Stable Homotopy Groups of Spheres: A Computer-Assisted Approach (Lecture Notes in Mathematics, 1423) 3540524681, 9783540524687

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Lecture Notes in Mathematics Edited by A. Oold, B. Eckmann and F. Takens

1423

Stanley O. Kochman

Stable Homotopy Groups of Spheres A Computer-Assisted Approach

Springer-Verlag Berlin Heidelberg NewYork London ParisTokyo Hong Kong

Author

Stanley O. Kochman Department of Mathematics, York University 4700 Keele Street, North York, Ontario M3J 1P3, Canada

Mathematics Subject Classification (1980): Primary: 55045 Secondary: 55T25. 55S30, 55050 ISBN 3-540-52468-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-52468-1 Springer-Verlag New York Berlin Heidelberg

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PREFACE

This work develops the theoretical basis for an efficient method for the S

inductive calculation of the stable homotopy groups of spheres, n.. the steps of this method are algorithmic and are done by computer. apply this method to compute the first 64 stable stems.

Most of We will

This method is based

upon the analysis of the Atiyah-Hirzebruch spectral sequence: =

H BP n

H.BP and n.BP are well known. h:n.BP oo = n,O

E

®

n

S t

=> n

n+t

BP.

Moreover, the Hurewicz homomorphism

H.BP is a monomorphism.

Therefore, E

hen BP) which is also well known. n

If n

S

oo

n,t

=0

if t

0,

and

is known for t < T then, with

t

the exception of one step, it is algorithmic to deduce the composition series r

r

I mage l d : E

r,T-r-+l

r

E

'O,T

], 2

oS

r- oS

S

T+1, of n . T

The determination of n

S T

from this composition series, the solution of the "additive extension problem", is accomplished using Toda brackets.

A distinctive feature of this method is that all the hard computations are done by computer.

This includes the determination of differentials using

Quillen operations and the computation of Er+ 1 = Kernel [dr:E r N,t N,t

1 / Image [dr:E r Er N-r,t+r-l N+r,t-r+l

r E 1. N,t

On the other hand there are two key steps which require human intervention in S

the computation of each n : T

(1)

the matching of the I ist of "new" elements in degree T+1 which are hit by differentials with the list of "new" elements in degree T+2 on which nonzero differentials originate;

(2)

the solution of the additive extension problems.

IV

Chapter 1 is devoted to the exposition of the background of this computation and to a detailed description of the method we will use.

Even the most

experienced reader should read the exposition of our notation for elements of the stable stems at the end of that chapter.

In Chapter 2 we develop the

three and four-fold Toda bracket methods which are used to solve extension problems.

In Chapter 3 we give a global computation of the spectral sequence

in the first eight rows.

In higher rows our computations are inductive and

rarely achieve a global understanding of the rows beyond the range of our computations.

In Chapter 4 we recall some facts about the Image of J and use

them to compute all the differentials which originate on En,o for n

70.

r

Chapters 5 to 7 contain our calculations of the first 64 stable stems. Chapter 8 we identify the elements e as well as the Mahowald elements

n

E

4

n

E

S 30

5

and e

and

32

S

E

E

n

S rr

62

S 64

In

of Arf invariant one The new proof that e

s

exists and has order two is based upon Mahowald's ideas [34A] and the computations of this paper. Mahowald sent to me.

It is a rewording of a detailed proof which

We also show that

has order four.

We conclude with

Appendices 1 - 4, 7 which contain tables that summarize and give references for all the computations of this paper.

In the fifth appendix, we discuss the

Fortran computer programs which are used in this computation. program listings is available from the author.

A copy of the

The most important output of

these programs is contained in the last sections of Chapters 4

7.

The

sixth appendix depicts the mod 2 Adams spectral sequence through degree 64.

We will work exclusively at the prime two. primes.

Our methods, however, apply at all

Of course, the computations at odd primes would be very different

from these computations at the prime two. involved at the prime two reached 2

32

,

the limit of the computer, requiring

the use of some multiprecision arithmetic. would involve much larger numbers.

In addition the size of the numbers

The computations at odd primes

v I wish to thank The University of Western Ontario and York University for their support of this research as well as the University of Toronto for their hospitality during my sabbatical leave there.

In addition, the Natural

Sciences and Engineering Research Council of Canada supported this research through Operating Grants as well as an Equipment Grant which allowed the purchase of the IBM PC/AT computer on which the calculations were performed. Last, but not least, I am very grateful to

Mark Mahowld for detecting errors

in earlier versions of this paper, for his ideas on 8

5

and for his assistance

in constructing the Adams spectral sequence tables in Appendix 6.

TABLE OF CONTENTS

Preface Chapter 1:

iii Introduction

Section 1.

History of the Problem

Section 2.

The Brown-Peterson Spectrum

Section 3. Chapter 2:

1

and QUillen Operations

3

The Inductive Procedure

5

Toda Brackets

Section 1.

Introduction

12

Section 2.

Definitions

12

Section 3.

Properties of the Toda Bracket

20

Section 4.

The Atiyah-Hirzebruch Spectral Sequence .... 25

Chapter 3:

Low Dimensional Computations

Section 1.

Introduction

Section 2.

d

2

35

Differentials and 4

the Determination of E Section 3.

d

4

35

Differentials and 6

the Determination of E Section 4. Chapter 4:

d

8

'"

Differentials and the Seven Row

..

39 48

The Image of J

Section 1.

Introduction

72

Section 2.

ImJ and the Adams Spectral Sequence

72

Section 3.

Differentials Originating on the 0 Row - Theory

Section 4.

80

Differentials Originating on the 0 Row - Computation

Chapter 5:

S

The Japanese Stems (n •

Sect ion 1.

N

Introduct ion

86

31) 99

VIII

Section 2.

S The Toda Stems (rr , 9

:!i

N

:!i

19) ............ 99

Section 3.

S The Oda Stems (rrH' 20

:!i

N

:!i

31) ............ 104

Section 4.

Tentative Differentials

Chapter 6:

H

S, 32 The Chicago Stems (rr

N

H

Section 1.

Introduction.. "

Section 2.

S Computation of rrH' 32

Section 3.

Computation of

Section 4.

Tentative Differentials

Chapter 7:

The New Stems

S

(n: , N

113 45)

:!i

139

S

39

n: H'

46

:!i

N

:!i

38 ............. 139

:!i

N

:!i

45 ....... , ..... 149

N

:!i

162 64)

Sect ion 1.

Introduct ion

Section 2.

Computation of

Section 3.

S Computation of rrN' 51

Section 4.

Computation of

Section 5.

S Computation of rr H' 61

Section 6.

Tentative Differentials .................... 253

Chapter 8:

212 1[

1[

S

46

N'

S

:!i

56

N'

:!i

N

:!i

50 ............. 212

N

55 ............. 220

N

60 ............. 230

N

64 ............. 242

The Elements of Arf Invariant One

Section 1.

Introduct ion

Section 2.

The Existence of e

Section 3.

The Existence of e

284 285

4

289

s

Appendix 1 :

The Stable Stems

Appendix 2:

MUltiplicative Relations

Appendix 3:

Toda Brackets

303

Appendix 4:

Leaders

306

Appendix 5:

The Computer Programs

312

Appendix 6:

The Adams Spectral Sequence

317

Appendix 7:

Representing Maps

327

Bi bl iography

294 '"

297

328

CHAPTER 1:

1.

INTRODUCTION

History of the Problem

The calculation of the stable homotopy groups of spheres is one of the most central and intractable problems in algebraic topology. [57] used his spectral sequence to study this problem.

In the 1950s Serre In 1962, Toda [60]

used his triple brackets and the EHP sequence to calculate the first 19 stems. These methods were later extended by Mimura, Mori, Oda and Toda [44], [45J, [46], [50] to compute the first 30 stems.

In the late 1950s the study of the

classical Adams spectral sequence was begun [1].

Computations in this

spectral sequence are still being pursued using the May spectral sequence and the lambda algebra.

The best published results are May's thesis [39] and the

computation of the first 45 stable stems by Barratt, Mahowald, Tangora [10], [37] as corrected by Bruner [16].

The use of the BP Adams spectral sequence

on this problem was initiated by Novikov [49] and Zahler [62]. spectacular success has been at odd primes [42].

Its most

A recent detailed survey of

the status of this computation and the methods that have been used has been written by Ravenel [55].

An exotic method for computing stable stems was developed in 1970 by Joel Cohen [19].

Recall [20] that for a generalized homology theory E. and a

spectrum X there is an Atiyah-Hirzebruch spectral sequence: 0.1.1)

2

E

N,p

=H(X;E) N

p

E

X.

N+p

Joel Cohen studied this spectral sequence with X an Eilenberg-MacLane spectrum and E equal to stable homotopy or mod p stable homotopy.

His idea was to

take advantage of the fact that in these cases the spectral sequence is converging to zero in positive degrees.

Since the homology of the

Eilenberg-MacLane spectra are known, one can inductively deduce the stable

2 stems.

This is analogous to the usual inductive computation of the cohomology

of Eilenberg-MacLane spaces by the Serre spectral sequence [17].

In that

example, however, all the work can be incorporated into the Kudo transgression theorem.

Joel Cohen was able to compute a few low stems, but the computation

became too complicated to continue.

His method was discarded since the Adams

spectral sequence computations seemed much more efficient. Nigel Ray [56] used this spectral sequence with X

= MSU

In 1972, however,

and E

= MSp.

He took

advantage of the fact that H.MSU and MSp.MSU are known to compute the first 19 homotopy groups of MSp.

Again this method was discarded since David Segal had

computed the first 31 homotopy groups of MSp by the Adams spectral sequence and his computations were extended to 100 stems in [31].

My interest in Atiyah-Hirzebruch spectral sequences began in 1978. In a joint paper with Snaith [32] we stUdied the case where X is BSp and E. is stable homotopy.

The methods we developed there, in particular the use of

Landweber-Novikov operations to study differentials, were clearly applicable to a wide class of examples.

In 1983, I observed that if Joel Cohen's method

were applied to the case where X is BP and E. is stable homotopy then the computations would be greatly simplified over Cohen's case because of the sparseness of H.BP and because Quillen operations could be used to compute the differentials.

So, I began computing at the prime two.

I soon discovered

that the computations became too complicated to do by hand, but since they were mostly algorithmic they could be done by a computer.

Using an IBM PC/AT

micro-computer I was able to compute the first 64 stable stems.

This work is

the account of that computation. Kaoru Morisugi informed me that in 1972 he attempted to use this method to

S

compute n. at the prime three, but he became bogged down with technical problems.

3

2.

The Brown-Peterson Spectrum and Quillen Operations

In this section we list some of the basic facts about the Brown-Peterson spectrum BP.

The notation introduced here will be used throughout the

computation. Let MU denote the unitary Thorn spectrum. n.MU is isomorphic to

By the Pontryagin-Thom isomorphism.

the ring of bordism classes of compact smooth

manifolds without boundary which have a complex structure on their stable normal bundles.

Using the Adams spectral sequence, Milnor [43] computed n.MU

to be a polynomial algebra over Z with one generator in each even degree. Brown and Peterson [15] discovered that when the spectrum MU is localized at a prime p, it decomposes into a wedge of various suspensions of a spectrum SP. This spectrum defines a generalized homology theory BP. and a generalized co-

• homology theory BP.

We list several basic properties of BP at the prime two.

The standard references are the expositions of Adams [1] and Wilson [611. (1.2.1)

N

There are M E H.BP of degree 2(2 -1) such that M N

H.BP (1.2.2)

1 and

0

= Z(2l[M1 •.•• MN , •.•

The Hurewicz homomorphism h:n.BP

].

H.BP is a monomorphism.

Henceforth we consider h as an inclusion. (1.2.3)

Define Y

N

E

N

H.BP of degree 2(2 - l ) recursively by Yo x-;

Y

The Y /2, N N

2::

N

= 2M

N

-

2

k=l

M .y2

k

N-k

k

.

1, are polynomial generators for H.BP.

are in the image of hand n.BP = Z(2l[V

1

•..•

'Y

N

2 and for N2::1:

' .•• ].

Moreover. all the V

N

The V are called the N

Hazewinkel generators [22], [23] . (1.2.4)

•

BP BP is the algebra of BP-operations.

fop any spectrum X including BP.S are natural.

