direct consequence of Corollary 11.3. -» Sn

n = 2, 3, ...

for

(1 1 .2)

and Lemma 11.1.

There exists a sequence of mappings such that

Fn : EnSO(n)

Fn+1 | En+1SO(n) = EFn

and

Fn* ° En = Proof. By use of homotopy extension theorem and Lemma 11.1, we have a sequence of mappings - EGn . Let formula

an

Gn

of Lemma 11.1

be a homeomorphism of

an((r, y t , ,

FR = GR 0 an


*n (Pn , Pn"1) I E11 v eV

which

*n (SO(n+l), SO(n) )

V r.n- 1 1"1) _____ zli*______ > 7r2n(EnSO( n+ 1) , EnSO( n))

V „ , _ *Pn(^(Sn+1), Sn) 2nv that En i

* 2 ^ ^ ^ ' EnPn_1)

It follows from the commutativity of the diagram

( % Fn + ^ *

is mapped onto a generator of

«2 (n(Sn+1), Sn). Thus we have

the following Corollary. Corollary 1 1 .5 .

(n0Fn+1)* :

-> n2n(n (sI1+’)» Sn)

is an isomorphism onto. Next, we use the following notation of the space of paths. n(X, For amapping

: I 1 -* X | *(0 )

A) = U

g

(t(^) = xQ) .

and

g :(CS1, S1, eQ) -> (X, A, xQ) , we define a mapping lQ ) ,

n'g : (S1 , e0) -» (Q(X, A), by the formula(nfg(x))(t) = g(d?(x, S1 x I1 -» CS1

A

t)) , (x,

aQ ( I1) = xQ,

t) g S1

is a mapping which defines the cone

Then it is verified that

x

I1, where

d1 :

CS1 .

' is independent of homotopy

and it

defines an isomorphism A' : *1+1(X, A) * Let

f : (X, A)

-» (Y, B)

*l+1(x, A)

A)) .

be a mapping, then the diagram

---- ------ >

A))

f* v *1+1(Y, B) is commutative, where formula

n> v ------2------- > ff.(n(Y, B))

fif : n(X, A)

(flf(4))(t) = f(J0(t))

for

fi(Y, B)

is a mapping defined by the

I € fi(X, A), t € I1.

Applying the commutativity for the mapping

%^n+i

: (E11?11, E^P11-1)

-> (n (Sn+1) , Sn) , we have that (1 1 .5 )

n(n0 Fn+i)*

: ff2 n - 1 (n(EnPn > e V 1-1))-* *2n_1(n(fi(Sn+1) , Sn))

Is an

isomorphism onto. Consider the union

X U CA

of

X

and

CA

in which

A

and the

RELATIVE J-HOMOMORPHISMS

base of

CA

are identified to each other.

i :X

ft(EX)

U CA -» ft(EX, EA)

i(d^(a, t))(s)

= d^(a, (1-t)s + t).

In particular, we have (1 1 .6 )

Thenthe canonical injection

is extended to an injection i :X

by the formula

119

an injection

i : En_1(pn+lc_1 U CP11'1) -> n(EnPn+k'1, EnPn~ 1) ^ Where we use the

Identification

En-^pn+k-i u C P n_1) = gn-Tpn+k-i y CE^'p11"1

By shrinking the subcomplex

Pn_1

of

of (i.16) .

pn+^ 1; we obtain a cell

complex pn+k-i = sn y en+i y In particular from

is an n-sphere.

pn+k_1 u CPn + 1

to a point, (“ 11 .7 ). ( 11

by shrinking

y en+k-iB

The complex CPn"1.

P^ +^ -1 is also obtained

Since

CPn_1

is contractible

then we have

The shrinking map of

pn+k 1

u CPn

1

onto

p^+k-1

is a homotopy

equivalence. Next, for a mapping associate the mapping

g : (ECSi_1, ES1”1, eQ) -4 (X, A, xQ) , we

ftQg : (CSi_1, Si_1, eQ) -> (ft(X), ft(A) , £Q) . Then,

by taking their homotopy classes, we have an isomorphism ftQ : *i+1(X, A) * jr± (ft(X) , ft(A)) .

The commutativity of the diagram *1+1(X, A)

I

%

.

it±(fi(X) , f!(A))

fl0

.

ir^(JJ(Y) , n(B))

f*

V

*1+1(Y, B) holds for a mapping usual

ftQ

if

f : (X, A)

-* (Y, B) . This

ftQ 0

coincides with the

A = xQ .

Now, we denote that c£+k

= n( nk (Sn+k), Sn) .

It follows from the homotopy exact sequence associated with the pair (11.8)

(ftk(Sn+i *i+k(Qn+k }

*i-i -i^n+k+h-1 v -k-i-pn+k+h-K n+k } ' n+k * 1 ^ ' By use of the composition of these isomorphisms, we have from the -1V^n+k+h-1 homotopy exact sequence of the pair ( E ^ ) that the , E11-i^n+k-i 1p n : following sequence is exact. ^(jjn-ipn+k-1) (ii.ii). . .

for ,k E-

^(jji-i^i+k+h-i)

1T,n+k-1 .(E1 p: i-r “ 1 ’*n i < 4n+k-3, where

,+k-i-nn+k+hn+k

I£ and A^ k ^ k are defined by the formulas I£ = p* ° j* and Ak = 3 • (Ek » p*)_1 Pn+k“i u cpn 1 t,e a homotopy inverse of (11.7) . Let q : P:,n+k-i n Then define a mapping ~n+k : E11-1Pn' q£+k n nn-1 ji-ir>n+k-l E11 (P; by the formula f!?+k = ^(^J? Fn v) o ° i 0 x 'n+k' j- o E q :E P;n CPn 1) -* n(EnPn+k 1, Enpn_1)

a(nk(sn+k), sn) = Q^+k .

121

RELATIVE J -HOMOMORPHISMS

Lemma 11.6.

The following diagram is commutative.

^ n - ’pn+k-1) ! ^ ^ n - i p n . k + h - i ^ ^ ^ k - ^ + k + h - ^

,fn+k+h>. 1 n '*

( ^fn+k) n

(En-ipn+k-i}

( mk) v n \

/^n+k+hx u n+k \ V

V *±
*i + 2 ^ 2n+1

is an isomorphism onto for

this result with that of

to

(fti*)"1(f^+ 1)^

Z and it is generated by the classof the

(ft

for

The group

iQ : S2n_1 -> ft2(S2n+1).

-io*^ L2n-1^

Since

is a representative of

,

Then the above diagram is commutative.

isomorphism onto to

^(ntts11),, Sn)) (n hn)* .'\ ( n 2(S2n+1))

(ft h^)*,

we have that

then it fol­

i < Un-3. f*

is an isomorphism

and it gives an isomorphism of the 2-primary component

Since

fti*

is an isomorphism,

then

(f^+1)*

Consequently the theorem is proved for the case that Applying

the five lemma to the diagram of Lemma 11 .6,

we have that theassertion of the theorem for tion for

Combining

(f^+k)*

is equivalent k = i. where

h = 1,

implies the asser­

(f^+k+1)* . Then the theorem is proved by induction on

k.

q.e.d. In the proof, we have a homomorphism H0 . fi'2 • (n

. (fli);1 : * ^ 0

which is an isomorphism for odd

n

or

for

- *i+2(38n+1)

i < 3n-2

isomorphism of the 2-primary components for even

n.

and it gives an Then it follows from

RELATIVE J -HOMOMORPHISMS

the definition of

H

and

A I

123

that the following diagram is commutative.

*i+2(Sn+1)

«i(Sn)

i+2

,„2n+1< >•

¥e have also In the above proof that (11.13).

H~

( jC -*- 1)^ In the following, we shall apply the theorem for some special cases

which will be needed in the next chapter. Let

x e Tr2n+2k-2^En"1pn+k"1^ be class 1-nn+k-1 e2n+2k-1 = En-1pn+k E11 p|n

attaching map of

the (2n+2k-1)-cell

"j (En_1P^+ ”1'“ *n l£”1) '— ,

-- * E 'M **i'“

(En”1P^+^”1) _

of (11.11),

i < 4n+k-3,

Lemma 11.8.

^ *nW 7

2n+2k-l

' i+k (S

. . .

we have the following lemma.

Let

i < 4n+k-3.

i). Ak(Ek+1 a) = X o a

for

a e *±_1(s2n+k~2) ,

ii). Assume that

IJ (a) Ek+1 a» 2n+k-2 a* c *._l(S; ) and

c ^(E^P^),

For the exact sequence

and p e

a'

3 = 0

(Si_1),

for then

of o Ep € — i*{X, Q-*, p} • Proof.

Consider the following diagram.

i^n+k-i *.(En-1p£+k, E21-1 P^‘

,n-1r,n+k-1 -> * i _ P £

A

^( C S where

f

2n+k-2v

),

is a characteristic mapping of the cell

Ir-

Ak - a . (E

o p )

1

e2n+k 1 , p ’ = p o f

. Except the triangle of the right side,

tivity holds for the other three triangles and the square. commutativity of the right triangle.

Thus

oc) , x o a = f (a) = a^.(E.k+1 „

and then i) is proved.

and

the commuta­

It follows the

124

CHAPTER XI.

Since

= Ek o

I

°

,

then

Ek : jti(S2n+k_1) -> *i+lc(s2ni'2k~1)

Since

Ek+1a' = 1^ (a) = Ek(p^ j^(a)). is an isomorphism onto for

i < 2(2n+k-i)-i, then it follows that p j (a) = Ea». Considering that En-ipn+k _ -gn-ipn+k-1 y Qg2n+2k-2, we have a coextension a e jt.CE11-1 n n x pn+k) Qf a ,_ By (1.18), p*(a) = Ea' for p* : «1 (En- 1 p£+k) -> n1 (S2n+2k-1) . Since

p* : nl(En_ 1 p£+k, En' 1 p£+k" V

isomorphism onto for

i < 4n+k-3,

then it follows that

p~1(Ea1) . From the exact sequence of the pair have that

a - a is contained in

jti(En"lP^+k"1) o Ep,

taining

{x, a r, p) then

1.8,

o Ep

is a coset

.

of a subgroup con­

a o Ep e - i*U, a', p} ,

and

ii) is

q.e.d. Proposition 1 1 .9 . Let

a € 1

we

and thus

proved.

for

= j*(a) =

(En“1P^+k, En-lpn+k 1 ^

a o E p e - i*U, a' , p) + i^*j_(E n " 1P^+k"1 ) Since thesecondary composition

is an

i* 7ti(En"1P^+k_1).By Proposition

a o Ep e - i*U, a', p]

we have that

**(S2n+2k>1)

2n+2k-1

i < 4n+k-3

and

k > 0.

Assume that

Ak(a) = i# (p) in *L_ 1 (En" 1 p£+k‘1) 2n - 1 and p e. Then there exists an element

7

of

such that A(E2 a) = Ek_1r Proof.

ni-i(^n+ 1 ) >

Set

«' = ( f ^ £ +\

and

H(r) = + E2p .

a € *1 +k( 0 £ £ +1)

and

(3' = (f£+\ a

*

then

H0 (a') = + E 2 a, HQ(p') = + E2p

and

Ak(a') = i #(p'),

by (1 1 •1 3) and Lemma 1 1 .6 . By the commutativity of (1 1 .1 2 ) and ( 1 1 .1 0 ), Ik(A(+ E 2 a)) = Ik(A(H0 a')) = Ik(P,#“ ') = Ak«' = i#p ' and Then

Ik.,(A(+ E 2 a)) = 1^1^ a(+ E 2 a)) = I,'(i#P') . 2 I ^ ^ a CE a)) = + Ij (i^pr) = o by the exactness of the sequence

*i_i(Q^+1) -*

1 (Q^+k) ->

ness of the sequence

of (1 1 *9).

It follows from the exact­

Tti+/ sn+1) -» *i+^( Sn+k)

there exists an element

y'

of

5rj_+‘ |(Sn+1)

E k ”1 (r 1) = A ( E 2 a)

of (1 1 .8 )

such that .

By the commutativity of the diagram (1 1 .io), we have that

that

RELATIVE J -HOMOMORPHISMS

125

Then it follows from the exactness of the sequence (11.9) that there exists 8

an element

of

+ p 1 = I 1(7 1) + a 1 (5 ) .

such that

commutativity of (11.10),

A., (5) = I 1Pk_ l^( 5 ).

we have that

1-1(7* + Pk- 1*^5^ '

k - 1 > 0 ,then

Ek’ 1pk- 1^( 8 ) = 0

of (11.8), and thus we have, by setting

Ek'’r = A(E2a) In the 7 = 7 '

We have Since

+ p* =

by the exactness

1 ,(7 ) = ± P' .

and

Then

we have, by setting

that 7 = A(E 2 a) and 1,(7) = +— i *P 1 = + . 1 — P' H(?) = ^ ( 1 ^ 7 )) = + H 0 (p’), hy the commutativity of (1 1 .1 2 ). 2 2 HQ(p') = + E p, then we have that H(7 ) = + E p. Finally we remark 7

that the element

can be chosen from

>

then the proof of the pro­

position is established.

q.e.d.

As is well known, in the real projective space cidence number of is zero. Thus, if 2n- 1 2n as S v S and 2n- 1 2n S e for x e£+ 1p£ +1

for

Thus

Pjc_ 1^( 5 )> that

7=7'+

k = 1 , i is the identity.

case

By the

(11. 1*0 .

P00 = U Pn ,

the in­

e 2m_1 with e2m is + 2 , and that of e2m with e 2m+1 n is even then En~ 1 P^ +1 has the same homotopy type if

n

is odd then it has the same homotopy type of

= + 2t2

of odd

n,

H(A(E 2a)) = +_ 2a

Applying Lemma 1 1 . 8 and Proposition 1 1 . 9

1#

we have for

a e

**n-2 and for odd

, i
then there

exists an element

A(ETa) = E 2p Proof, i).

and

2a = 0

Assume that p

of

for an element

a

such that

H(p) e Cn2n+1, 2 L2 n+2 > e 2 q;}2 -

By (1 1 .1 6 ), we may assume that

e2n v S2n+1U e2n+2.

By shrinking

S2 n_ 1

En_ 1 Pn+ 1 * En~ 1 pn+1

iiasa similap

structure as

wg see that the characteristic class

X

to a point,

for

En 1 P^+^ = En_ 1 P^ +3

K( 0 ) of ( 1 1 .1 6 )

e 2n+2

S2 n - 1

U

becomes . Then

is of the form

RELATIVE J-HOMOMORPHISMS - 2 L2n+1 + 1 ’ vhere By the assumption

7 G *2n+i (En”1pn+1^ is

2a = 0,

127

acoextension of

12n_r

we have that

X

o Ecu = En_1P^+1

° Ea)

for the injection

i ’ of

into

En_1P^+2 . By Proposition 1.8,

for the injection

7 o E(a) = i" (&) * i" : S2n~1 En_1P^+1 and for an element

5

of

i ^2 L2n-i 9 Tl2n-1 , a)‘ Thus \ o E(a) = i*(e>)

for the injection i of S2n_1 into 2n— 1 5in :ti_1 since (2l2n-i> T12n-v tains

2^i_1(S

^2n-l f

'

2n _ i

).

En_1P^+2 . Remark that we may choose a

Further, we have that

Applying

coset

a subgroup which con-

- (2L2n_1, 12n_-,, a} =

Lemma 11.8, we have that

A 0(E^a) = X o Ea = i (5) . 3 * Then it follows fromProposition 1 1 . 9 that there exists anelement |

p

of

such that and H(p) = +_ E 2 5 2 is contained in + E f2L2n_1J ^2n_i >

A(E^a) = E2p The element T12 n - 1 >

2

+_ E 5

C ^2 L2n + 1 > T12n+v e2q:}2*

e2n+2

of

lsProved-

En_1P^+3 is a coextension of

C En 1P^+2, where

e2n+1

E *-2L2n-i'

n = 3 (mod h ), the characteristic class of

ii). In the case that the cell

Thus

. 2

is attached to

S2n_1

2l2n by

of ii) is similar to that of i).

S2n_1 U e2n+1

^n-i*

Th©n the proof

q.e.d.

We know the following examples in the previous calculations. Examples for Proposition 1 1 .1 0 . i) .

(a, p) = (r]12,

nu )^ (n20, e 1),

0

(ei2> a '

o eu ) , etc. ii) . 0 v 15^

(a, p) = (l6,

(l22>

o

(tu , aQ o ^

+ 7 Q + eg),

(n2^,

> (a lk> ag 0 v 15 + ag 0 s 15). e t c -

Examples for

ii) of Proposition 11.11:

(a, p) = (n^, 2 o q ° v^),

(vn > a8 ° ei5} ’ Lemma 11.12. i). Assume that “ e ,ri+k+2(sn+k+1)

—

(i < 4n+k-3)

=

€ *i(En'1p£+k) . Then

for

128

CHAPTER XI.

H(o) = + E 2 ( I £ a ’) for

and

(f£+k+\(or'

• p') = Ik+,(“

Ek+2p ')

p' e ^ ( S 1) . ii) . Assume that

2n~ 1 )

p €

I> .(a) = (f?+k+1) i P

k+i and the injection homomorphism

7 of

(En_1P^+k). Then there exists an element a = Ekr

Proof,

and

,

a €

for

i+k+1 (Sn+k+1) 2n-i i «i_i(S' i-i Iti+-|(Sn+1) such that

E(y) = + E 2 p .

i). By the commutativity of (11.i2), (1 1 .1 0 ), Lemma 11.6

and by (1 1 .1 3), we have H(cr) = HqI, (a) , H 0 I ^ I k + ,(a) - H 0 I^(f£ +k+1 )*(«•) = + E 2 (I£.(a')) .

It is easily verified that the following diagram is commutative. j

*i+,(X)

(X, A) a
Ik+i > Ek+1p 1) = (n*

«,(n(X, A)) i

Then, by the definition of

J,

.k+1 ) ( a

= i * ( n k + 2 (a

Ek+1p ')

Ek+2p ')) = i# (nk+2a= p') p , _ (fn + k + i ^ (aI)

■ = (f?+k+V ( a '

ii).

By Lemma 11.6,

° P')

P'

•

Ik+1(a) . (f£+k+1)# (i# p) = i#((f£+1)*P)

.

By the commutativity of (1 1 .1 0 ) and by the exactness of (1 1 .9 ), = x; - °Then it follows from the exactness of the sequence ( 1 1 .8) that there exists an element

7 1

of

"1+1

(Sn+1)

such that

Ekr ' = a.

By the commutativity

of (1 1 .1 0 ), i*(f£+1>,p - w * )

- W ® V )

- i.i, r' •

Then it follows from the exactness of the sequence (1 1 .9 ) that there exists an element Denote that

5

of

*i+k+1 (Q£+k+1) such that

7 = 7 ' + p1 (5 ),

= 1^7') + A1(5) = (fj^+1)*p,

A^s) = (f£+1)*p - I / 7 1). I., (7 )= 1 ^ 7 ’)

+

(&)

by the commutativity of (1 1 .1 0 ).