= n.BP

These operations act on BP.X

and BP.KZ

= H.BP.

These operations

In particular. they commute with the Hurewicz homomorphism h.

4

•

(1.2.5) The r

w

BP BP

= n.BP[[

r

w

I w is a finite sequence of nonnegative integers]].

are called the Quillen operations [54].

They have the following

properties. (a)

The rare Z

(b)

If f:X

w

hor (c)

(2)

Y is a map of spectra then f.orw r

w

-module homomorphisms.

w

oh.

If X is a ring spectrum and A,B E BP.X then we have the Cartan formula r (A'B) = \' , r (A)'r .. (B). w J.., w=w +w" w' w

In [32] we showed how Landweber-Novikov operations act on the AtiyahS

S

Hirzebruch spectral sequences for n.BU and n.BSp.

The following theorem

shows that the QUillen operations act on Atiyah-Hirzebruch spectral sequences for BP.X.

THEOREM 1.2.6

Let F be a ring spectrum.

Consider the Atiyah-Hirzebruch

spectral sequence for F.BP: 2

= H BP

E

@

F

F

N, t N t

Then each Quillen operation r

w

N+t

of degree K induces a map of spectral

sequences:

These r

w

have the following properties:

(a)

The rare Z

(b)

The r

w w

(2)

-module homomorphisms.

are natural with respect to maps of spectral sequences induced by

maps of spectra. (c)

The r

w

satisfy the Cartan formula

r (A'B) = \' w L (d)

W=W'+W U

The action of r

w

r ,(A)'r .. (B) for all A, BEEs. tv

Cd

on £2 is given by r

usual Quillen operation on H.BP. r odS for all s w

1.

w

@

1 where the latter r

w

is the

5 (f)

The action of r

(g)

The action of r-

(h)

The action of r

w w w

on

,d

H.(

on the

S

S

)

is induced by the action of r- on E w

induce an action of r

W

on E""

=

S

lim E

.

•

--7

on E"" defined by (g) agrees with the action of

r-

w

on E""

induced by the usual action of the Quillen operations on F.BP PROOF.

Since r

r :LKBP

--7

w

w

BP.

E

BpkBP, we can represent r

w

by a map of spectra

Since the Atiyah-Hirzebruch spectral sequence is natural we

have an induced map of spectral sequences.

All of the properties are

immediate except for the Cartan formula (c).

It follows from the observation

that the following diagram must commute up to homotopy: BP

r

w'

1\ r " w

In this diagram

is product map of BP and

wedge summand k

k'+k" and T is the switching map.•

3.

1

is the pinching map.

In each

The Inductive Procedure

In this section we will describe in detail the inductive procedure that we will use to compute the stable stems.

However, before we apply this procedure

in Chapters 5 to 7 we will digress to compute the first eight rows of the spectral sequence in Chapter 3 and to study two of the basic ingredients of our procedure:

Toda brackets in Chapter 2 and the image of J in Chapter 4.

This section concludes with an exposition of the notation that we will use to denote the elements of

n;.

Consider the Atiyah-Hirzebruch spectral sequence: (1.3.1)

H BP N

IS!

n

S t

=> n

N+t

BP.

6 Since H.BP is zero in odd degrees we see that in this spectral sequence: r

E

.

= 0 if N is odd.

N •

d 2r +1 = 0

0.3.2)

E2r +1

and

= E2r +2 for all r.

The Hurewicz homomorphism is given in terms of this spectral sequence by the following commutative square:

--------------.7

nNBP (1.3.3)

1

N

t"

>

EOO

H BP

-) E2

N.O

N.O

Since h is one-to-one. it follows that: oo

EN.t

0.3.4)

EIX>

(1.3.5)

-.0

{

if t

0

n BP N

2(2)[\"

if t

*'

0

and

0

···yN····)·

Thus. there must be nonzero differentials originating on the 0 row so that each monomial K(2- e y e (1 ) ••• ye(M») in divisible by 2

e

1

!'I

survives to Eoo if and only if K is

where e = e(l)+···+e(M).

We will prove in Chapter 4 that. in

our range of computations. all nonzero differentials which originate on the

o

row land in ImJ

@

H.BP.

We will assume that ImJ is known.

The first step

in our analysis of the spectral sequence (1.3.1) will be to compute all these differentials which originate on the 0 row in degrees 2 through 70.

This

computation is entirely algorithmic. is done by computer with no human assistance and is carried out in Section 4.4.

The purpose of this computation

is to record the cokernels of all of these differentials.

The behavior of the following elements in the spectral sequence is the key to

the determination of differentials which originate above the 0 row.

7

(a)

5

have order q and let V E H BP. Assume that: 2N 2 2r E survives to an element of E for some 2 r00; 2N,t 2N,t

DEFINITION 1.3.6 E

Let

E R

t

(b)

if r =

(cl

2s we know all differentials which originate or land on elements of E

00

then V = 0; 2k, t

which have a where N'

in Z

q

=N

if r /leg) then -(f A g)

=

(f /I g) A h

(-f)e(-g) 0

sw

PROOF:

0

=

g)

(1

sw

/If)

0

(1

sw

/I g).

h) = (f A g) o (f /I h l .

= f A ( -gl.

The proofs of these properties are straightforward and are left to

the reader .• NOTATION: f

1

/1.•. /1

In view of property (e) above, -f

(-f) /I... A f k

where M(f )

t

= min(M(f

1

/1..• /1

f

will mean

t

), ... ,/l(f )).

k I t

We state next a useful technical result which says that A can be 1\ defined from any fixed set of

THEOREM 2.2.6 for 1

oS

t.

where {G 10

oS

oS

i Ij

the given {G

Assume that is defined. t

AI j

t

.

Let G

i-l) i

represent X.

Then any element Z of has a representatives 1

i < j

oS

11

i

1-1, I

PROOF.

representatives of Xl'" ,X

Let {A 10 IJ

oS oS

t, (i,j) oS

t

1

Gat

(O,t)} is a defining system which contains

t}.

i < j

oS

is a representative of Z.

t, (i,j)

(O,t)} be a defining system such that

By induction on k = j

i

1, we construct a

18

defining system {G H

[Doma l n l C 1\ G

IJ

Ir

given,

)

rJ

and

/

I

=

Ir

for i < r < j.

rJ

and H

st

G )

IJ

r J'

define a homotopy H =

UJ - 1

r=l+l

(H

Ir

1\ H

rJ

Define G

= H

IJ

IJ

IDomain (G

IJ

A

) from I

IJ

1--1,1

the G

are

1-1,1

both represent XI'

By the induction hypothesis the

extension property, there is a homotopy H A...

and G

1,

i < r < j, agree where their domains intersect and thus

homotopies H 1\ H Ir

=

is homeomorphic to some (DN,SN),

IJ

it has the homotopy extension property.

such that

have been constructed for

st

Since (Domain G ,Domain

IJ

When k

1-1,1

k and assume that the G

to G

IJ

can be found since A

1-1,1

1 :!5 t-s < k.

from A

IJ

H 1\ H

and the H

Let j-i

homotopies H

IJ

to

of A

J

I

G

IJ

By the homotopy

•

which extends both Hand

J

This completes the inductive step.

x {l}).

Thus we have constructed a defining system {G } and a homotopy I

t

1

Ur=l

(H 1\ Or

J

Aat to Gat .•

H ) from rt

Observe that the three-fold Toda bracket

O.

4

O.

l+N +N 2

3

X '1[S 4 1+N

O.

2+N 3

(e)

S

If Y E 1[l+N +N then Y 2

X

X.

X

X. 4

1

(g)

2

3

3

Y

1

+

Y

2

such

that

X .y 1

1

o and X .y 4

2

O.

19

PROOF:

We use the smash product and the smash product Toda bracket of

Definition 2.2.2 throughout the proof. (a)

Let

G

G

1 2,

G

2 3,

G

3 4,

13,

. 4

defines 0 in

1

2

be a defining system for

24

.

There are other choices G'

we can find a G

+ ·X

i '

34

,G

*

02

(0,4)} be a defining system for

,G

13

,G

03

to a defining

}

(-l,3)} of ,X3,X/.

Assume that is defined and that X • X = O. o 1 2 3 3 4

Then

'X c (_l)NO)+l is defined and contains (_U ,X ,X >. 01234

23

34

G

,

(i,j)

*

represent

X

2,

S

1J

,X ,X> C 01234

(d)

This identity follows from the identity in (c) by Theorem 2.3.4(a) .•

(_llN(O)+l.•

31

THEOREM 2.4.5 Let X E (a)

is defined if and only if X·M

projects to d (b)

8(X·M

M).

10 2 projects to d (X·M M).

10, Then X·M2M survives to E and 1 2

Moreover,

1 2

Let G:SV A SU

In this case

.

Moreover, vX is divisible by l/.

1 2

Assume that is defined.

PROOF.

8

survives to E

1M2

is divisible by v. S

SU represent X E

We use the smash

product Toda bracket of Definition 2.2.2 throughout this proof and the notation of the proof of the preceding theorem. (a)

XM 6

to E

=d d

4

survives to E

1M2

if and only if l/X

O.

if and only if vX is divisible by l/, i.e. 0

6(XM)M

2

6(XM)

1 E

and d

(v) ,

if 0 E survi ves Then XM

4

1

B

vx

B

Let M represent

4 and that we have found for

c {Z vM MO 8

3.

Figure 3.3.2 shows that all such cycles are in the image of d S = {Z(2l

of H.BP which

N>

2M2

1

44 6

The final application of our two lemmas is to compute E

We begin by

.

strengthening the two lemmas to obtain a global calculation of all d and all d

4-boundaries.

DEFINITION 3.3.9

4-cycles

First we introduce an important algebra.

Let B 0";1 - - - - . 7 ) 0"2 +M2 _ _ _--') 0"2(M M2+M M4)

0"M 0";1 2;1 2+M5+M M 1

3

321

2+M ) 1131312 4>. +M M +M M +M M2M 10M M ifM +M , 2

(viii) > + > + M M 2 y22 2. 2>2, {2 (20-){

2V >+2Y +M2M 40"'M 2> 2y 4>2 32. @ 2 Y 8 M3M 2+2V22+85 1123 131 1221 221 1 123 2>3+ y32+ y 2>3+ 2>4 4> 4> ) i; 8 2> 4> 2> ,

(2)

= Z

[ 2, , b

fb+l rrS . a

00

implies that 0 is an elements of the Toda bracket.

This explains

why we require the stronger hypothesis in (b) when i equals 4 or 5.

THEOREM 4.2.4

(a)

contains 0 for N

(a), (c), (d), (e), (k),

Theorem 4.2.3(a) with

I'

=2

l. (1)

From Figure 4.2.2. we see that

applies to show that each of these Toda brackets ESN+S, 4N+2,

has an element which projects to zero E8N+14,4N+5 00

co

'

(d). (e). (k l ,

v which

N

Thus,

can

From Figure 4.2.2, we see

that there is now no possibility for to contain a nonzero element. In (L), ll = v = {OJ while N

N

be an element of . N

N+l

)

'*

O.

Hence

x-i can not

Thus. we see from Figure 4.2.2 that there is now

79 no possibility for

to contain a nonzero element.

All the other

triple products in (f) - (j) contain zero by (a) - (e) or must equal zero by Theorem 4.2.3(a) and Figure 4.2.2. Assume that N is odd in case (f). case (i) when N is odd. Let r

=2

Let r

=3

in cases (f), (g), (j) and in

in case (h) and in case (i) when N is even.

From Figure 4.2.2, we see that Theorem 4.2.3(b) applies to show that each of . . . 8N+I0 4N+2 these Toda brackets has an element WhICh pr-ojects to zero In Eoo ' , E8N+16,4N+I E8N+10,4N E8N+12,4N+2 E8N+12,4N in case (f), 0)

,

(g), (h),

'00

('X)

'

with N odd,

(1)

(1)

00

'00

with N even, (j), respectively.

With four

exceptions, E is zero in each of these degrees in higher filtration degrees: oo

However, 1}O'.N+I' 1}'lN+1 ' 1}O'.N+I' 2

1}'lN+I is in the indeterminacy of

elements of n

S 26

,

n

N is 2,

4 or 6.

We shall see that there are only two

S and nS that are not contained in (1),v), and (1),v) is 42

58

contained in the indetermincacy of and 2C(42].