By

the

then we have that

commutativity of (1 1 .1 2 ) and by (1 1 .1 3 ), TJ /fn+1 V

RELATIVE J-HOMOMORPHISMS By the exactness of (1 1 .8 ),

we have

E k (r) = Ek (?') + E k~1(E p 1# (6 )) = Ek(r ') = a .

q.e.d. Proposition 1 1 .1 3 . H(a) = E5p

and

an element

7

2p = 0

Let

n be odd and

a a *?*2

for

and

of

such that

2 a = Er

O and H(r) = E p o

Proof. By Theorem 1 1 .7 ,

i < 4n-2.

P e

Assume that

*i-32 * Then there exists

p p-n_1 mod 2E it^_1 .

(fn+2)* : *i-i(En~ 1 pn+1^

*1-1

is an isomorphism of the 2-primary components (an isomorphism for 2n+3). Then there exists an element

a 1 of

(En_1P^+1)

i+2 =

such that

( C V ' - V 0*) • By i) of Lemma 11 .12, then I^Qf1

E 5p = H(a) = + E2(I1,a ’) .

E 2 : ir^(S2n+1)-» Iti+2 (S2n+3) = + E3p.

+ 2t2n-i‘

Since

^

n

Since

is 5131 isomorphism onto, and hence

is odd, then

En"1p£+1= S2n"1 Ux e2n

of Lemma

i*(EP 0

e ^^n-l'

2 a' = i*(Ep o 5

for an element

Pn — 1

of

•

Ep> 2ti-25 * Thus

).

It follows then

Is(2«) - (f£+2)*(2“ ,} - ( C 2)*

such that

2a =

E7

+ 25)-

and

exists an element

of mod

q.e.d. (a, y) = (a', a"), (k? , 7 g « v2^) , (p", p"').

Examples:

Proposition 11.14. = E5p

and

3

Let n = 3 (mod 4)

07 = 0

E 25 = a o E 5r

5

of

and i < 4 n- 2 .

P e it2^ 2

for a €

Then there exists an element

]

7 e it^ " 2

and

Cf„+3)# :

Then there exists an element

a*

.

such that E 2r ) 2

and H(s) e (’l2n+1>

Proof. By Theorem 11 .7 ,

Assume that

.

(En_ 1 p£+2)

is an isomorphism of the 2-primary components (an isomorphism for 2n + 5 ).

7

H(7 ) = _+ E 2(Ep °n^ _ 2 + 25) = E^P 0

2E2ir|"~’.

H( a)

X =

ni_2 + 2 5 )

Applying ii) of Lemma 1 1 .1 2 , we have that there *1+1

for

11 *8,

2af = a r o 2Li_1 € + i*C2L2n_i, Ep, By Corollary 3.7,

i < 4n-2,

of

iti_1(En~1P^+2)

i + 3=

such that

( f ^ V 1 - I3(a) • By i) of Lemma 1 1 .1 2 , E 5p = H(a) = + E2I^(a’). < 2(2n+3)-i,

then

E2 : «j_+1 (S2n+3) -» ITi+ 3 (S2n+^)

Since

i < 4n-2

is an isomorphism onto.

130

CHAPTER XI.

E 3p = +_ I^(a’) . By (1 1 .1 6 ) , the characteristic class of

Thus

En" 1 P^ +2 - En_ 1 P^ + 1 En_ 1 P^ +1

x = ±1 ^2n-l

is

for

inJection

e 2n+1 = S2 n_1

i*

into

. By ii) of Lemma 1 1 .8 , we have a' ° Er e - i* (1* ^2n_-,, 3, r) C V ^ n - l '

for the injections

i" : En"1P^+1 C En-1P^+2

^

i = i" 0 i'.

and

By i) of

Lemma 11.12, I3(« • E 5 r) - (f£+3),(«' ° Er) e (fS+3 )»i,(nan-1, P, r). Then it follows from ii) of Lemma 1 1 . 1 2 that there exists an element jri+l(Sn+1)

5

of

such that a o E5r = E 25

and

H(5 ) e + E2^ ^ ^ , 5

that we can choose Proposition H(a) = E^p of

for

^

C ^2n+1> in

ir^![

1 1 .1 5 .

e2^ 2 • The proof of the fact

will be left to the reader.

Let

a e *i+3

n = 2 (mod 4) and P e *1-2*

q.e.d.

i < 4n-2.

Assume that 7

Then there exists an element

such that 2 a = E2?

H(r) € fn2 n+i'

and

2 ti ] 2

'

Proof. By a similar discussion to the previous two Propositions, we have an element'

e

1(E11-1P^+2)

(fn+3)*“ ' = I3^“^

such that

^

E3p = i

By (1 1 .1 6 ), the characteristic class of x = i 1 * ,,2n-i + i2 *^2 t2 n^

S2n

fo:r’ the injec^ 1 01 1 3

i,

of

S2 n _ 1

and

i2

of

En- 1 P^+1. By i) of Lemma 11.3 and by the exactness of (1 1 .1 1 ),

into

x . p = a 2(E3p) = a2(+ IpO') = Since

I2 e 2 n+1 = En~ 1 P^+2-En_ 1 P^+l is

i - k < ta- 6 < 2 (2 n-2 )-1 ,

morphism onto.

Let

p’ = E

_o

p.

0 = X O p = (iu

0

E2 : it±_1+(S2n_2) -> «i_2 (S2n)

factors of

i^

is an iso-

By (1 .7) , n2n_, + i2 # (2 t2n)) o E 2 p '

= i^Clgn., * E 2 P') ± i 2 #(2E2 p') By Theorem 4.8 of [2 6 ],

.

±2^

and

.

are isomorphism into disjoint direct

Jt1 .2 (S2 n ' 1 v s2n) . it follows that 'I2n - 1

° E2p' = 0

and

2EV

= B 0 2 ij__2 = °-

By Proposition 3.1, p ° ti1_2 = e2p» ° i\±_ 2 =

n2 # P T = n2no e 3p' = o.

Apply ii) of Lemma 11.8 tothe relation

Ij, or' = + E3p.

Then

131

RELATIVE J-HOMOMORPHISMS

2a' e - i ^ i 1# n2n_1 - 12* 2 i 2n> ^ > 2Li-2^

c i; iuflan.,, P, 24.2) ± 1; W for the injection

i f of

En_ 1P^ +1

into

2W

E 'p '' 2 ti-2] >

En 1P^ +2 •

Since

3 0 r\^_2 =

it follows from Corollary 3.7 that ( 2l2 n ,

eV,

Since

. 2tgn o rt._l(S2n) +

2 C . _ 2)

i -1 < 4n -1 ,

(S2n) = ^ ( S 211) . 2L1_1. 2t2n.

2n

then Since

)

(S2n) • 8^.,.

is stable and

i; \ = 0,

then

2L2n 0

*1-1

i^ i,# t,2n_1 = ± i; i2#

Therefore we have that *L 2 * ^ 2 l 2 n ’ 2i±-2^ = -*•* 12*^2L2n 0 *i-i^S

=

ii.^an-i° rti-i(s2n))

The secondary composition

£n2n_-,,

^

•

2Li-2^

is a coset of a subgroup con-

Or~]

taining

n2n_i 0 *1-1(s

) •

Thus

2 a ' e i * U 2n_i>

for the injection Now, by existence of an

i

of

S2n_1

into

2li-2^ En 1P^'1'2 •

a similar way to the proof of Proposition 11 .14 , wehave the element

y

of

2a = E 2 j and

such that

H(r) € E2{n2n_■,, P, 2i._2} c (T)2n+ 1 ’ ®2p’ 2ti}2q.e.d. Proposition 1 1 .16 . Let H(a)

=E^p

of

]

a

for

e *^*2 and

p e *1-1'

i < 4n- 2 .

Thenthere exitst

Assume that an element

Proof.

°mod 2 * i + V

H(r) = ^n+i

‘ +_ H( 2 a) .

By Propositions 2.5 and 2.7, H(a(E^P)) = _+ 2E^p =

follows from the exactness of(4 .4) that there exists an element

*i+1

7

such that a(E5p) _+ 2 a = E7 and

It

n = 2 (mod4 ) and

7

of

such that E7 = a(E5p) _+ 2a . By Theorem 1 1 .7 , there are elements

it?^1

(f^+2)*a' = l^a

such that (f£+2)* V

and

a' e it^_1(En 1P^+1)

(f^+1)*7* = 1^7-

=

Lemma 11 *6

= I2E7 = I2(A(E5p)+ 2of) = I2AH0(f^+1)^E3p + 2(f£+\

We have

by (11.10) a'

by (11.13)

= x 2 p u ( C \ E3p± 2(f£+2)* a ’

(11-12)

and

7 ’ e

132

CHAPTER XI.

= A2(f£+1)#E 3f3 ± 2(f£+\

by (11.10)

= (f^+2 )_^(a2 (E3 3 ) ± 2a')

by Lemma

11.6.