Thus,

2

,1»

The two exceptions are C(42]

and

6

,1»

contain O.

= because

=

=

these properties. 4C(42].

(1),V'O'.N'V>,

Thus, each of these Toda brackets contains O.

, respectively.

Now consider (f) when

,vA[19],v> However, v'rr

S 28

= vA[8]C[20] = O.

Hence 2A[31]

(V).

0.1

The computations of Section 4 show that we have the following leaders.

Since

this is the last table of leaders of this chapter, we include the leaders of all degrees.

113

Degree

Leader

Row

Degree

9

33

2o1} M5M 1 3

22

62

11

33

2/3 M 1 1

23

33

14

34

23

35

15

37

23

63

16

34

A[ 16]tfM 1 2

24

60

2M 1 2 3M 4vC[20]M 1 2 20 1 2M1 15M 1} A[23]M 1 2

17

51

0:

28

36

A[8]C[20]M

18

38

C[ 18]

30

34

A[30]M

55

18 /3 2M1

31

33

31

35

1}A[30]M 1 2 A[31]M 1

Row

19 21

33 FIGURE 5.3.7:

4.

l l

1 2 tf1 1 1M2

MUM 2 1 2 1 2

1 2

)

3M

vcr 18]M

1 2

Leader 7M2 A[23]M

1M2

2 1

Leaders in Rows 1 to 31 of Degree at Least 33

Tentative Differentials

In this section we give the tentative differentials determined by the differentials on leaders of degrees less than or equal to 32 which were determined in this chapter.

We omit the differentials originating on the 7

row since they were determined in Chapter 3.

Recall that these differentials

are tentative in the sense that they are only valid under the assumption that there are no hidden differentials interfering with the computation. We order the differentials by row for easy reference.

We use the same

notation as in Section 4.4 to display the bases in the various bidegrees. In a Z2-vector space we omit the group in front of each basis element, and monomials which are to be added are bunched together. first basis element in degree (28,9) below.

For example, see the

114

a

DEGREE 9:

and

1

The leading differential d

14

M) =

( 2 uM7 1 2

3 1

VA[19]M

determines tentative

differentials by assigning the following values to monomials of degree 29 of [2

2

1

)

(j)

2 a ] 21

@

8:

monomials are assigned

o.

given by the table below.

W and

12

12

are assigned 1 and all other

The kernel of these tentative differentials is In this table as well as in the following one, the

monomials with an even factor of M have coefficient a 1

1

while the monomials

with an odd factor of M have coefficient 1

DEGREE (18,9)

(28,9)

DEGREE 6 100

(22,9)

9 100

(26,9)

5 3 0 0

DEGREE 11 0

o

0

(24,9)

6 0 1 0

5 0 1 0 7 2 0 0

11 1 0 0

(30,9)

5 1 1 0

15 0 0 0

(32,9)

6

7 0 1 0 6 300 13 1 0 0 (36,9)

350 0

11 0 1 0

(42,9)

(34,9)

(38,9)

1 0

o

0

11 2 0 0

14 1

5 210 15 100

9 300 15 100

3 3 1 0

6 2 1 0 (40,9)

4 3 1 0

9 1 1 0

13 2 0 0

550 0

11 300

13 0 1 0

7 0 2 0

6 5 0 0

11 1 1 0

14 0 1 0

15 2 0 0

7 0 o 1 7 5 o 0 15 0 1 0 13 3 0 0 15 0 1 0 5 6

(46,9)

o0

14 3 0 0 3 700

(48,9)

(44,9)

5 120 15 0 1 0

6 3 1 0

9 2 1 0 15 0 1 0

510 1

6 120

11 4 0 0

13 1 1 0

320 1

6 1 0 1

5 4 1 0

9 5 0 0

115

(52,9)

(54,9)

11 2 1 0

14 1 1 0

5 2 2 0

6 4 1 0

7 600

9 3 1 0 15 1 1 0

11 0 2 0

22 1 0 0

1 6 1 0 520 1 13 2 1 0

3 3 2 0

503 0 13 2 1 0

520 1 5 7 0 0 13 2 1 0

5 7 o 0 7 4 1 0

9 120 13 210

11 0 0 1

11 5 0 0

13 2 1 0 23 1 o 0

330 1

5 0 1 1

6 0 3 0

5 5 1 0

7 2 2 0

9 1

6 7

(56,9)

3 5 1 0

o

1

0

11 3 1 0

14 2 1 0

15 4 0 0

21 200

27 0

3 111

6 0 1 1

3 6 1 0

5 320 703 0 15 2 1 0

7 2 o 1 7 7 o 0 15 2 1 0

6 5 1 0

11 120

12 3 1 0

13 500

21 0 1 0

25 100

5 5 7 7

130 3 0 1 0 1 1 5 1 0

11 1 14 5 (60,9)

o

(50,9)

o o

530 7 0 1 7 5 1 13 3 1

1 1 0 0

1

11 6 0 0

0

22

o

(58,9)

o

0

1 7 1 0 7 5 1 0 6 3 2 0

15

o

2 0

1 0

23 2 0 0

3 2 3 0

340 1

5 1 1 1 15 o 0 1

6 130

6 3 0 1

5 6 1 0 15 5 0 0

11 410

13 1 2 a 15 0 a 1 15 5 a 0

a

21 3 a 0 23 a 1 a

9 7 15 5

a a

0 0

15 0 a 1 15 5 a 0 23 a 1 a

14 3 1

116

27 1 0 0 The leading differential

1

1

determines tentative

differentials by assigning the following values to monomials of degree 27 of [2

2

1

)

ttl

2 a: J

assigned 1. below.

21

®

B:

ex M M is assigned 0 and all other monomials are 2

113

The kernel of these tentative differentials is given by the table

The new leader is

2 . .5-

cTM M . 1 3

DEGREE

BASIS

DEGREE

BASIS

DEGREE

BASIS

(24,9)

5 0 1 0

(26,9)

720 0

(28,9)

5 300 7 0 1 0 11 1 0 0

(30,9)

5 1 1 0 15 0 0 0

(36,9)

0 0 0 0

(38,9)

3 3 1 0 9 1 1 0 13 2 o 0

(40,9)

550 0 13 0 1 0 4 3 1 0

(42,9)

7 0 2 0

(44,9)

512 0 700 1 750 0

(46,9)

510 1 11 4 0 0

(48,9)

320 1 9 5 0 0 610 1

(50,9)

5 220 7 600 11 020

(52,9)

1 7 13 23

1 1 1 0

0 0 0 0

3 5 5 7 9 11

(54,9)

330 501 910 670 14 2 1

1 1 1 0 0

501 1 15 4 0 0

7 220 21 200

3 1 1 1 13 5 0 0 601 1

532 703 720 770 11 1 2

6 4 2 1

21 2 0 0 27 o 0 0

(56,9)

3 5 11 15

5 2 0 1

0 1 1 0

3 2 0 0 3 0 7 0 0 4 1 0 120 5 0 0

5 0 3 520 741 11 0 0

0 1 0 1

0 0 1 0 0

117 (5S,9)

21 1 0 1

23 2 0 0

5 1 3 0

530 1 7 7 11 11

0 5 1 6

1 1 0 0

1 0 1 0

15 0 2 0 14 5 0 0

22 0 1 0 (60,9)

3 2 3 0 340 1

340 1 5 1 1 11 4 1 15 5 0 23 0 1

5 6 1 0 11 4 1 0

13 1 2 0 15 0 0 1

21 3 0 0

23 0 1 0 27 1 0 0

1 0 0 0

21 3 0 0

23 0 1 0 6 130

DEGREE 9:

7jA[S]

IJ

3

The leading differential d 2(A[S]M ) 1

= 7jA[S]

determines tentative differentials

with cokernel Z 7jA[S]M 0 8, and the 7jA[S]-leader is 7jA[S]M . 2 1 1

=

The leading differential d 4(1J2 M3 ) 1

3

-

1

1 2

with image Z 7jA[S]{M ,M ,M M} 2

Z

2

1 2

)

1

0

0

7jA[S]M determines tentative differentials

8.

1

The remaining elements are

8, and the new 7jA[S]-leader is

= C[20]

The leading differential

.

determines tentative

differentials which are a monomorphism on Z (7jA[S]M 2

1 2

3M

1 2

)

0

8.

Thus, there

are no remaining elements.

DEGREE 11:

13 1

The leading differential d 16 ( f3 MIl) = v2C[20]M3 determines tentative 1 1

1

differentials by assigning the following values to monomials of degree 33 of

13 Ml 1 is assigned 1 and all other monomials are assigned O. 1 1

kernel of these tentative differentials is given by the table below. lO

13 1 -leader is 13 1 M1

.

The

The new

118

DEGREE GROUP GENERATOR

DEGREE GROUP GENERATOR DEGREE GROUP

(20, III Z

100 0 0

(22, III Z

2/

9 100

(26,11) Z 2

1/ 6 0 1 a 2/ 10 1 0 0

2

2/ 11 1 3/ 14 a

Z

2 2

(28, III 2

(30,11) 2

2

2

2

2

(34, III 2 2

2

2

2 2

(38, III Z 2

(40,11) 2 2

2 2

2

4

2

2/ 11 100 4 1 1 a 3/ 14 a a 0

2

2/ 12 1 0 0

2

2/ 7 3 0 0 1/ 13 1 0 0

2

1/ 7 110 6/ 14 100

Z

2/ 14 100

(36, III Z

1/

1/ 9 3 2/ 15 1 3/ 12 2

7 11 13 4 10 14

2 3 0 3 1 2

2

2

2/

830 0

Z

3 o 0 3 1 0 1 1 0 2 0 0

2

aa aa

2

a a a

2

a a

z

1/ 9 1 1 2/ 10 3 0 6/ 12 0 1

1 0 1 1 1 0

0 0 0 0 0 0

Z

2/ 11 3 0 0 1/ 14 2 0 0

2

1/ 7 2 1 0 2/ 10 1 1 0 1/ 14 2 0 0

(42, III 2

6/ 5 3 1 6/ 11 1 1 1/ 600 6/ 14 0 1

2

2

2

2

(44,11) Z II 410 1 2 61 6 3 1 0 6/ 12 1 1 0

2

2

5/ 13 1 0 0 1/ 10 2 0 0

6/ 15 1 1/ 12 2

0 0 1 0

2

(32,11) 2

z2

2

2

2 4

z2

6 3 0 0

3/ 9 1/ 2 3/ 8 2/ 12

6/ 2/ 4/ 1/ 6/ 7/

6 2 0

2

2

aa a0 aa

2/ 6 2 1 2/ 10 3 a

3/ 6/ 6/ 1/ 3/ 6/

2

(24, III Z

a0 aa

4

2

8 100

14 000

7 300 6 110

2

2

a

2/ 10 1 0 0

4

4

GENERATOR

1/ 730 0 5/ 13 1 0 0

2

1/ 10 0 1 0 3/ 14 1 0 0

2

2/ 2/

4

2

9 3 o 8 1 1

2/ 15 1

a a

aa

1/ 10 3 a 6/ 12 a 1

a a

6 2 1 0

4

2/ 6/ 4/ 2/ 1/

7 11 13 10 14

2 3 0 1 2

1 0 1 1 0

0 0 0 0 0

0 0 0 0 0 0

2

1/ 13 0 1 0 2/ 14 2 0 0

2

2/ 5 3 1 0 2/ 11 1 1 0 6/ 12 3 0 0

21/8210 2 5/ 12 3 0 0 3/ 14 0 1 0

2/ 12 3 0 0

2

7 11 13 6 10 14

2 3 0 0 1 2

1 0 1 2 1 0

2

14 2 0 0

4

5 11 12 14

3 1 3 0

1 1 0 1

0 0 0 0

2/ 13 6 5/ 10 2/ 12

3 3 4 1

0 1 0 1

0 0 0 0

4

2/ 11 1 1 0

1/ 4 6 0 0 2 3/ 10 4 0 0

Z 2

21

119 2

(46,11) 2

2

2

2

2

(48, 11) 2 2

Z 2

Z 2

(50,11) Z 2

Z

2

2

2

(52,11) Z

2

2/ 13 3 0 0 1/ 10 4 0 0

2

2

1/ 4 4 1 0 1/ 6 120 3/ 10 2 1 0 1/ 14 3 0 0

2

1/ 13 1 0 1/ 10 2 1 0 5/ 14 3 0 0

2

1/ 4 2 2 0 6/ 610 1 3/ 6 6 0 0 6/ 8 3 1 0 1/ 10 0 2 0

Z

1/ 2/

6 6 o 0 8 3 1 0

2

2

2

7/ 13 3 o 0 1/ 15 0 1 0 1/ 12 1 1 0

2

2/ 6 1 2 0 6/ 10 2 1 0

2

2/ 14 3 0 0

2

Z

2

2/ 11 2 1 0 2/ 15 3 0 0 6/ 14 1 1 0

Z

3/ 7 1 o 1 6/ 15 1 1 0 1/ 4 2 o 1 6/ 4 7 o 0 6/ 6 4 1 0 3/ 10 001 6/ 10 500

Z

2/ 15 1 1 0 1/1000 1 3/ 10 5 0 0

Z

2/

8 120

Z

3/ 5 7 0 0 1/ 11 5 0 0 3/ 13 2 1 0 1/ 230 1 1/ 4 5 1 0 1/ 810 1 3/ 26 0 0 0

Z

Z

2

4

2

Z

2

2

2

2

Z 2/ 13 3 0 0 4

2

8

2/ 8 5 o 0 2/ 10 2 1 0 2/ 14 3 o 0 1/ 7 3 1 0 5/ 13 1 1 0

6 120

4

2

6 3 1 0

4

5/ 6/ 7/ 5/ 1/ 7/ 3/ 6/

7 120 950 0 11 2 1 0 15 3 0 0 610 1 8 3 1 0 10 0 2 0 14 1 1 0

2

7/ 7/ 1/ 5/ 6/

11 15 8 10 14

0 0 0 0 0

Z

9 5 0 0

Z

1/

2 3 3 0 1

1 0 1 2 1

2

2

4

2/ 15 3 0 0 1/ 7 1 o 1 2/ 15 1 1 0 2/ 4 7 o 0 2/ 6 4 1 0 2/

Z

2

2/ 4/ 6/ 2/ 3/ 6/ 2/

7 9 11 15 6 10 14

120 5 0 0 2 1 0 3 0 0 6 0 0 0 2 0 1 1 0

2/ 2/ 1/ 2/

11 15 10 14

2 3 0 1

1 0 2 1

0 0 0 0

1/ 7 120 6/ 15 3 0 0 6/ 14 1 1 0 6/ 15 1 1 0 1/ 4 7 0 0 6/ 8 120 3/ 10 5 0 0 3/ 12 2 1 0

6 4 1 0

1/ 9 3 1 0 4/ 15 110 2/ 10 500

Z

2/ 12 2 1 0

Z

7/ 5 700 3/ 9 1 2 0 3/ 11 500 5/ 13 210 1/ 401 1 3/ 4 5 1 0 6/ 6 2 2 0 6/ 810 1 3/ 14 4 0 0

Z

2

4 2

4/ 15 1 1 0 2/ 10 5 0 0 2/ 12 2 1 0 2/ 15 110 7/ 5 7 0 0 3/ 7 4 1 0 6/ 9 120 7/ 11 5 0 0 2/ 13 2 1 0 1/ 4 5 1 0 7/ 6 2 2 0 6/ 14 4 0 0

120 2 2

2 2

2

2/ 7 4 1 0 4/ 9 1 2 0 2/ 11 5 0 0 1/ 6 2 2 0 2/ 8 1 0 1 3/ 14 4 0 0 5/ 26 0 0 0

2

6/ 5 7 0 0 4/ 13 2 1 0 2/ 8 1 0 1 7/ 14 4 0 0 5/ 26 0 0 0

2

1/ 7 4 1 0 4/ 9 1 2 0 4/11500 7/ 13 2 1 0 1/ 14 4 0 0

21/9120 2 3/ 26 0 0 0

Z

26 0 0 0

2

(54,11) 2

2

Z

2

2

1/ 5 5 1 0 21/ 2610 1/ 6 7 0 0 2/ 12 5 0 0 2/ 14 2 1 0 2/ 24 1 0 0

Z

1/ 14 4 0 0

3/ 2 1/ 1/

2/ 2/ 3/ 1/ 3/ 1/ 1/

2 2

2 2

Z

2/ 5 5 2/ 9 1 6/ 14 2 6/ 20 0

Z

0 1 0 0 1 0 0 0 0 0

1/ 9101 2 2/ 11 3 1 0 2/ 6 7 0 0 6/ 10 1 2 0 6/ 12 5 0 0 6/ 24 1 0 0

Z 2/ 14 2 1 0

1/

5510 11 3 1 0 6700 12 5 0 0 14 2 1 0 24 1 0 0

Z

7 7 0 0 0710 4 1 3 0 4301 6 5 1 0 10 1 0 1 10 6 0 0 12 3 1 0 14 0 2 0 22 2 0 0

2

1/ 3/ 1/ 7/ 2

0 1 0 0

551 910 4 3 2 6 0 3 620 6 7 0 10 1 2 12 5 0 20 0 1 24 1 0

2/ 10 1 2 0 2/ 12 5 0 0 2/ 24 1 0 0

4 6/

(56,11) Z

1 0 1 1

2/ 1/ 2/ 1/ 1/ 1/

2/ 3/ 2/ 2/

2 2/ 24 1 0 0

4

1/ 6700 1/ 14 2 1 0 2/ 24 1 0 0

4/ 2

2

2/ 10 3 1 0

2/ 13 2 1 0

4

2

7 7 0 0

6/ 11 1 2 0 1/ 4130 7/ 4 3 0 1 3/ 6 0 1 1 7/ 6 5 1 0 5/ 10 1 0 1 1/ 14 0 2 0

7/ 22 2 0 0

2/ 5 7 0 2/ 7 4 1 4/ 9 1 2 2/ 13 2 1 3/ 14 4 0 7/ 26 0 0

0 0 0 0 0 0

2/ 11 5 0 2/ 13 2 1 5/ 14 4 0 5/ 26 0 0

0 0 0 0

Z

14 4 0 0

4

2

2/

5 5 1 0

2 7/

9 1 0 1

1/ 6201 5/ 6 7 0 0 2/ 10 1 2 0 5/ 14 2 1 0 2/ 24 1 0 0

Z 2

2/ 2/

6 0 3 0 6 7 0 0

Z 2/ 10 2 2/ 12 3/ 20 2/ 24

Z 4

Z

4

1 5 0 1

2 0 1 0

0 0 0 0

1/ 6 0 1/ 6 7 1/ 14 2 2/ 24 1

3 0 1 0

0 0 0 0

2/ 11 3 1 0

Z 3/ 5 3 2 0 21/ 7030 4/ 7 7 0 0 3/ 11 1 2 0 2/ 13 5 0 0 3/ 15 2 1 0 1/ 25 1 0 0 1/ 4 3 0 1 5/ 6 0 1 1 1/ 6 5 1 0 3/ 10 1 0 1

1/ 12 3 1 0 7/ 14 0 2 0 3/ 22 2 0 0

121

Z 2

Z 2

Z 2

Z

(58,11) Z

5/

5 3 2 0

2/ 1/ 7/ 6/ 1/ 2/ 6/ 6/ 6/ 7/ 6/

7 0 3 0 7700 11 1 2 0 13 5 0 0 6 0 1 1 6 5 1 0 10 1 0 1 10 6 0 0 12 3 1 0 14 0 2 0 22 2 0 0

3/

3/ 3/ 1/ 6/ 2/ 1/ 6/ 7/

7 7 0 0

15 25 6 10 10 12 14 22

2 1 5 1 6 3 0 2

1 0 1 0 0 1 2 0

0 0 0 1 0 0 0 0

7/ 13 5 0 0 4/ 15 2 1 0 4/ 25 1 0 0 1/10600 2/ 12 3 1 0 2/ 22 2 0 0

1/ 41/ 1/ 1/ 6/ 2/ 1/ 2/

Z 2

Z 2

Z

Z 2

5320 7030 7700 11 1 2 0 15 2 1 0 25 1 0 0 14 0 2 0 22 2 0 0

Z

5 3 0 1 7 5 1 0 1/11101 1/ 13 3 1 0 1/ 4111 1/ 4610 1/ 8700 2/ 14 0 0 1 2/ 14 5 0 0 3/ 22 0 1 0 1/ 26 1 0 0

Z

1/

2 3/

2

4

2

2/ 4/ 2/ 6/ 4/ 2/ 6/

5 7 7 11 13 12 14

3 0 7 1 5 3 0

2 3 0 2 0 1 2

0 0 0 0 0 0 0

Z

6/ 2/ 7/ 6/ 3/

7 10 10 14 22

7 1 6 0 2

0 0 0 2 0

0 1 0 0 0

Z

4/

2

2

7 0 3 0

2/ 4/ 4/ 6/ 6/ 5/ 2/

7 15 25 6 10 10 14

7 2 1 5 1 6 0

0 1 0 1 0 0 2

0 0 0 0 1 0 0

2/ 1/ 6/ 2/ 5/

7 10 12 14 22

7 6 3 0 2

0 0 1 2 0

0 0 0 0 0

2/ 4/ 6/ 3/

13 15 25 22

5 o 0 2 1 0 1 0 0 2 0 0

2/ 11 1 2 0 1/ 14 0 2 0 7/ 2/ 5/ 2/ 1/

13 15 25 12 22

5 2 1 3 2

o

0 0 0 0 0

Z

1 0 1 0

11 11 4/ 3/ 5/

7 13 25 14 22

0 5 1 0 2

3 0 0 2 0

0 0 0 0 0

Z

2/ 3/

530 701 7 5 1 11 1 0 4 6 1 8 7 0 12 1 2 14 0 0 14 5 0 22 0 1 26 1 0

1 1 0 1 0 0 0 1 0 0 0

6/

2/ 2/ 2/ 6/ 7/ 7/ 6/ 2/

Z

2

4

1/ 13 500 2/ 25 100 3/ 22 200 14 0 2 0

4

Z 2

1/ 6/ 6/

2/ 6/ 6/

5/ 6/ 5/

7 7 11 13 8 14 20 22 26

0 1 1 5 1 0 101 310 700 500 300 010 100

122 2 2

2

2

2

2

2

4

2

4

2/ 4 6 1 0 2/ 6 3 2 0 2/ 14 5 o 0 2/ 26 1 o 0 2/

2

2

5 1 0 3 1 0 5 0 0 0 1 0 1 0 0

2

2/ 8 7 0 0 2/ 12 1 2 0

Z

2/ 6/ 6/ 2/ 6/

7 13 14 22 26

6 3 2 0

6/ 13 3 1 0 1110410 2/ 12 1 2 0 11 14 5 0 0 3/ 20 3 0 0 2/ 22 0 1 0 1126100

2

1/ 5 3 0 1/ 11 1 0 11 870 5/ 22 0 1

1 1 0 0

Z

1/ 3/ 1/ 1/ 6/ 1/ 2/

0 0 0 0 1 0 0 0 0

2

7 13 4 6 14 14 20 5/ 22 2/ 26

5 3 6 3 0 5 3 0 1

1 1 1 2 0 0 0 1 0

The leading differential

2

2 2

4

2/ 2 2/ 2/ 11 11 7/ 11 3/ 2

2/ 12 1 2 0 6/ 26 1 0 0

2/ 1/ 7/ 3/

13 14 14 22

3 0 5 0

1 0 0 1

0 1 0 0

Z

4

11 1 0 13 3 1 870 14 0 0 14 5 0 20 3 0 22 0 1 26 1 0

1 0 0 1 0 0 0 0

2/ 14 5 0 3/ 20 3 0 6/ 22 0 1 5/ 26 1 0

0 0 0 0

2/ 13 3 1 0

5/ 20 3 0 0 5/ 26 1 0 0

6 320 26 100

4

l G) M = A[8]C[20]M

1 1

1

determines tentative

differentials by assigning the following values to monomials of degree 31 of lG M is assigned 1 and all other monomials are assigned O.