Then it follows from Theorem 11.7 that i 7 1 = A 2 (E3 p) + 2 a' . By (1 1 .16 ),

En- 1P^ +2 = S 2n' 1 v S2n Ux e 2n+1

for the injections

i

S 2n_1

of

and

i'

x = i# 1, 2n_ 1 + i; (2 i2n)

for

of

S2n

S 2n_1 v S2n .

into

By i) of Lemma 11.8, A 2 (E30 ) = X O P = (i# T)2 n _ 1 + i^

= M^an-i where we remark that decomposition i#

and i^

p

• e) i ^ (2p)

are isomorphisms into. + i^a^

It is obvious from the

for aj

e *2^ 1 Ij

(S2n+3)

2 a ’ = i*(2 a1 *) _+ i^(2 p).

a £ € jt2^

and

that

= + E 2 (I1,Qf') = + E 3 a^.Since

E 3p = H(Qf)

Ijar =Ea£. i-i
1 2 . Assume that

a c *9 ^+ 1 0 'then there exists

anelement

*2n+ 7 such that i A ( v 2n+9) = 2a - E 3p

and

H(p) = v2n+, .

H(a) = p

of

133

RELATIVE J-HOMOMORPHISMS

Proof. By Propositions 2 . 5 and 2 .7 , H (A (v 2 n+9 ^

= —

*

It follows from the exactness of the sequence (4.4) that there exists an element

= 0

p'

of

*2n+9

2a ' Ef3' = - A ^v2n+9^ '

suoh that

by Propositions 5 . 8

and

(it.it)

that E : * 2 ^ 7

*2n+8

onto.

Thusthere exists an element

5.9-

811(1 E : n2nt8

p

of

H(p) = 0 ,

Assume that

then

*211+9

*2 n +7

i Now we shall show that the assumption diction.

n2nt9 = n2nt8

Then it follows from the exactness of are homoMorP11131118

suchthat

A(v2n+9) = 2“ ' E3pH(p) = 0 implies a contra­

p = Ep"

pM e *2 n+ 6 ‘

for som'e

By the exactness of (1 1 .8 ), 1^(80 - A( + v2n+9)) = 1 ^ 0

= I^V'

- 0.

we have also, I 4 A ^v2 n+ 9 ^ = ^4

v2 n+7 ^ = — ^4 A * V fn+4^*v2 n +7

” i XU Pi,< 0 ■ i M

O

onto, byTheorem of

by (11.12)

v 2n +7

by (11.10)

Ai*(v2n+7’ 1

(fn+^ * Ait(v2n+7') = -

Thus

, v 2n +7

.

= ±

^ (1 1 -1 3)

2Ilt(a) •

by Lemma 11-6

Sincethis

ls an isomorphism

11.7, then it follows thatthere exists

n2n+5(En" 1pn+3)

an

element

a*

3U0h that A4 < v 2 n + 7 }

= 2a'

By i) of Lemma 1 1 .8 , 2 a f - A ^ ( v 2 n + 7 ) - X 0 v 2n+2

for the class

x

of the attaching map of

e 2 n+3 = En” 1 P^+i

Sq^e11 = en+^

Sq^Sq^ = Sq^ Sq2 + Sq^Sq1 ■S ^ S n - l

but this contradicts generates

H^M, Z2) = 0

for

of Adem.

+

=

Therefore H(p)

then we have that

and

Sq*^11 =

Then we have 0 ’

0. Since

V2n+1

H(p) = v2n+1 • q.e.d.

Remark that

Lemma 1 1 .1 7 still holds for

n = 4 (cf. Theorem 7.3).

CHAPTER XII.

2-Primary Components of i).

*n+k(Sn)

for

16 < k < T9-

Some new elements.

First we have Lemma 1 2 .i. =

mod 2

and

Proof. E(crn o e * 22

There exists an element E£ ' = a' o

® e

Consider the composition

* 22 7 2 *2 3 *

' of

mod

such that

a" 0 e 1 3 e * 21 .

) _ Ecr" 0 e 1 ^ = 2 a ’ 0 e 1 ^ = cr1 ° 2 e 1 i+ = °*

H U ’)

Lemma 5.H, E : jt21

Since

isan isomorphism into (Theorem 1 0 .5 ), it follows that

-» = 0.

a" ®

Then the following secondary composition is defined: {a\ e13, 2 i2 l } 1 Since

2 *2 2 (S^)

€ it2 2 (S6 )/(a" o

+ 2 *2 2 (S6)).

contains the odd component of

choose an element

«2 2 (S^) ,

* 22 n {aM, el3, 2 L 2 i^i'

t;' of

^

then we

may

Proposition 2 . 3 and

Lemma 5.4, HU')

€ H(a",

e13, 2 l21) 1 C(Ha”,

e 2 L2 2 ^1

The secondary composition

a cose' *:2cJ*0 lT23 E

+ «23(S7) . 2l23 = o' O E2«23 + 2n23(S7) = 2n23(S7). Thus

E?' =

belongs to Let

a' ° • e15 mod 7 *23,then E£ 1 = a ’ 1

E? 1and o' 7 mod 2 * ^ 3 .

2*23(S7) .

Since

0 r\^ 0

°

°

q.e.d.

be an element of the secondary composition (^3, 2^i2 , 8ai2^1

and denote that iln = E11” 3

for

Lemma 1 2 .2 . H(IT^) =

n > 3

and

JT = E°°il3 .

2~n = 2TT = o

and

for

n > 3.

H[(i^,2 1 1 ^, 8 a12)l

Proof. H(JT3) €

CHm-3 ,2 t12, 8 a 1 2 ) 1

C

2.3

by Proposition

= {a"’,2 1 12, 8 a 1 2 ) 1

by Lemma

6.5.

(cr"1, 2 l12, 8 awhich is a

p^" is, by its definition, contained in coset of a" 1 °

0 8 a1 3

E(a"’ 0 v"12) =

= {a"' o v

U * 1U51

8tl6’

2 0 1 6}1

as follows.

Denote that for

Tn - E11 Lemma 12.4. and

2£

H(ki') =

n > 5

mod E3*2^,

H(7»)

2\x' = ^

H(?5) = 8p',

o ^

we have

e H{m- f,

hL,k ,

C {H|ir,

W u }1 ^-cr1

n

by Proposition 2.3

= Cn5, and

7^

by

=E 27 3e E 2{^3 ,

C {m

which is generated by (Theorem 7 . 1

^

-

(7.7)

2 i 12, 8 ct12} 1

^

l4’tyii4. ) 1

by Propositions 1.3 and 1.2. ^} 1

isa

^5 ’ E4 ? + “?5 ° 1+0 1 5’ o 7 ^, 0 ^ °

°

Thesecondary composition

^li4 >

Cm-^,

and Theorem 7.3).

0

Obviously,

^ 6 ° a 15 = 0 *By (7 .1 0 ) and Lemma 10.7 , = \

"f

= E 2n 1 . Proof. First,

4ct^

and

v^ ° aQ

o4a^

coset of

and

v5 Ut) ° 4a '15 " =4(v^ o aQ) o 5 °

ay

° z i ° °i5 = °-

Next we have H(h3 o 7 12) = H(h3 o e i2) = 0 .

(12.1) .

By Proposition 2 . 2 and Lemma 6.5, 0 7 12.

By Lemma 5.14,

E(a"' 0

**(^3

0 ~ 2 ^= H(m-^)

is an isomorphism into (Theorem1 0 .5 ), then

a"' 0 *"12 = °*

Similarly

H(n3 0 e 12) =

itfollows that

0 .

By the exactness of the sequence (4.4), we have that 6 12

are in

ct" 1

2) =■• 2a" 0 ~ 1 3 = a” 0 271 3 = 0 . Since

E : jr2Q -» jr21

2

0 v"-,2 =

Eit1^. Then we have obtained that

n3 0 7 1 2

and

138

CHAPTER XII.

and thus the relation

_

op

H((i’) = ^5 m0(i E * 1 9

Proved.

Next, we have 2 (i’ = 2 t3 ° n 1 e 2 lj • tn ',

and

by Lemma 4 . 5

C (2 m 1, U l lU» U o 1 * } 1 2 € n3 • E 2 ( h 3 , 2 t , 2 , 8 0 , 2 ),

2 i3 • ^

2

C n3 • (^5 , 2 L -\k’ 8 ° U , 1

by Proposition 1.3

c (if ° ^ 5 . = (2u '

by Proposition 1 . 2

!tC,llt , 1

by ( 7 . 7) . of

(2 m>, 1 h iU,

The secondary composition

is a coset

Sp. ' • En2? + n?5 * ^15 • *5

As is seen in the above discussion, we have = 0(Theorem 7.1), 2^' = Ti2 0

2 (i' o

0 ^-,5

= 0*

13 2jr2i

Since

= (i’ o E 2 *2;j = 0 . Therefore we conclude that

.

Similarly, we can prove that H(T5), 8p' e {8o9, 8l1g, 2 6 and mod

Proof. H(o?6) e H(v6,

2 a^

containing

have that

a ” 1 (v^

0 (eg

by Proposition

+ vg))is a coset of

an - Then the lemma is proved

ii).

The groups

By use of

^n+iJ Sn) for

2.4. 2jt]q = ^2 cTii^

q.e.d. 16

the elements introduced in

are stated as follows.

. 2

1

= A~ 1 (v^ 0 (eg + vg)) 0 alQ By (7.17), we

= E°°ffg

2

.

+ v^, cr1

~a

< k < 19

and

2 < n

< 9•

the previous i), our results

II

ro —d VJl

}

^5

o

l D

@

)

o

{ 8 Tl

~

8 Z

©

‘ 5 Z

II

ro on -P"

{ e5u o ( e a )v )

9 5} ©

o

o 9 U) © [L i o

:z © 8z © 5z ~

CS1

©

N ro

©

©

©

VJl

LT'C

II

ro vji OO

(SI ro

©

(SI ro

©

(SI -p-

©

N ro

©

ro

(S 3

©

N ro

©

N 00

n

©

3

©

w

©

i

©

©

-p-

j

ro 00

8

tSI

(SI

©

(SI ro

©

“

■— •

©

mi on

© ©

ro -p-

8

©

(S 3 ro

©

(SI -p"

©

©

CD B

4

O

i?