The

1 1

kernel of these tentative differentials is given by the table below. 1

-leader ia

DEGREE

GROUP

8-

GENERATOR

DEGREE

8 100

(24,11) 2

11 6 0 1 0 e 2/ 10 1 0 0

(28.11) Z

(30,11) Z 2/ 12 1 0 0 2

Z

(22,11) 2

2

2/

The new

Z

GENERATOR

DEGREE

6 200

(26,11) 2

2/ 11 1 o 0 11 4 1 1 0 3/ 14 0 0 0

Z

GROUP 2 2

4

630 0

GROUP 2 4

GENERATOR

2/ 10 1 0 0 2/ 11 1 0 0 2/ 14 0 0 0

(32,11) Z 3/ 7 3 0 2 4/ 13 1 0 11 6 1 1 7/ 10 2 0

0 0 0 0

123

Z 4

(36,11) Z 2

(38,11) Z 2

(40,11) Z 2

1/ 7 300 7/ 10 2 0 0

(34,11) Z 2

Z

2/ 10 3 0 0 6/ 12 0 1 0

Z

2

6/ 7 4/ 13 1/ 6 6/ 10 6/ 14

0 0 0 0 0

Z

Z

3/ 6/ 6/ 1/ 3/ 6/

Z

0 3 1 2

1 1 1 0

2

9 3 o 0 8 1 1 0

2/ 6 2 1 0 2/ 10 3 0 0

13 4 10 14

Z

2

2/ 2/

Z

7 2 1 0

Z

2

2/ 9 300 6/ 15 100 1/ 2 3 1 0 3/ 8 1 1 0 7/ 12 2 0 0

11 3 0 0

1/ 7 1 1 0 7/ 10 0 1 0 3/ 14 1 o 0

2

0 0 0 0

2 0 0 1 2

1 1 2 1 0

4

2

1/ 7 2 1 0 2/ 10 1 1 0 1/ 14 2 0 0

Z

2/ 11 3 0 0

(42, 11) Z

Z

6/ 5 3 1 0 6/ 11 110 1/ 6 0 0 1 6/ 14 0 1 0

Z

2/ 12 3 0 0

Z

Z

2/ 11 1 1 0

6/ 13 3 0 0 11 410 1 6/ 6 3 1 0 7/ 10 4 0 0 6/ 12 1 1 0

Z

2/ 6 3 1 0 2/ 12 1 1 0

2/ 8 5 0 0 2/ 10 2 1 0 2/ 14 3 0 0

Z

2

(44,11) Z

2

(46,11) Z 2

Z 8

Z 2

11 7 3 1 4/ 13 1 1 7/ 10 2 1 1/ 14 3 0

4

2 4

2

2

4

2/ 6 1 2 0 6/ 10 2 1 0

6 2 1 0 2/ 7 2 1 4/ 11 3 0 4/ 13 0 1 2/ 10 1 1

2/ 5 3 1 0 2/ 11 1 1 0 6/ 12 3 o 0 5 11 12 14

(48,11) Z

2/ 7 120 4/ 950 0 6/ 11 2 1 0 2/ 15 3 0 0 2/ 660 0 2/ 8 3 1 0 6/ 10 0 2 0 6/ 14 1 1 0

Z

2

2

310 110 300 010

11 4 6 o 0 7/ 6 3 1 0 3/ 104 o 0

2/ 13 3 0 0

Z

2/ 14 3 0 0

Z

6 120

2 4

0 0 0 0

0 0 0 0

Z

4

2

2/ 15 1 0 0

4

Z

8 300

2/ 14 100

4

Z

4

2/

5/ 6/ 4/ 2/ 1/ 7/ 2/

7 120 9 500 11 2 1 0 15 3 0 0 610 1 10 0 2 0 14 1 1 0

0 1 0 0

Z

2/ 11 2 1 0 2/ 15 3 0 0 6/ 14 1 1 0

Z

1/ 7 120 6/ 15 3 0 0 7/ 10 0 2 0 2/ 14 1 1 0

Z

2/ 15 300

11 6/ 3/ 6/

4 2 2 610 6 6 0 8 3 1

2

4

4

124

(50, III Z 2

z2 (52,11) Z 2

Z 2

Z

(54, III Z

Z

Z

2

2

4

4

4/ 15 1 1 2/ 10 5 a 2/ 12 2 1

a a a

z2

2/

a

a

z4

2/ 15 1 1 0

5 7 a a 9 120 11 5 a 0 13 2 1 a 4 a 1 1 4 5 1 a 622 0 810 1 14 4 0 0 26 0 0 a

z

7/ 5 7 a a 6/ 9 120 1/ 11 500 3/ 13 2 1 0 11 4 5 1 0 6/ 6 2 2 0 2/ 810 1 2/ 14 4 a a 3/ 26 0 a a

0 0 0 1

z

2/ 5 7 0 0 2/ 7 4 1 0 4/ 9 1 2 0 6/ 11 5 0 a 6/ 14 4 0 a 2/ 26 0 0 a

0

z4

2/ 13 2 1 0

z

2/ 2/ 3/ 2/

10 1 2 0 12 5 0 a 20 a 1 a 24 1 a 0

z

2

2/ 2/

6 0 3 0 6 7 0 0

Z 2

2/ 14 2 1 0 2/ 24 1 0 0

3/ 710 1 4/ 15 1 1 a 1/ 420 1 6/ 4 7 a a 6/ 6 4 1 a 2/ 10 a a 1 3/ 10 5 a a

Z

2/

8 120

Z

3/ 5 7 a a 6/ 7 4 1 a 41 9 120 7/ 11 5 a a 3/ 13 2 1 0 11 230 1 11 4 5 1 0 7/ 6 2 2 a 7/ 810 1 5/ 14 4 a 0 6/ 26 a o 0

z

7/ 2/ 3/ 5/ 1/ 3/ 6/ 6/ 7/ 5/

2/ 5 7 0 a 2/ 7 4 1 0 4/ 9 120 2/ 11 5 0 a 4/ 13 2 1 a 11 6 220 11 26 0 0 0

z

6/ 5 7 a 6/ 11 5 0 2/ 13 2 1 2/ 810 2/ 14 4 a

a

2

2

2

2

11 710 1 2/ 470 a 2/ 6 4 1 a 7/ 10 a a 1 5/ 10 5 a a

2/ 12 2 1

a

z4

2/ 11 5

2/ 2/ 11 11 2/ 11 7/ 7/ 5/ 5/ 1/ 1/

5 5 1 9 1 a 261 432 6 a 3 620 6 7 a 10 1 2 12 5 0 14 2 1 20 0 1 24 1 0

0 1

Z

2/ 551 a 2/ 9 1 0 1 6/ 14 2 1 0 6/ 20 0 1 0

6/ 11 3 1 11 4 3 2 11 620 11 670 11 10 1 2 2/ 24 1 0

0 0 1 0 0 0

a 0 0 1 0 0 0 0 0 0

2/ 11 3 1 0

Z 2

Z

4

2/ 10 1 2 0 2/ 12 5 0 0 2/ 24 1 0 0

6/ 2/ 1/ 1/ 6/ 6/ 6/ 3/ 6/

9 1 o 11 3 1 6 a 3 620 6 7 0 10 1 2 12 5 0 20 0 1 24 1 0

2

2

2

a

2/ 10 3 1

2

Z

1 0 0 1 0 0 0 0 0

2

Z

4

6/ 2/ 1/ 6/ 6/ 6/ 3/ 6/

6 4 1

9 1 o 11 3 1 620 6 7 0 10 1 2 12 5 0 20 0 1 24 1 0

1 0 1 0 0 0 0 0

125

(56,11) Z

2

Z 2

Z 2

Z

Z

4

4

11 5 3 2 0 2/ 7 0 3 0 6/ 7 7 0 0 3/ 11 1 2 0 6/ 13 5 0 0 11 15 210 2/ 25 100 11 o 7 1 0 2/ 413 0 11 430 1 5/ 601 1 5/ 10 1 0 1 7110600 6/ 12 3 1 0 7/ 14 0 2 0

Z

5/ 2/ 7/ 7/ 6/ 1/ 2/ 6/ 5/ 5/ 11

5 7 7 11 13 6 6 10 10 14 22

320 0 3 0 7 0 0 1 2 0 5 0 0 0 1 1 5 1 0 1 0 1 6 0 0 0 2 0 2 o 0

Z

4/ 2/ 6/ 2/ 6/

7 10 10 12 22

7 0 0 101 600 310 200

Z

2

2

4

2/ 11 1 2 0

4/ 2/ 4/ 2/ 6/ 2/ 6/ 1/

7 15 25 6 10 12 14 22

7 2 1 5 6 3 0 2

0 1 0 1 0 1 2 0

0 0 0 0 0 0 0 0

(58,11) Z 2

3/ 6/ 3/ 7/ 2/ 11 7/ 2/ 5/ 7/ 6/ 2/ 2/ 11

5 3 2 0 7 0 3 0 7 7 0 0 11 1 2 0 13 5 0 0 4 130 430 1 601 1 6 5 1 0 10 1 0 1 10 6 0 0 12 3 1 0 14 0 2 0 22 2 0 0

Z

2/ 4/ 2/ 6/ 4/ 2/ 6/

5 7 7 11 13 12 14

320 0 3 0 7 0 0 1 2 0 5 0 0 3 1 0 0 2 0

Z

1/ 11 7/ 7/ 6/ 2/ 7/ 2/ 6/ 5/

5 7 7 11 15 25 10 12 14 22

3 2 0 0 3 0 700 1 2 0 2 1 0 1 0 0 6 0 0 3 1 0 0 2 0 2 0 0

Z

4/ 2/ 4/ 2/ 2/ 6/ 6/ 2/ 3/ 6/

701 7 5 1 11 1 0 13 3 1 4 1 1 12 1 2 14 0 0 10 3 0 22 0 1 26 1 0

1 0 1 0 1 0 1 0 0 0

2

Z 2

2

4

Z 2

5/ 5/ 7/ 7/ 6/ 1/ 7/ 11 6/ 6/

5 3 2 7 0 3 7 7 0 11 1 2 13 5 0 430 6 0 1 10 1 0 14 0 2 22 2 0

2/ 4/ 6/ 3/

13 15 25 22

5 o 0 2 1 0 1 o 0 2 o 0

4/ 4/ 4/ 6/ 6/ 6/ 7/

7 15 25 6 10 12 22

0 2 1 5 1 3 2

11 4/ 6/ 6/ 6/ 2/ 6/ 5/ 5/

7 7 15 25 6 10 12 14 22

030 7 0 0 2 1 0 1 0 0 5 1 0 6 0 0 3 1 0 0 2 0 2 0 0

2/ 7/ 6/ 2/ 2/ 2/ 2/ 3/ 7/ 6/ 11 5/

530 7 0 1 7 5 1 11 1 0 13 3 1 4 6 1 8 7 0 10 4 1 14 0 0 14 5 0 20 3 0 26 1 0

3 1 0 1 0 1 0

0 0 0 0 0 1 1 1 0 0

0 0 0 0 1 0 0

1 1 0 1 0 0 0 0 1 0 0 0

126 Z 2/ 2 6/ 6/ 2/ 6/

7 13 14 22 26

Z 2/ 4 6 1 2 2/ 632 2/ 14 5 0 2/ 26 1 0

5 1 0 3 1 0 5 0 0 a 1 0 1 0 0

2/ 8 700 2 2/ 12 120

Z

Z

4

1/ 11 2/ 11 1/ 1/ 7/ 11 6/

DEGREE 14:

530 11 1 0 13 3 1 411 6 3 2 8 7 0 10 4 1 20 3 0 22 0 1

Z

2/ 12 1 2 0 2 6/ 26 1 0 0

Z

1 1 0 1 0 0 0 0 0

Z 2/ 14 5 0 2 3/ 20 3 0 6/ 22 0 1 5/ 26 1 0

0 0 0 0

4

Z

4

11 7/ 6/ 11 7/ 1/ 2/

530 11 1 0 13 3 1 8 7 0 14 0 0 14 5 0 22 0 1

Z

2

Z

1 1 0 0 1 0 0

0

6 3 2 0

2/ 13 3 1 0 7/ 20 3 0 0 3/ 26 1 0 0

4

Z

2/

0 0 0

6 320 26 100

4

2/ 11 1 0 1

A[14]

The leading differential d

12(4vM3

differentials with image Z A[14] 2

2, 2 elements are Z A[ 14] {M M, M M} 1212 2 The leading differential d

M) = A[14]

1 2

B.