»-3

-3 ON

(SI

tSI

©

ro

*

0

©

vo

ts i

o

(SI

©

(SI ro

1= l

© ©

N

© j'

ro

®s 1= 00

0

—3

©

-VI

1= 1

O

-d




5

and

en •

n > 3. 2

-

e^,

e 3 ° t^q e [e^, 2 L ^ >

&3 o t^q € e 3 o E{2 l1q, v 2 q

By Proposition 1.4,

for

2

e 6 3 o E{2 t10, v1Q, rit^} . The composition

, hi6K

vn^i 0 ^ 1 8

By (7 .6 ),

e3 »

b 3 » E{2t1Q, v1Q, tii6}

is a

2

coset of the subgroup e 3 0 Eit]° 0 ^ 3 0 TJ|8 2 3

Thus Since

E

:

t]3 0 e ^.

By

= {s3 o

= e 3 ° e ii* 5

-»

o n1 } = 0

Lemma 6 .4 and

= e 3 ° e 11 + e3 0 al! o ^18 = e3 o e11* ov 1 1 = e 3 e 1 1 = ° r^g = ^ 2

(v 5

that v3 = _+

2

2

{v^, 2 l \\> v 1 1

the secondary composition 1

° v2 2

1•

for

©

(v5

. n8 °

Mg} *

z2 ©

is contained

Z2 ,

H(v5

o Og

o ( j . = 0 . Thus the secondary composition

contains

v 5 0 ag 0 V 1 5

^9

HCv2, 2 tn , v^ l ) 1 =

in

H(v^ 0

or

v5 ° °8 ° V U

see that . v 15)

(5.9),then we

e Cv5’ 2t11'vn ]1 ° v 18 2 "3 C {v 5 ,2i1 1 , v ^1 ) 1 2

= {v5, 2 tn , Tin

ir^g =

= Vg

and

Cv2, 2 1 ^ , v^1)l

V5 ° CT8 ° V 1 5 + V5 0 T18 0 ^9 *

°V 1 8 = v 5 0 ^8 °^ 1 7 0 V 1 8 = 0

and thus

n ^ 3* Next conslder>

2

» v 15)

^13 =

have> by Proposition 2 . 6

(vj+o 2 l 1 0 ) o v 1 2 . In Theorem 7*7 and its proof, we » o8

e3 o

Lemma 1 0 .7 ,

en 0 v "n+ 8 = en 0 en + 8 = En 0 ^n+ 1 5 = nn 0 ®n+ 1

_

-

= e3 ° ^e 11 + a11 ° ^18^

Consequently we have

and by (5.13),

° n20 = £ 3 # n2 = ^5 0 ^

Proposition 3.1,

is an isomorphism, then it follows that

e3 ° V11

A

by Lemma 1 0 . 7

Since

o ijg o have that

by Proposition 1 . 2

_ o v1 2 ) 1

by

Lemma 6.3.

By (7 .6) and Proposition 1 .2 , e5 ° v 13 e 2

C

v5* 2I11' n11* 1 ° v 13 Cv5 > 2t11’ n11 ° *'12]1 •

_

The secondarycomposition {v^, n-j-j 0 V-|2^1 2 1 0 5 — 2 1 v5 ° Elt20 + "12 0 I1 2 0 v13’ by E(v5 0 E*2(P

is a coset of the subgroup 0

2

= v 6 0 n12 ° M13 = °-

144

CHAPTER XII.

by Theorem 7.3 and (5-9). E(*^2 o r\}2 o v ) = Ecr” 1 o i\}2 o v 1 3 = 0 . *5 6 2 Since E : « 21 - > * 2 2 is an isomorphism into, then it follows that o E*20 + * ?2

v® 5

0 ^1 2

0 *13

= °*

o „ ^^

Therefore we have that

vn . an+3 . v 2+10 = en . vn+8 - r,n . en+1

and

for

0 aQ °

n > 5-

q.e.d. 11 8 A : * 2 ^ -» *2b>

In the above computation of is an isomorphism into. that

O

(1 2 .5 ).

E : *25

we see ^^at this

A

It follows from the exactness of the sequence (4.4)

Q *26

is orrto*

Proof of Theorem 1 2 .7 . 2 The result of is a direct consequence of (5 .2 ) and Theorem 1 2 .6 .

Consider the exact sequence 21 * 19 By Lemma 5.2, we have that

of (4.4).

and r\^ o e^

o

generates

and

E

is an isomorphism into.

*30

has at most 1 6 elements.

2 which is not contained in

JL 20 ^ 20 A(i^ ® a^) = a(t^ o gg) = 0.

*21 (Theorem 1 2 .6 ), then

By Theorem 1 0 .5 , By Lemma 1 2 .2 ,

Ejt2^,

since

A

is trivial

* 2 0 88 Z2 © Z2*

jl

Since

Thus

is an element of order

H(m^3) ^ 0 . By Lemma 1 2 .3 ,

is an element of order 4 which is not contained in

E^ ^

e1

Z2 © Zg. Then

it is an algebraic consequence that 3 20

= Ce'} © C?3) © (r)3 .

o o13) « Z k © Z 2 © Z2.

From this, (5 .6 ) and from Theorem 10.3, the result on

« 21

follows.

By (1 2 .2 ) and by the exactness of (4.4), we have an isomorphism

The group

Q

*|3 2

E : *ai/A*23 2 is generated by a and

"22 k ^,

by Proposition 2 . 5

A(cr^) = A(a^) o a11(_

and

= x(v^ o a 1 o a ]k) + Ee' o a ^ k

by (7.16)

= M Ly) o X'j

by Proposition 2 . 5

= + (2vu - Ev f)° Krj

by (5 .8 )

= ± (vu 0

0

~ £*)

= + (2 (v^ o Krj) -s') where

x

• by Theorem 1 0 .5 . We have

is an odd integer. It follows that 5 0 22 - {v5 © ,8} © ( , 5) © ( n 5 0

by Lemma 1 2 . 3 by Lemma 4 .5 ,

^

© Z ^ © Z^,

2-PRIMARY COMPONENTS OF

*n+k(Sn )

FOR 16 < k < 19

H5

and that (12 .6).

A : * 23

the kernel of

rt2 i —

** Z 2 '

Next we prove H(a(E©)) = 0 ' .

Lemma 12.11. Proof. Let

a

(A(a13), v ^ , T2 i^i*

be an element of

Proposi­

tions 1.2 and 1.3, Ea

e E{A(ct^3 ),

v ^ g,

^21^1

x,

by Theorem 12.6.

Since HE

= 0,

(mod 2)

it follows that

subgroup

o ti23 = t|13 o MiU a n23 = and

it22 o r\2 2 ,

composition. Thus

Ea = Ep o rj23. then

By (12.3),

a-po^22

Since

^ 0.

Thus

x = 0

is a coset of the

is contained in thissecondary

2

^ ) =

2 (a - p o

and it vanishes under the

By the exactness of (4 .4) and by Theorem E : * 23 -» *2l+

is

a * 23 = (a(E 0)}

Therefore, we obtain that

H(a(Eo)) = 0 t.

Consider the exact sequence 11 A A 5 5 E 6 H *24 *22 24 *22 23 of (4 .4). By use of Proposition 2.5, we have

11 *23

A (0 ’ 0 ^23) - a (o') 0 n21 = Ha (Eo ) 0 ti21 = 0 2 \ V18) " A(an ) 0 v2g

and A(a11 0

= v5 ° e8 ° v?6 + v5 ° 78 ° A s = v5 0 e 5v ' 0 ;n 0 Vi9 + v? 0 ;8 .

by (7 .17) 2 v 16

fey (7 .12)

= 2 v5 0 ''ii 0 v 19 + v 5 ° *8 ° v?6 2 = v5 ° v8 0 v16 ' — 2 2 By Lemma 10,.1, v5 0 v8 0 ^16 ~ 2^^5 0 *8^ mod v^ 0 « n • By (10 .7 ) Lemma 5 .14 , (T.16), and Theorem 7 .3 , 2 , v5 0 Cn = v5 O E2(vg . ?9) = v5 O 2E2(ct" 0 a 13* 0 4Ea 1 0

= 8(

0 aq ) 0

and it

a(E0) = a - p o ^22 q.e.d.

=

= °-

® r\2 ^) =

H(o! 0

(a(ct13), ^3, ^21^1

is an element of order

has at most two elements.

and an integer

xH(a’ o n11+ «> ^3)

=

H(a - p 0 ti22) = 0 1 and

7 .6 , we have that the kernel of

and

o

a - p ® ti22

suspension : E(a - p o r\2 2 ) =0.

pejr22

for an element

By Proposition 2.2, Lemma 5.14 and by Theorem 7.4, H(a1) o

^22^1

o ti23 .

Ea = Ep o r\2^ + x a 1 o ^ ^ 0 t\23

Thus

^ 0 1 3 ^ ’ v 1 9*

^

= (°, v 19, r\2 2 ) 1 =

°i 5

= 0

.

0.

146

CHAPTER XII.

Thus we have the relation

{v6

E n 22 =

ti23

since

°

A(a^

V

©

and

© {,16 0 ^7 0 °1 6 ] ~ Z 2 © Z 2 © Z2 ’ 2

and

2 o Vlg) = 2v^ o K q ,

cr.j. j o v^g

11

n 2k ’

generate

Theorem 7 .7 . We have

also (12.7).

the kernel of

”* * 2 2

A :

0 ^2 3 ^ = W

M

A (E0) 0 ^ 3 ^

z2. *23 =

Theorem 7 .6 . It follows from Lemma 12.11 and

% Z2'

from the above exact sequence that n22 = (A(Ee)} © E * l 2 ~ Z2 © z2 © z2 © z2. 1^ ir2^ = CEe} = Z2, then itfollows from the exactness of

Since

(4 .4)

that

E*23 ‘ E^n22 * Z 2 © Z 2 © Z 2, (12.8)

-> tc^

E : ir|^

is onto

and that the following sequence is exact: n __ , P 6

0

> Ejt23

__

7

H

> 24

13

* 24

A

6

* 22

It follows from (12.4) and Theorem 7 .4 that the lasthomomorphism

2

has the kernel {4 ^ v30^1 ° 8 l 34 = al6 o E{2a22, v29, 8 2) op C a 16 ° E,t33 = ^°1 6 0 ^23 ]•

Thus

Lemma 6.4

by Lemma 10 .7.

The secondary composition a ^ o E{2a22, t)2

Thus

by Proposition 1.4

= al6 o

=0

mod a ^ o (;23#

By

by Proposition 1.4

Proposition 1 .4 ,

16512 e to12’ v 19> °22) 0 16t30 'l9' °22' 1®t293 We may identify

(v19, a22, 16l 29)

by (9-3), it contains

x£ 19

for an odd integer

x.

a12 ° (v19, a22, 16i29]

is a coset of

Therefore

Q12 0 *30 ° i6l30 = al2 0 16iT30 = 0 * 1^ll2 = xa12 0 ^ and the last relation of the lemma is proved. q.e.d.

2 -PRIMARY COMPONENTS OP

Lemma 12.15.

*n + k (Sn )

16 < k < 19

FOR

i) . There exists an element

155

1u

of

such

that H ( cdi U )

= v2 ?.

ii). There exists an element H(e*2) = v23 Proof. By (10.21),

and

s*2

1 2 such that

of

E2e*2 = to, u . ti30.

a (v 2^)

= 0.

By the exactness of the sequence

(^.M, it follows the existence of an element

cd1^

suchthat

= V2 2

and

7 = tj 2 ^ .

H(oo1

= v27*

n = 11 , a = cd^,

Next, apply Proposition 11.1U for the case Then there exists an element

e*2

of

12

tt2 ^

such

that and H (£12> e fl23> v 2h ’ 'l27)2 H(e*2) = v23* Then the lemma has been

e2£12 = “ lU 0 ^30 By Lemma 5-12, it follows that proved.

q.e.d. Denote that

and

ton = En ~ll+ lU

and

e* = En-12 e*2

for

n > 12

and

e* = E°° e*2 .

1, t^*, con

and

e*,

By use of the elements

cu = E°° co^

our results are

stated as follows. Theorem

IO 1 * 12.16.

_ f{ A ( rc \ 1 © {a,Q f rr „ *2° = a (o21)} - n

^ 10

*£♦16 = t o n * " n +75 “ Z 2

*30

I 2E

n -

11,

12,

} = Zlg © Z2,

13,

° ^21^ ^ Z8 © Z2 ’

=

*3?

■ fn*'} ©

{®15)©

*32

=

lEl’*' 5 © t£0l65© (/ O ^ LA} ©

o 55u O £ lD } ©

(^*3} © {LSU o ,^U} =

• iix m & am o

2 -PRIMARY COMPONENTS OF

*n + k (Sn )

1 6 < k < 19

FOR

157

Then it follows that

(12.19)

*26

a :

and

is an isomorphism into,

*27 = ^an

0 ^18^ ~ Z2*

Next consider the exact sequence 23

A

^8

of (4.4).

= 0

E

11

*29

* *27

a(v^)

23

* *28

* 2 9 = ^v23^

by Proposition 5.9*

By ii) of Lemma 1 2 .1 5 ,

H

12

* *28

= aH(b*2) = °-

exactness of the sequence that

E

Proposition 5 .1 1 .

Then it follows from the

is an isomorphism onto and

12 r 1 ry *28 " 12 ° ^ 19 ^ 2 ' by Propositions 5.8 and 5 .9 - Then it follows from the exact-

*29 = *30 = 0 ness of{ k . k )

thatE : *28

12

1^

* 2 9 is

anisomor,Pbism onto and thus

*29 = ^a13 0 ^20^ % Z2 * 27

rt3l = 0

by Proposition 5.8. n __ > J 3 J L

and

the

element

a(v2^) . Since

= +_ 2cdi1+ + x a ^

o |i21

Ba (

v

2 ^)

^14

= _+ 2v2y =H(+ 2cdi1+),

for some integer

8colU = ± A(

is generated by

is onto and

14 *30 % Zg © Z2 if

a ll+ o ^ 2 1 *Furthermore,

cdiJ+

H

27 30 ‘

H

30

29 By i) of Lemma 1 2 .1 5 , we have that

Then we have an exact sequence 14

x.

= °*

Consider

then

a(v2^)

We have then

29) + x (^°iU 0 ^21) = A (l29)

= A (l29) ° 129 = AH( t)* ') . r)29 = 0 , by Lemma 12.14.

Thus

"30 = Cffilt} + Cct1U 0 ^211 = Z8 © Z2We have obtained also (12.2°) .

The kernel of

A

: *32

Proof of Theorem 1 2 . 1 7 for

29^ = ^29^ ’

*30

n < 16.

We prepare the following lemmas. Lemma 12.18. There H(X) = v 2^

exists an element and

E 3X - 2V*6 =

\

Lemma 12.19.

There are elements

X 1,

1^

n = 12, € *29

such that E 2X,’ = 2\, H(xf) = e 21 mod v21 + e2i> E £ ’ " 2I •)2 = ± ^(^25^ '

such

that

+ A( v 33) .

This is just the Lemma 11.17 of the case

€ *28

of

H( | ’) = v"2) + e21J

and and

a = '1 6 *

|

1 58

CHAPTER XII.

EX" = 2\', H(x") = n19 ° e20 mod I19 0 e20 + v?9 and

E|" = 2|', H(g") = y^9 + ri19 « e2Q . a = X

Proof. Consider Proposition 11.15 of the case that n = 10, 2 11 p = v 20 • Then we have an element x' of such that

and

2a, = E 2x.' and

H(x') e £n21, v 22, 2 i 28} .

By i) of Proposition 3.^ and by (7.6), the secondary composition

2 (^l* v22>

2i2Q}

By Proposi­

is a coset of

ti21

0 jt22

+ 2*29(S21) which containse21.

Lemma 6.k, we have that

tion 5 .15 , Theorem 7.1 and by

H(\f) = e21 mod-

v21 + £21 = ^21 ° a22* Next consider Proposition 11.16 of the case that and

a = i12

n = 10,

p = a2Q mod 2a2Q. Then we have that there is an element

g'

of

such that E| ’ - 2g12 = + a (cj25)

and

2*29 = ° ‘

By Theorem 7 . 1 ,

mod 2*^.

H(|T)=

°a22 Thus ^ =^210a22

= *21 + e21

by Lemma 6. b . Apply Proposition 11.13 to the case x"

Then we have the assertion for

and

|M,

a = x 1 or

n = 9

and

since

2E2*2g = 0

7.2.

|T .

by Theorem

q.e.d. Now, we prove

(12.21).

E :

1

and the kernel of ° ii17

Jg E

is an isomorphism into if

is generated by

if

the first

for

n = 13

and

generates 2 Lemma 12.18, a (v 2^)

k

A7t2^|9 = 0

12.

Since

for

generates

n = 2^

+ a9

0

then

then 5 .9 ,

By Propositions 5.8 and

Ait2^|9 = 0 for n = 15. By 2 2^ v2^ generates jt^, then

By Lemma 12. 14 , a(cj23) = aH( 112) = °-Since 2n+1 A*n+19 = 0 for n = 11. By Lemma 12.19 and

that

H(xr)

and

H(t')

A«^|^ = 0

span the subgroup

is spanned by the subgroup and A^^)

10 < n < 15,

By Lemma 12 .14- a(v31) = AH(v*g) = 0.

= a H(\) = 0. Since

the exactness of (^ .*0 H(|")

n = 1U.

we have that

Theorem 7.1, the elements

ated by

A jt2^|9 = 0 for

assertion of (12.21) is proved.

A*2^!9 = 0

and

= ^9 0 ^10 0

n = 9. Proof. ¥e shall show that

ct23

a Ch ^)

10 < n < 15,

for

span

n = 10.

(v^) + f*n1^ 0 e 2Q} i±^

21

*29'

Similarly

of

(Theorem 7.2), then

and it has at most two elements.

f°llows from

jt2q.

H(xn )

Since

A*2q

Sener_

By Theorem 3 .1, we have

th e r e l a t i o n

E (tj9 o n 1Q 0

By Theorem 1 2 . 7 , E.

0^)

0 n 1Q 0

cj19

=

+

(b.b),

T hus, b y th e e x a c t n e s s o f a ( m-19)

(r\2

=

o ^

# a Q = E(c?9 0 nl6

o t] ^

o

0

° ^1 ? )

but i t

•

v a n is h e s under

we have t h a t

0 n 10 0 cr19 + a 9 0 ^16 0 ^ 1 7 *

By th e secon d a s s e r t i o n o f ( 1 2 . 2 1 ) ,

(12. 18)

q .e .d .

and b y Theorem 1 2 . 7 ,

we have t h a t Tt^rj

= E tt ^g = { a 1 0 ° ^17 0 ^ 1 8 ^ © ^v 1 0 0*1 3 5 © {^ 1 0 } =s^2 © ^ 2 © ^ 2

By ( 1 2 . 1 9 )

and ( 1 2 . 2 1 ) we

have t h a t

Thus we have th e r e s u l t s f o r

* 2g

E: ^7

-» * 2g

*

i s an isom orphism o n to .

i n Theorem 1 2 . 1 7 .