@ @

determines tentative

Since

*

0, the remaining

2. B, and the A[ 14]-leader is A[ 14]M 1

4(A[14]M2) 1

= vA[14] determines tentative

2, 2 differentials which are a monomorphism on Z A[ 14J{M M, M M} 2 1 2 1 2

@

B.

Thus,

there are no remaining elements.

DEGREE 14:

2

o:

differentials by assigning the following values to monomials of degree 28 of 2

Z o: 2

@

H.BP:

The elements Z

2

(j2 @

(j2 M3

and (j2M14 M2 are as's i. gne d 1 and

(j2M?1 1S

ass i. gne d 0 .

in degrees less than 69 with a representative in

H.BP are given by the table below.

The new (j2-leader is

1 2

127

DEGREE

BASIS

DEGREE

BASIS

DEGREE

BASIS

(20,14)

4 2 o 0

(24,14)

2 1 1 0

(36,14)

12 2 0 0

(40,14)

14 200

(44,14)

4 6 0 0

(48,14)

2 5 1 0

4 2 2 0

(50,14)

4 2 0 1 4 7 0 0 12 2 1 0

(52,14)

2 130 230 1 401 1 4 5 1 0

20 2 0 0

(54,14)

6 2 0 1 6 7 0 0 14 2 1 0

DEGREE 15: The leading differential d

2(A[14]M 1

) =

differentials with cokernel Z

2

The leading differential

1 2

differentials with image Z

2

Z

2

3M} 3, 14]{M MM ,M 11212

@

1

1

determines tentative B.

@

)

=

®

B.

1

1

is

determines tentative

The remaining elements are

B, and the new

The leading differential

The

14]-leader is

3. 14]M 1

2C[20] determines tentative

differentials which are a monomorphism on Z

2

3, 3M} 14]{M MM ,M 11212

®

B.

Thus, there are no remaining elements.

DEGREE 17: Since

16]

Z

DEGREE 17: Since

2

v A[

• ,17

with a representative in

vA[14]

14J

ZvA[14J @ 2

4

0, the only element of E

H.BP is zero.

®

2

'#

'#

6 0, the only element of E

H.BP is zero.

·,17

with a representative in

1

.

128

DEGREE 18:

C[18J

The leading differential

)

1

=

4C[18J determines tentative

differentials with cokernel [2 (C[18JM )

® 8J $ [2 C[18J 8 1 4

The leading differential d

4(C[18JM2)

= vC[18J

1

differentials with kernel [2 C[18J{M ,M} 8

1

2

®

8J.

®

determines tentative

8J

[2 (2C[18]){M M,M3 M} 8] @ [2 (2C[18]) ® 8J. 4 1212 2 . 12 12 12 In Sect ion 3.4 we computed the Image of d : E. E. . However, that @

,7

,18

computation was done in three stages so that the global image of these d

12-differentials

Therefore, we give the computer

is hard to unravel.

calculation of the cokernel of these differentials in the table below.

The

new C[ 18J-Ieader is 2C[ 18]M . 2 DEGREE

GROUP

(6,18)

2

GENERATOR

DEGREE

GROUP

o 0

(10,18) Z

(14,18) 2 2 21

4 1 0 0

(18,18) 2

(20,18) Z

011 0

Z

(22,18) 2

2

2 2

21

21

0 1

21 2 3 0 0

(24,18) Z 21 2

2 1 1 0

(26,18) Z 41 3 2 21 o 21 4 21 10

1 2 3 1

1 1 0 0

0 0 0 0

(28,18) Z 41 5 2 4/ 7 41 11 2/ 4

3 0 1 1

0 1 0 1

0 0 0 0

(30,18) 2

2

21

o5

0 0

Z 2/ 12 1 0 2

a

2

2 2

GROUP

GENERATOR

DEGREE

21

2 100

(12,18) 2

21

o

21

7 100 4 2 0 0

Z

11

3 0 0

2

21 2 21

3 100 6 0 0 0

21

6 100

2 2

4 0 1 0 8 100

2

5 0 1 0 9 100

2

21

4 0 1 0

Z

Z

21 2 61

3 300 6 2 0 0

2

2

21 2 61

3 1 1 0 6 0 1 0

Z

Z 21 2

4 3 0 0

4

11 o 2 1 0 11 4 3 0 0 21 6 0 1 0 21 10 1 0 0

21

5 300

5 1 1 0 6 300

Z 2/ 2

6 3 0 0

3 2 1 0 7 300 a3 1a

Z 21 3 2 1 0 2 61 10 2 0 0

Z 21 7 0 1 0 24/11100 2/ 14 0 0 0

(32,18) Z 2

21 7 100 61 10 0 0 0

4

2

Z 21 2 61

GENERATOR

Z

8

11 7 0 1 0 5/ 11 1 0 0 11 4 1 1 0

129 Z

2

Z 2

(36,18) Z 2

Z

2

Z 8

2

2

Z

Z

Z

2/

8

Z 2

(44,18) 2 2

Z 2

2/ 7 1 1 0 6/ 10 0 1 0 6/ 14 1 o 0

Z

2/ 350 4/ 5 2 1 2/ 15 1 0 2/ 640 6/ 18 0 0

0 0 0 0 0

Z

8 1 1 0

Z

1/ 5 2 1 0 7/ 9 300 2/ 15 1 0 0 2/ 18 0 0 0

(38,18) Z

2/ 10 300

Z

12 010 16 100

(40,18) Z

2/

4

2

6 1 1 0

2/ 4/ 6/ 2/ 6/

2 3 0 1 2

1 0 1 0 0

Z

(34,18) Z

2/

2 5

o0

Z

2/

421 0

Z

2/ 5 2 6/ 9 3 2/ 15 1 2/ 18 0

1 0 0 0

0 0 0 0

Z

Z 2 2

Z

1/ 7 2 1 0 1/11300 1/ 13 0 1 0 3/ 17 1 0 0 1/ 4 3 1 0

(42,18) Z

2/ 3 1 o 1 6/ 5 3 1 0 2/ 11 1 1 0 6/ o 7 0 0 2/ 4 120 2/ 600 1 6/ 8 2 1 0 6/ 12 3 0 0 6/ 14 0 1 0

Z

7 500 13 3 0 0 15 0 1 0 o5 1 0 4 6 0 0 10 4 0 0 12 1 1 0 22 0 0 0

2

2

0 0 0 0 0

6/ 4/ 6/ 2/ 6/ 6/ 2/ 2/

7 11 13 17 14

2

2/ 7 3 0 0 6/ 10 2 0 0

2 2

2

2

2

2/

2

2

2

2

2

Z 2

o

1 2 0

2/ 14 1 0 0 4 2 1 0 8 3 0 0

4

2/ 15 1 0 0 1/ 12 2 0 0 2/ 18 0 0 0

o

2/

0 1

2/

4 500

2/

6 2 1 0

1

2 3 1 0

2/ 3 3 2/ 6 2 2/ 12 0 2/ 16 1

1 1 1 0

0 0 0 0

2

2/

2 1 2 0

Z

2/

2

o

1

2

Z

2

Z 2

4/ 5/ 7/ 2/

11 13 17 14

3 o 0 0 1 0 1 o 0 2 o 0

Z

2/ 2/ 6/ 2/ 2/

5 11 12 14 18

3 1 3 0 1

0 0 0 0 0

Z

2/ 2/ 2/ 6/ 6/ 2/ 2/

5 11 4 6 12 14 18

3 1 0 1 1 0 120 5 0 0 3 0 0 0 1 0 1 0 0

Z

1 1 0 1 0

2

2

Z

2

2

Z

Z

2 2 4

Z 4

Z

2/

2/ 12 0 1 0 2/ 2/

3 120 6 0 2 0

2/

4 3 1 0

2/ 10 1

0

2/ 8 2 1 0 2/ 12 3 0 0 6/ 18 1 0 0 2/

6 5 0 0

2/

0 7 0 0

1/ 8 2 1 0 1/ 12 3 0 0 2/ 11 1 1 0 6/ 14 0 1 0

2/ 12 3 0 0 4/ 4/ 6/ 2/ 6/ 6/ 6/

7 500 13 3 0 0 19 1 0 0 410 1 6 3 1 0 12 1 1 0 22 0 0 0

Z 2

Z

2

2/ 4/ 6/ 2/ 6/ 6/

7 13 15 19 10 12

500 3 0 0 0 1 0 1 0 0 4 0 0 1 1 0

2/ 12 1 1 0

130 Z 2

(46,18) Z 2

Z Z

(48,18)

2 2

z

2

z 2

z z2

2/

6/ 13 1 1 a 6/ 14 3 0 a

z

6/ 13 1 1 a 2/ 6 120

z2

2/

2/

4 4 1 a

z2

2/ 14 3 0 a

2/ 2a 1 a a

2/

2 7 a a

z

2/ 7 3 1 a 6/ to 2 1 a

z2 z4

1/ 5 4 1 6/ 7 1 2 5/ 9 5 a 6/ 15 3 a 6/ 8 3 1 2/ 14 1 1

2

2

4

2/ 19 l o a 6/ 22 a 0 a

8

6 3 1 a

4/ 5/ 11 1/ 2/

13 15 19 12 22

3 a 1 1 0

0 1 a 1 0

a a a 0 0

a 3 2 0

4 4 1 a 8 5 a a

6/ 7 1 2 a 2/ a 1 3 a 2/ la a 2 a

z

2/ 3 7 0 0 6/ 7 120 2/ 15 3 a a 2/ 6 6 a a 2/ 10 a 2 0 2/ 18 2 a a

Z

3/ 3/ 2/ 1/ 2/

Z

2/ 7 1 2 a 6/ la a 2 a

z 2

2/ 11 2 1 a 2/ 15 3 a a 4/ 21 1 a a

11 15 21 8 18

2 3 1 3 2

1 a a 1 a

a a 0 a a

2

2

2

a a a a a a

z2

2/

a 3 a 1

z2

2/

2 5 1 a

z2

422 a

Z

2/

6 1 a 1

z2

2/ 14 1 1 a

z4

2/ 15 3 0 a 2/ 18 2 a 0

z

2/ 2/ 2/ 2/ 2/ 2/ 2/ 2/ 6/ 6/ 6/

3 15 o 4 4 6 1a 12 16 18 22

5 1 1 a 7 4 5 2 3 0 1

a a 1 a a a 0 0 0 a a

z

2/ 2/ 2/ 2/ 2/ 2/ 2/ 6/ 6/ 6/

Z

6/ 6/ 11 7/ 6/ 6/ 11 2/ 2/

2/ 4/ 2/ 2/ 6/ 2/ 6/ 6/

7 15 4 8 10 10 16 22

1 a 1 1 1 a 7 0 a 120 0 0 1 5 0 0 3 0 0 loa

z

2

(5a,18)

z

a a a a

2/ 7 5 a 4/ 15 a 1 1/ 4 6 a 6/ 22 0 a

2

z 2

1 1 1 3 a 1 0 1 0 1 a

2

2

3 15 4 4 6 la 12 16 18 22

4/ 3 2/ a 2/ 4 2/ 10

5 1 a 7 4 5 2 3 0 1

1 1 3 a 1 a 1 0 1 0

0 a a a 0 a a 0 0 a

5 1 a 6 1 0 700 500

2

2

2/ 8 1 2 a 6/ 12 2 1 0

0 a 1 a a a a a a

z2

2/

2 3 2 0

Z

2/

4 7 0 0

Z

Z

3 5 1 15 1 1 420 470 641 la 5 0 12 2 1 18 a 1 22 1 a

Z

2 2 2

2/ 12 2

0

2/ 22 1 o 0

131

Z

2/

Z

3 5 1 0

4 2/ 15 1 1 0

Z

2/

3 5 1 0 0 6 1 0 4700 2/ 10 5 0 0

1/ 4030 2/ 6 4 1 0 1/ 8 1 2 0 2/ 10 5 0 0 6/ 16 3 0 0 6/ 18 0 1 0

Z

The leading differential d

12 2 1 0 16 3 0 0

4

6(2C[18]M 2

2/ 15 1 1 0

4 6/ 18 0 1 0

4 1/ 1/

)

= A[23]

determines tentative

differentials by assigning the following values to monomials of degree 24 of

Z C[18] 8

H.BP: C[18]M3 is assigned 1 and C[18]M is assigned 2.