C o n s id e r th e e x a c t sequ en ce 11 *28 o f ( ^ . M , w here H( b * 2) = V23 = 03^0

E

12 29

H

23 f 2 , 29 “ l v 23j

*

i s an isom orphism in to , b y ( 1 2 . 2 1 ) .

and

2 ti30 = 0

E

E 2e * 2

= cu1 ^ o t^ q . By ( 1 2 . 2 1 ) ,

im p lie s

4

9

22^^ 27 tt3q = jt3 ^ = 0

-

th a t

En2 8 28

©

By Lemma 1 2 . 1 5 ,

th e r e l a t i o n

2 e * 2 = 0. T hus,

2 ] "“

U *

Z2

©

Z2

©

b y P r o p o s it io n s 5 .8 and

Z2

© Z 2 -‘

5.9. 1122

and

1u

Ev

31

0

E

.15 32

-»

*

By Lemma 1 2 . i k and P r o p o s i t i o n 22 .. 22>•,j H(t}*'

T!31)

3

= x\‘29 \9

= 2^ * 6

n32) -

0 ^32^ = °1D*.

= E * 3 2 © ^ * 6 0 '‘ 32 J ““-^2 ^ ^2

In th e group form

th e n o1

INI

^ ^2 ^ ^2 ^

>15

n

jr^o = (o)1l+} © { a ^

+_201^ + x u ^

= ^

•

Then Then 1we have

z2 © z 2 © Z 2 © z.

( n*< o

P r o o f o f Theorem 1 2 . 1 6 f o r

of a

H(n* 6 o 0 n32) H(nif6

and

tsi

*33

-3

*3 q jt ^o

15 29 ■ ' 32 JL rl n*3 16 JL f ,2 • * 33 — ^311 i • * 3 3 —* {n3

ro © ESI ro © INI ro ©

and

o

2(t^*1 0 113-1)

= E « ^@

r e s u lts fo r

and ( 1 2 . 2 1 ) , we have e x a c t seq u en ces 0

Obviously

I t f o llo w s from th e 11^^ ^ -4 and E : jc *30

1 ^ 1^

e x a c t n e s s o f ( ^ . M and from ( 1 2 . 2 1 ) t h a t E : -> 1 !(.!(. jt31 a r e isom orp hism s o n to . Thus we have o b ta in e d th e 1U and jr^ * 311 i n Theorem 12 1 2 . 117 7 .. By ( 1 2. 20)

2E2e * 2

o ^21 f o r

o ti22 0 ^23 = 1715 0 ^22

B ut t h i s c o n t r a d ic t s th e r e s u l t on

some 0^31 = 2

^2 ^ ^2

and Theorem 1 2 . 1 7 f o r

°

,

in t e g e r x . — 2a5l 5 ° ^31

th e ele m e n t If

£

a ( v 29)

is

0 (mod 2 ) ,

- Ei^ v 29^

i n Theorem 1 2 . 1 7 .

a ( v 2 9 ) = ± 2 CD1 ^ ‘

x

n > 17.

Thus

0^31

= *“**

CHAPTER XII. Consider the exact sequence 29 a 14 E 15 H 29 *32 * *30 * *31 * *31 of (4 .4). 12.14 ,

1x^2

and- *3^

H(r]*')= r|29 *31

are generated by

and

By Lemma

2ti* ' = 0.Then it

= tl*'} ©

t“ 15) ©

In the exact sequence 31 A

15 31

33

Q1 tt^3 and

v a n d

"21 *32 are generated by

12.14 and by Proposition 2.2,

AH(n*g o n32) = 0

follows that

( ° 15 ° ^22)

E

2 t]^1

16 32

= z 2 © z2 © z 2 .

,31

H

32 '

and

t]^1

H( tj*^) = *13-,,

respectively.

By Lemma 2 anc^ M 1!^) =

2t1*6 = 0

. Thus we have

"32 = ('Ite1

©

»23]

© f“l6} © {°16 0

Consider the exact sequence _33 35

A

17

16 E

*

of (4 .4 ) . By Theorem 10.10, E :

-» *33

element

p e

is onto.

33 A 34

H

* 34

33

16 E ,17 H *32 * 33

a : tt33 -> jt^

H(p) = t} ^

a (ti33)

h

16 31

is an isomorphism into.

By i) of Proposition 11.10,

such that

,3 3 A 33

= H(rj*T).

a (ti33)

= Ep

Thus

for an

Thus

E ii*' mod

2 2 14 a (^33) = Erj* ’ 0 r^2 mod E .

and

Then it follows from the exactness of the above sequence that n33

and

= (l t 7 ) ©

{ m 17} ©

t o l7

°

|J2 U )

”

Z 2

©

Z 2©

Z 2

*34 = ^ * 7 0 ^33^ © ^*7^ © ^ 1 7 ° ^2b ° ^25^ © ^v 17 °K 20^

© ^17^

- z2 © z 2 © z 2 © z 2 © z 2 .

Next consider the exact sequence _35 A 17 E 18H 35 A 17 36 3U 35 35 33 of (4 .4) . By ii)of Proposition 11.10,a( 1^) 3 e jt^2

such that H(p) = A( t35)

and

a

^ 35^

E ,1 8 _______ k n 31 *=Ep for an element

= H(r|*g) . Thus = t!*7

mod

= ^*7 °^33

E 2tt^ mod

E:T32

*

Then it follows from the exactness of the above sequence that "3® = f“ l8> © (o18 0 * 2^ and

~Z2 © Z2

= Ca(i37)} © (e*g} © (ol8 13

= pn 0 T'n+16

0 ^n+7

+ ^ 17

a

o

0 r\ e < a ,

2a,

and Lemma 5 • 1^

By Lemma 6.5

a

o M

The composition a

generated by 0 Gg © t\ = 0

and

o t] =


n

mod

ctr

o < 8a,

2 1 , ri >

C a

o .

o

81,

t) > .

Then

by i) of

r] >

a ° Gg °n

e o i\. By Lemma

which is

10.7, we have that

By the anti-commutativity of the

( io a = a°fi = por) E°° : it^g

(3 -9 )

.

o n = p 0 tj.

-> G ^ and

a °

= a

cj0 .

2a,

8l > e

composition operator, we have

i)

p e < o,

By Lemma 10.9

p

Then

mod

n > 18Proof.

a

. 0n+g = nn . pn+,

r,]s » p13 = ct12 .

and

ii) . A( t35) =

an » ^n+? = ^

-»G ^

= riop.

By

Theorem

are isomorphismsinto.

Is proved immediately. Next let

n > 18.

The secondary composition

is a coset of the subgroup by the compositions

aR °

on . v 3+ 7>

e 'n ° ^n+ 15 ‘ By { 7 -2 0 ) ’

+ *^+15 0 ^n+15 °n ° ^ n+7,

an ° vL j

" °-

*n • ^ 7 an 0 V 7

{an ,

^n+i1^

which is generated 0 en+8 ’

en 0 TW l 5

° £n +8 = CTn ° sn +7

0 "n+15 = 0 by Lefflma 10-7' Sn 0 ^ 1 5 = ^n• £n+l = vn 0 °n+3 0 Vn+10 by Proposition 3 .1 , Lemma 12.10 and (7.20). Thus (an , 2or + ^n+iu^ = ’’n + {an 0 ^n+71 = ’’n + {pn 0 T'n+15) • By Theorem 12‘6> 1* S 0 ^n+7

for some integer

0 ^n+7 for some (mod 2) and follows that

integer

x.

Assume that

A( l35) s cjd1 7 +

for n >

mod a ^ 1Q

and

18 .

for

n* = yan

Thus

x^ 0

From Theorem 12.16,

o n 2j+.

19

mod an

then

y.But this contradicts Theorem 8.^.

tj* = cDn mod an o Mn+7

iv) . The groups

x = 0 (mod 2) ,

q.e.d n > 10 . 12

0

Lemma 12.21. There exists an element cd’ of jt31 such that 2 _ E cd 1 = 2coli+ o v3Q and H(o>') = e23 mod v23 + e23-

it

1 62

CHAPTER XII.

Proof. By Proposition 2.2 and Lemma 12.15, we have that v3Q) = v27 • APP!y Proposition 11.14 for the case that v3Q

and

7 = 2 l2g • Then there exists an element

E2o)1 = 2o3i1+ o

and

n = 11 , a = ^ ° 12 of * 31 such that

o>'

H(cd') = e2^ mod v2^ + e 2 ^,

H(cd1^ o

by a similar way to the

proof of Lemma 12.19.

q.e.d.

By use of the elements

co', x, x 1,

x",

our results are

stated as follows. Theorem

12 .2 2 .

rt^g « ZQ © Z 2 © Z 2 : generated by

^10 ° ^11’ ir1 1 « Zg © Z^ © Z 2 : generated by

x ’, |»

29

12

and

and

ojln2,

l12, EX', E| ’ and r\}2 o (1^

*30

88 Z 3 2 © Z U © Z U + Z 2 : generated by

«n

= U n} © lEn_1 3x) © U n . ^n+1} = Z8 © Z8 © z2

n + 18

XM,

=13 , I1*,

for n

15

16

*3 ^

= (v* 6^ © U 1g) ©

*17

= {V*r^} © {|17} © {7117 0

J 8

= {v*g} © U*g +

35

*36

J 9 37 n+1 8

(Gi8;2)

© (Ii g ° ^ 17} = z8 © z8 © Z8 © Z2’ q

)

Zg © Zg © Z2, |18) ©

= ^v* 9 ^ © ^ vt9 + ^19^ © ^ 1 9

Theorem 12.23.

^9)

0 ^ 20 ^ 58 Z8 © Z2 © Z2*

= (v*} © {^n ° ^n+1) - Zg © Z2 = {v*} © {ii o (I]

°

for

n > 20,

and

Z g © Z2. *2° = ^10^ © ^ 10^ * Z2 © Z8 >

11 *30 12

*31

= {*-1 0 ^29^ © ^ ' 0 ^ © ^ 1 1 } © ^ 1 1 }

Z2 © Z2 © Z2 © Z8*

= {0)»} © (|12 o n30) © (Ex 1 o n30} © {Ei’o n30} © {512} © Cf12)

% ^2 © Z2 © Z2 © Z2 © Z2 © Zg, ^©{^13

0 Tl3 ‘ | ^ © { ^ i 3 ^ © { ^ i 3 ^ 558 ^2 © Z2 © ^2 ©

*32

=

*33

= ^m 1U 0 V30 ^ © ^ 14^ © ^ 14^ ~ 2 ^ © Z 2 © Z g ,

*£+19 = Ca3n 0 vn+16} © (5n} © {^n} ** Z2 © Z2 © Z8 nf9

^

n =15, 16, 17,

= {A(tM )} © {5 20) © (f205 w Z © Z2 © Z8'

*n+19 = ^ n J © ^ n J % Z2 © Z8

n > 18

and

n ^ 20,

and

(G19;2) = (5 ) © {p * Z2 © Z g . First we prove (12.22) . E : *^+19

-» *£+20

is an isomorphism into if

Zg,

9 < n < 11.

~ Z8 © Z

2-PRIMARY COMPONENTS OF

*n+k(Sn )

FOR

16 < k < 19

163

Proof. By the exactness of (4 .4), it is sufficient to prove that A*n+2i = 0

for

n = 9

a = r^g ® e ^.

that

and H(p)

of Proposition 11.11 for the case

9 — n — 11 * Apply

e (21.J

n 2Q o e 2 1 ).

follows from Lemma 9.1 that

p

Then there exists an element

H(p)

The group

generates

jt^ q

of

jt^o

i s stable.

= ^19^*

such

Then it Art3o =

Tkus,

A H n ^ = 0. Next apply i) of Proposition 11.10 for the case

a = n2o # Then there exists an element 21 n2g. Since t)21 o ^22 generates

p

of

n = 10

such that

by Theorem 7 .3 , then

and

H(p) = 12 1 0 2 1 1 1 A^., = aHjt^1

= 0. Apply ii) of Proposition 11.11 for the case that n = 1 2 a = 8a22. Then there exists an element p of *^ 2 such that 2 l2 ^ f

8cJ2^}.

11 and H(p) e (n23>

It isverified from Lemma 6.5 that H(p) = n23 mod{v23) + (^23 0 e2i+5*

By Proposition 2.2 and by Lemma 12.14 , we have H(li2 0 n30) = *23 0 ^30 = v 23 + ^23 0 e 24 * By Proposition 2.2 and by Lemma 12.21, we have 3 h( \X ’

00

^ 20

(^2 0 ^ 14^ = ^ 1 0 ° a19 = ^ 1 0

*#6 CO

°1 3

II

and

• ?18

II

2 © ( ( EX' )

+ {Eft 1J + ( I ) )

^ ^2 © ^ 4 © ^4 © ^32^ * Consider the above second sequence.

Obviously

2(i 12 o t^ q) = 0.

Then the result * ’2 = E k ’ q © ( l 1 2 = r,3 0 ) © { » ' }

of Theorem (12.26).

2o>'

= 0.

Proof.

E.

0t]^0 Since

A* 3 1 = ( E X'

1^3

2*5

^

on3 1 ) •

= 0

and

Thus the elements

elements in the kernal

of

Thus the second

Next we have

TCVI

v20, n23 } 0 “ 25

*13 0 n 31 e (°i3, v20 * °23) *

(a13 ’ v20, n S3

composition The secondary > 20 13 a13 ° *32 + * 24 0 ^24 20 C CT13 ° *32 + Thus A (23 0 1’

C

n30} •

span a subgroup of 4

A*^

assertion is proved.

and

•

has 4 elements, then it follows from the exactness of the

sequence (4 .4) that

€

+ (E ft'

EA(a2^) 0 tj^ 1 + 2(|13 o ti31) = 0.

andE g f r^0 25

o n3 0 )

By Lemma 12.19, we have

E(E|1 o ti3q) = _+ EX'

= Z2 © Z 2 © Z 2 © Z g © Z 2 © Z 2

12.23is proved by the first assertion of the following

31

» y24^

by Proposition 1.2.

t»l 3» ^20 f ^23 0 a24 ^ ^ c°set of 20 ru T o 13 ^24 0 a25 0 a 25 = a13 0 *32 + l An517 0 a2^ = 0 E

Since

a2^

by Theorem 7 .6 .

generates

In the computation of

A(v 2^) = ± 2coll;. Then we have 2 2 E (2a)1) = 2E a)' = 4^ ^ ° V30

then the last asser *3^, we obtained a re­

by Lemma 12.21

= A(2v 29) 0 v3Q = a (2v 2^) = 0

by Proposition 2.5.

2 -PRIMARY COMPONENTS OF

*n+k(Sn)

FOR

16 < k < 1 9

167

By the exactness of (^.4-), it follows that 2cnf = x|12 o t)30 + yEX' ° for some integers

x, y

we have the relation that

x = 0 (mod 2).

and

z.

+ zEg 1 ° ^

#

Applying the homomorphism

23 *31 '

12

H

*31

0 = H(2a>') = xH( g12 o n3Q) = x(v23 + e23) . It follows O PQ H(oi’) = E p for p e *28. Then, by Proposition

Let

11.16 and i) of Proposition 11.10, we have elements

y

y'

and

of

a(E5£) +_ 2m' = Ey, H (7) = tj21 o E 2p, A(E5p) = E7 ’ and 2 *5 = t]21 o e p. By cancelling a (E^p ) , we have that such that

H(7!)

2a>' = +_ E(7 - 7 1) and H(7 - 7 ’) = 0. 11 -> it31 12 is an isomorphism into. Then it followsthat E : jt3Q

By (12.22),

+ (7 - 7 1) = y*-’ 0 ^29 + z|* 0 t129> 11 21 Applying the homomorphism H : j t 3 0 -»it3Q, we have o = y H ( x ’ o ti29) + zH( g ' o n29) . We know that that

H(x’ o t^29)

y = z = 0 (mod 2).

and

H( g' otj2^)

are independent.

Consequently we have

Thus we

have

obtained that

2o)’ = 0.

q.e.d.

By a similar discussion to the proof of the fact

E (2it3 0 . We have obtained that

Thus the homomorphism

A

is anisomorphism

into.

+ a (c?2^) = Eg' - 2g12. Then it follows from theexactness

of the sequence (^.*0 the result EK30 = U 13 0

© ( U 13l + (Esx')) - Z2 © (Zg ©

and the following exact sequence : 12 13 H 0 — » Er t 30 * 31 12 E * 0 -- > a25 * 33 * *31 2 We note that E X ’ is the element of order = 0.

25

Since

is generated by

2

v2^,

zu)

25 31 13 32

U, since

2 2 ^E x ’ £ [E (^n ° £ 1g)

then it follows from Lemma 12.18

and Lemma 12.19 that *31

= ^ 1 3 0 ^ © C s 13) © U ) « z2 © z8 © z8 .

It follows from (12.26) that -13 = 32

( E cd

1) © {g13

o

n31) © (a13) © {^13) - Z2 © Z 2 © Z 2 © Z 8 .

1 68

CHAPTER XII. By Proposition 2 . 2 and Lemma 1 2 .1 5 , we have

^(“ 14 0 v3o^ = v27 * Since

2

v2^

generates

27

then it follows from the exactness of (4 .4)

*33*

that the following two sequences are axact. 14 H 27 13 E 0 > *32 , 31 * *32 27 14 13 E it^ 0 * --> * (H(u>11( o v30)} 34 -- > 32 * *33 27 0 . Thus = °* Thus By Proposition 5 .9 , 32 “ „13 E 14 ©< {T114 ° ^15} 31 ~ 32

© t e lU)

By (12.27) and (12.26), we have *33 = ^1 4 ° V30 ^ ® ^ 14^ © ^ 14^ % © Z2 © Z8 * By Propositions 5.8 and 5 .9 , *33 = =0. By Proposition 5 .11 , the 29 2 group is generated by v2^. Then it follows from the exactness of (4 .4) that

E :*34

is 811 isomorphism onto and that the sequence *\k -- > 0

0 -- > {A (Vg9)} -- » ”3 3 ^ is exact.

By (12.27), we have that

*34 = ^ 1 5 0 v 3 i ^ © ^ i 5^ © f ? i 5^ % Z2 © Z 2 © Z g . ^1 ^1 31 0. * ^ = *31 *^ = 0.Then we have the following 35 = *36 = two exact sequences 16 15 E > 34 0' 3? * 35 E 16 H 15 0 ^ , ’