@

1

of these tentative differentials is given by the table below. C[18]-leader is C[181M

DEGREE

GROUP

(20,18) Z

2/

21/

(26,18) Z

6/ 41/ 1/

(32,18) Z 2

1/ 7/ 1/

4

8

MZ.

GENERATOR

DEGREE

GROUP

7 1 0 0 4200

(22,18) Z

3 1 1 0 0210 4300

(28,18) Z

3210

(34,18) Z

(40,18) Z 2

8

7 3 0 0

4

(38,18) Z

4/ 11 3 0 0 7/ 13 0 1 0

Z 8

2/ 11 8 1/ 12 6/ 14

1 2 3 0

1 1 0 1

0 0 0 0

6/ 18 1 0 0

7/

4 0 1 0 8 1 0 0

(24,18) Z

2/

5 3 0 0

(30,18) Z

1/ 7 0 1 0 5/ 11 1 0 0

1/

4 1 1 0

6/

7 1 1 0

1/ 4 2 5/ 8 3 2/ 10 0 2/ 14 1

1 0 0 0

2

2

(44,18) Z

2

GENERATOR

3/ 3/ 2/

5 0 1 0 9 1 0 0 2 1 1 0

2/

5 1 1 0

(36,18) Z 2/ 15 1 0 0 2 1/ 12 2 0 0 2/ 18 0 0 0

1 0 0 0

6 2 1 0

6/ 10 3 0 0

7/

GROUP

7 2 1 0

1/ 11 3 0 0

Z 4

2/ 7 5 0 0 4/ 15 0 1 0 11 4600 6/ 22 0 0 0

1/ 13 0 1 0 7/ 16 1 0 0

(42,18) Z 2/ 5 3 1 0 4 2/ 11 1 1 0

6/ 13 0 1 0 2/ 17 1 0 0 1/ 4310

2/ 10 1 1 0 6/ 14 2 0 0

4 1/

DEGREE

2/ 2

5/ 17 1 0 0

Z

GENERATOR

1/

4

5 2 1 0 9 3 0 0 2 3 1 0

5/ 3/ 2/

The new

1 2

0310 2/ 10 2 0 0

Z

The kernel

2

3/ 8 2 1 3/ 12 3 0 2/ 14 0 1 6/ 18 1 0

0

0 0 0

Z 4/ 13 3 o 0 8

5/ 15 0 1 0 1/ 19 1 o 0 11 12 1 1 0

2/ 22 0 o 0

132

(46,18 ) 2 (48,18) 2

(50,18) 2

4/ 7 3 1 0 4/ 13 1 1 0

2

7/ 5 4 1 0 2 2/ 7 120 3/ 9 5 0 0 2/ 15 3 0 0 2/ 2 5 1 0 1/ 4 2 2 0 2/ 8 3 1 0

Z

2

4

4/ 6/ 1/ 5/ 5/ 1/ 2/ 3/ 6/

DEGREE 19:

3 5 1 0 710 1 4 0 3 0 420 1 4 7 0 0 8 120 10 0 0 1 12 2 1 0 22 1 0 0

2

2

4

2/ 13 1 1 0

2

6/ 3/ 3/ 2/ 1/ 5/ 2/ 6/

7 1 2 11 2 1 15 3 o 21 1 o 422 8 3 1 10 0 2 18 2 0

0 0 0 0 0 0 0 0

Z

4/ 6/ 1/ 5/ 2/ 2/ 3/ 6/

3 5 1 710 420 4 7 0 10 0 0 12 2 1 16 3 0 22 1 0

0 1 1 0 1 0 0 0

Z

4

4

4

Z

1/ 7/

4 4 1 0 8 5 0 0

1/11210 3/ 15 3 0 0 6/ 21 1 0 0 1/ 8 3 1 0

6/ 1/ 5/ 2/ 2/

15 12 16 18 22

2/ 1/ 1/

o

4

1 2 3 0 1

1 1 0 1 0

0 0 0 0 0

3 5 1 0 6 1 0 4 7 o 0

A[19]

The leading differential

2

with image 2 A[19]{l,M ,M} 2

1

®

2

2M

Z 2

2

3M}

M ,M ,M 11121212

The leading differential d

)

A[19] determines tentative differentials

8.

The remaining elements are 2

e B, and the A[19]-leader is A[19]M

4(A[19]M2)

1

•

= vA[19] determines tentative

1

differentials which are a monomorphism on 2M 2,M3,M 3M} M ,M Z A[19]{M ,M 2 11121212

DEGREE 20:

@

B.

There are no remaining elements.

C[20]

The leading differential d

4(vA[14]M2 1

)

=

4C[20] determines tentative

differentials with image 2 (4C[20]){l,M ,M M} 2 11212 differential 2

1

= C(20)

2

B.

The leading

2C[20] determines tentative differentials with

1

image Z (2C[20]){l,M ,M}

®

@

B.

The leading differential

determines tentative differentials with image

133

2 C[20] 2

@

8.

The cokernel of these differentials is

2i1 3i1} 3 [2 C[20]{ M , M ,M 8 11212

o [2 C[20]{M ,M} 2

1

8] o [2 C[ 20 ]{ M2, M M} 4

8].

2,2,2.2.{M} [ZA[30] \OJ 2 [a{3' (1) 2 A[30] = + =

=

A[8J

2,A[8»C[20)

Therefore,

Thus, 2

by 5.6.

We shall see

By Lemma 3.3.14,

We showed above that 2B[40) e c

Thus,

155

modulo Z (vA[37]) 2

=

c

v

=

=

C

=

A[14]C[20]A[8]

Now =

which as we remarked above must be nonzero.

Note

that the four-fold Toda bracket above is defined by Theorem 2.2.7(b) because

o

and 0 =

E

A[39,2] E

= d 6 (A[14]C[20]M- 2 ),

Since A[39,2] and vA[39,2] E

= A[8]A[14]C[20] = 1/2C[20].

=

Thus,

vA[39,2]

[6.13]

=

Now

c

= O.

As we shall see, the only element 0 is A[39,2].

vA[37]

1/A[39,3] + 1/uA[32,1], 4C[20]2 = O.

= 1/A[39,2]

+ h1/A[39,3] + k1/uA[32,1].

S

E

rr;g such that

= 1/A[39,2]

and

2C[20]2

Thus 2C[20]2

E

13

0,

modulo Z (vA[37]). 2

= Since

Write Redefine A[39,2] as

A[39,2] + hA[39,3] + kuA[32, 1] so that 1/ times the new A[39,2] equals 2C[20]2. Note that vA[39,2] and uA[39,2] remain unchanged.

Let A[40,1]

By Theorem 2.4.2, A[40,1] E .

Then 2A[40,1] E 2

= u

+

v

=

[6.14]

3

1

Row

Degree

Leader

39

45

A[39,21M 2

40

46

41

47

42

46

-42

48

42

44

3 211M ,C[20] M 1 2 3 1/A[ 40, 1] M 1 2 C[42]M 1

4C[42]M

2C[20]2M 1

1 FIGURE 6.3.4:

Leaders from Rows 1 to 42 of Degree at Least 44

There are two leaders 01 degree 44 and live leaders of degree 45. proof of Theorem 6.3.3 we showed that vA[39,2] d

1,

4(A[39,2]M) = 2

1

= 1/2C[20].

Since r. 06'1 M8 123 (crA[30)+A[37] )M16

6M (crA[30]+A[37] )M 123

1

The leading differential d

6(A[32,

3

2M 2 1]M ) =A[37]M determines tentative differ1 2

1

2 entials with image 2 A[37]M ® B. The remaining elements from 2 A[37] ® B 2 1 2 2M} are 2 A[37]{M ,M ® B, and the new A[37]-leader is A[37]M. The leading 2 2 1 2 2 4(A[37]M differential d ) = + l]M determines tentative 2

1

1

-

2-

differentials which are a monomorphism on (2 A[37]{M ,M M} ® B,M M ,,,M ,M ,M M ,M ,M M ,M ,M M , 3

12

1213

12

6

2-

10-

4- -

1

13

123

12121231

12

2 . .2-

14

M ,M , MM, M M, MM ,M MM ,M ,M

M

.

M ,M

M

,M } ) .

123122

There are no remaining elements in degrees less than 68 from (2

2

1

DEGREE 38:

®

Bv

Since A[39,3] E

= S 13

= 0 because

= O.

Note that the four-fold Toda bracket is defined by

Theorem 2.2.7(a) because O'A[39,3] E =

= O.

= 0 and

=

. = O'

[7.3.] By 6.18,

Thus, 2vC[44] E vvA[3oJ

Therefore, B[47] is the only elements of CokJ Thus,

IJ

= 0,

and

which may not

IJM must be a boundary.

In

1

2A[30J

= 4vC[44] = vO'

must bound because

1

47

=

o.

There are only three leaders of degree 49 which do not clearly transgress: 2M, s 2M. A[39,l]M A[39,llM and A[39,3JM 12

or

1

1

then r.

L't

12

applied to d

8(A[39,

If d

8(A[39,

3M

1]M

12

2M IJM ) equals 12

) produces a contradiction

because there is no possibility for a hidden differential on A[39,

llM

1

215

Thus, A[39,1]M 1

2M 1 2

transgresses.

and

1

Therefore, 2C[20]2M M, 1 2

8(A[39,

1

5),d8(A[39,3]M2M}

llM

1

1 2

=

1 2

1

=

2, 1

to be boundaries is

1

1

= O.

A[32,1](M 132

Now

=

and

2M

the 34 row: 2M A[39,1]M

and

=

E

=

2

can not bound.

The only possibility for {d

2C[42]M, C[44]M

There is only one leader of degree 50 below

As we shall see in the proof of Theorem 7.2.4,

must bound and the only possibility is d

8(A[32,

2M

1](M

1 3

3

+M

2

) )

2M. = A[39,1]M Thus, can not bound and vA[45,l] is nonzero. 121

*

Therefore, 11M. 1

Thus, d

A[45, 11,

8(A[39,

1]

=

1

1

*

II and d

8(A[39,

5

11M

2M) d 8(A[39,3IM = 12

1

)

1

* and [7.4]

11 = We have thus proved the following theorem.

THEOREM 7.2.3

7[S = Z B[47] 47 4

@

Z A[471 2

@

Z 2

@

Z vC[44] 2

@

Z a 32 5

where 2B[47] The computations in Section 6 show that we have the following leaders. Row

Degree

17

51

32

50

38

50

38

52

39

51

39

49

40

52

Leader 2 M17 a1 1 3) 2M +M 132 3B M , B[381M 1 2 1 4M 2B[38]M 1 2

A[32, 1 I CM

1 2 2M A[39,llM 1 2 39, 31 B

A[40,1IM

42

Leader

Row

332,1] )M M , 1 2

44

52

4 C[441M 1

45

49

A[45, llM

45

51

46

52

"46

66

47

49

2 2) Z (D[45IM 1 1, 8 3, 8D[45]M A[45,2IM 2 1 3 llM 1 2C[44]M 7M 1 2 1

,

A[471M , B[47]M 1 1 2 vC[44]M 1

2]M , 1

47 51 1 2C[201 2M1M2 50 FIGURE 7.2.2: Leaders from Rows 1 to 47 of Degree at Least 49

216

There are nine leaders of degree 49 and four leaders of degree 50. d

6CC[20]2

i;

M) 2

=

d

= A[45,2],

lOCA[39,

l]WM ) 1 2

=

d

lOCITA[32,3)M2M)

i; E

by Theorem 2.4.2.

Then i;W can not bound, vi;

since i; is not divisible by v.

=

12(1T3M3M)

= d

1 2

12 [ CIT2M3 )ITM J, Theorem 2.4.6(c) implies that 1

A[32,3]

2

[7.5]

E

=

E

= A[32,3]A[19)

=

C

kfd

8(1T2A[30)M

kfd

8(4C[44)M

M ) + Indet

M) +

= 0;

since by Theorem 2.4.6(c);

1 2 1 2

Thus,

=

+

+

Then vi; E

0 and vi; E

1

=

Since A[32,3) = d

Note that

1

E

1 2

Assume that i; is nonzero.

Indet

Indet

since 4C[44]M M is a d

8-boundary.

1 2

S

By Theorem 2.3. l(b), there is Then vi; E

1

+ v

E rr

46

2

,u>

By Theorem 4.2.3 and Figure 4.2.2, it follows that C[44)

2D[45)

8D[45)M

3M

1

f------ 20[45]M

4-

A[ 52, 1) (

A[39,llM

v (- B[38]M

M17

'1 1

f------

1

( - - crA[32, l)M M

1

1jA [ 39 ' 3)+]M3M (-12 [ 1jcrA[32,lJ

2

1

2

2vD[45)

2

(

1

1j2A[45,2)M

1

vD[45]

vA[45,l]M

2

1j20[45]M

1 2

vC[44) ( - - C[44)M

1 2

1

2

1

A[47) (

20[45)

1

4M

3 2C[42)M

6

1

2M 1 2

2M A[39,llM f-----1 2 2 vA[45,1) ( - - A[45, 1 )M

3 1jA[ 40, 1) M

0[45)

5

f--

1

1

1j2C[ 44) ( - - 1jC[44]M

2

1

6

1jA[50,2)M

1

2M (-- A[36)M

1 3

3

(-- vA[45, 1)M

A[8]D[45] f - - 2vD[45JM

1

4

3

1jA[52,2) ( - - A[52,21M

1 3

vAl 50,1)

1jA[50,2) (-- A[50,2)M

vA[50,2) ( - - A[50,2]M

1

(---

1

2 A[ 50 , 1 ]M 1

2crC[44) ( - - 1jA[45,l)M

1

1

2 1

1

310

7i?M:

A[50, llM 2 vC[44]M M

4{3 M 1 123

lIA[S]D[4S]M

1 2

1

18

{3 M

1

16M: 2{3 M 212

A[ 56] (

2 1

2

-

II A[45,2]M M

1 2

2 A[52,llM 1

1

(

1

lIA[52,2]M

v A[50 , 2 ] 6 2C[ 44]M

lIA[S4,2]M

lIB[47]M M

A[57]

A[54,2]M

lIA[56]

1 2

lIA[54,2] (

60

1

61 {3 M

1

2 1

A[30] vC[44]W

1

2 l} C[44]M 1 2

A[62, llM 1

28[64,1] (

4

7M

MZM 1 223

8[64,2] (

1

7

7M

4C[ 18]M

A[32,

A[52,1]M A[59, l]W 1

28[34]M M 123

(

3]WM 1 2 3

A[62,4]M 1

28[64,2] (

1

8[60]M 2

l]M 1 1 1

1 2 2M p2A[50,2]M 1 2 2M3 2C[44]M 1 2 3M l}A[ 8]D[ 45] M 1 2 2 A[62,3]M 1 1 2 A[62,4]M 1

APPENDIX 5:

THE COMPUTER PROGRAMS

All of the computer programs used to make the computations of this paper are written in FORTRAN 77 [9).

They were compiled and linked by Ryan-McFarland

RMlFORTRAN version 2.00 on an IBM PC/AT microcomputer running DOS version 3.0. There are eight component programs which are linked into five programs.

The

scheme for linking them is given in the left columns of Figures 1 and 2.

The

files S70.FOR, MODB70.FOR and MODS70.FOR contain subroutines for manipulating the arrays which represent monomials, polynomials and bases of poloynomials. The boxes in the right columns represent files which store data.

The arrows

indicate the data which is required as input for each program and the data which each program generates.

For each of the five programs, we describe what

the program computes and briefly indicate how the program carries out the computations.

The 100 pages of complete program listings are available from

the author. PROGRAM I. generators 1

i

This program uses formula 1.2.3 to compute the Hazewinkel VI'

1

i

5, as polynomials in the M and to compute the M , N

5, as polynomials in the V

HAZEWINK.

In the second part of this program we consider all monomials ME in

the MN, degree ME all t

I

This information is stored in the file

" N

70, all Quillen operation rr' degree rr

(degree M )/2. E

Let U

N

= VN/2

degree ME' and

denote polynomial generators of H.BP.

We determine the coefficient C(E,I,t) of Ut in r (M ) written as a polynomial 1 r E in the U. N

This is accomplished by first computing r (M ) as a polynomial in I

E

the M using the Cartan formula and the fact that N

if s

k

k and

r (M ) = { M IsO

otherwise

Then we use the observation that when M is written as a polynomial in the U , N

N

.2 - 1

the coefficient of Ui 1

is 2

2

N-N-1

The values of the C{E,I,t) are used in

k

313

the second program to compute the d 2t-differentials which originiate on the

o row.

The C(E,I,t) are stored in the files STROMONl ....• STRDMON7.

PROGRAM II.

This program computes the differentials on the 0 row in all

2r Elements of E are wri t ten as 2t,O 2r polynomials in the U, and elements of E are written as polynomials N 2t-2r.2r-l 2r-differentials in the M. The d are computed by converting an element X in

degrees 2t less than or equal to 70.

N

2r E to a polynomial in the M using the information in the fi l e HA2EWINK. 2t.O N 2r(X), The coefficient of M in d written as a polynomial in the M , is N

I

2

k(rl-r

times the coefficient C(X; I. r-) of

in the U.

if in r (X), written as a polynomial 1

Here k(r) equals 1. 2.

N

2. 4. respectively.

I

if r is congruent modulo 4 to 1.

The C(X;I,r) are determined from the C(E.I.r) which were

stored in the files STROMON1, ... ,STROMON7 in Program I.

Now d

2r(X)

has been

determined as a polynomial in the M and is converted to a polynomial in the N

M.

When r

2 a

®

N

2 S

4s+1. we have only determined the summand of d

8.

22112"¥S_lMl

®

If s

4

®

(2

4s

$

= 3. 2

"d

+

When r

+

2(X)

4s+2 we have only determined d

M ) ® 8.

2(X)

in

in

8

S

8 s 4(X) +

8s 4

+ ( X) "

in

To complete the determination of the

8s1

coefficients in 2 8 s 6(X)"

8s

+

8 by carrying out the above procedure for finding "d

using k(4s+2) 2

1, we determine the summand of d

8s

M ® 8, we carry out the above procedure for finding 1

using k(4s+3) = 3.

We use elementary row and column operations to

2r keeping a record of the elements of E 2t.O 2r represented by each row of the matrix and the elements of Ezv-ar-, 2r-1 2r represented by each column of the matrix. The basis of Kernel d are the

diagonalize the matrix 0 of d

2r

appropriate powers of two times the row representatives of 0 and Cokernel d

2r

is the direct sum of the cyclic groups generated by the column representatives of 0 corresponding to the diagonal entries d

1.

The cokernels of these

differentials are stored in the files STRJCOKl, ... ,STRJCOK4 as polynomials in the M.

N

2r The E ,0< t 2t,O

35. are stored in the files STRBSGPl ....• STRGSGP6

314

as polynomials in the U . N

PROGRAM III.

This program reorganizes the data stored in

STRJCOKl, ... ,STRJCOK4 in the way that it will be used in Program V to compute differentials originating on the rows of CokJ. 2r 2t-2r,2r-l

E

That is,

1 in STRJCOKl, ... ,STRJCOK4 is stored in the

lexicographical order of (t,r).

The output of this program stores these

cokernels in INFILER where R = 2r-l.

We only need store this information for

23 because the cokernels turn out to be zero in degrees less than 70

R

when R > 23. PROGRAM IV.

This program computes images of differentials in the bidegree

(N,t) of a leader X on which a nonzero differential originates.

In order to

use Quillen operations to compute the tentative differentials determined by this differential on X in bidegrees (N' ,tl, N' 2 N,t

2s N,t

elements of E

which are homologous to zero in E

program is used to determine these elements. very short and is produced on the monitor. 2s N+2s,t-2s+1

E

N, we need to know all

on which the d

2s

for

1

s < t.

This

The output of this program is The input file INPIPE contains the

are to be computed.

Any of the files

INFILE3 •... ,INFILE23 or any of the files OUTDOMl •... ,OUTDOM9, OUTRANGl, ... ,OUTRANG9 produced by Program V can be renamed INPIPE and used as input for this program.

When the d

2s

originate on the 0 row, the required

information can be obtained from Program II. PROGRAM V.

This program is the analogue of Prgram II for computing the

cokernels of differentials d 2r which originate on the t row where t > O. 2r . ,t

input file INPIPE contains the E.

The

Any of the files INFILE3 •...• INFILE23 or

any of the files OUTDOMl, ... ,OUTDOM9, OUTRANGl, ... ,OUTRANG9 produced by a previous running of program V can be renamed INPIPE and used as input for this program.

2 r+ 2 ·,t

The E

are stored in OUTDOMI and the E2 r +2

·,t+2r-l

are stored in

315

A sequence of differentials of this sort d2 r ( k l , 1

OUTRANG1.

k

computed with one run of the program where t is fixed and r(1) 2r(k)+2

The E.

,t

are stored in OUTOOMk and the

E2r(k)+2

9, can be ...

are stored in

• ,t+2rlk)-1

OUTRANGk.

I.

I HAZEWINK.FOR I

+ S70.FOR

II.

I HAZEWINK I

< >

I STROMON1 I ,... , I STRDMON7

-1//

STRBSGPl

I ,... , I STRBSGP6 I

B70.FOR

+

I-S-T-R-J-C-O-K-I-I , ... ,

I

STRJCOK4

1 --

S70.FOR

1_STRTABLE

III.

COMPACT. FOR

+

1 INFILE3 I , ... , I INFILE23 I

S70.FOR FIGURE 1:

COMPUTING DIFFERENTIALS ORIGINATING ON THE 0 ROW

reg).

316

IV. DETGENLT.FOR

+ +

MODB70. FOR

I

f-(- - - - -

I INPIPE I

MODS70.FOR

V. SSHOM.FOR

+ +

MODB70.FOR

1/ I

MODS70.FOR

FIGURE 2:

INPIPE

I

•I

OUTDOMl

OUTRANG1

I···· ·1 I···· ·1

OUTDOM9

OUTRANG9

COMPUTING DIFFERENTIALS ABOVE THE 0 ROW

APPENDIX 6:

THE ADAMS SPECTRAL SEQUENCE

The tables below depict the S

sequence for R..

-term of the classical mod 2 Adams spectral

The notation is standard:

multiplication by h

o'

vertical lines represent

lines of positive slope represent multiplication by h 1

and lines of negative slope represent nonzero differentials.

If solid

vertical lines from both A and B land on C, this indicates that C If these lines are dotted, this indicates that C = hoA + hoB. extensions given by multiplication by 2 or

= hoA

h B. o

Nontrivial

are denoted by dotted lines.

To

make the tables readable we do not label any of the elements or include lines indicating multiplication by h labels indexed by bidegree. left to right.

In each beidegree the elements are labeled from

For each infinite cycle we use the symbol X

the name of the element projects to X.

Instead, each table is followed by a list of

a

to indicate

in the Atiyah-Hirzebruch spectral sequence which

We also include tables giving products with h:a'

These tables are based upon the tables of E:a of the Adams spectral sequence of Mahowald [55] and Tangora [59]. The differentials in degrees less than or equal to 45 were computed by Mahowald and Tangora [37], Barratt, Mahowald and Tangora [10] and Bruner [16].

The differentials in degrees 46 through 59

confirm the tentative differentials given by Mahowald in [55].

The

differentials in degrees greater than 59 are tentative in the following sense: (1)

some are consequences of differentials in lower degrees;

(2)

some are consequences of the computation of

(3)

some are the most reasonable choices among several possibilities which

S

R , n

n

:S

64, in this paper;

agree with the computations of this paper. The use of these tables in Sections 7.5, 8.3 and Appendix 2 do not rely on any of the choices described in (3).

318

The tables below include the following entries which were accidentally omitted from the tables in [55J and [59J: d

3(hU

h

2R

5

a

= hx'

in (53,11),

a

= Ph5 i

in (62,12),

hR = S

in (54,11),

all

h p a

=p

4r

4s

in (62,23),

d

4(gm)

= Pgj in (54,15),

h h Q

132

2h = ha 0 32

in (65,9)

and several products which are marked with asterisks in the tables below.

7 6 5 4 3

•

2 1 0 0

1

2

3

4

5

7

6

9

8

10

11

12 13

14

15

Notation: (0,1)

h

(6,2)

h

(9,5)

Ph

(14,4) d

a 2 2

a

f-2 f1

f-

V

2

0:.

1

f-A(14)

Multiplication by h (9,3)

h

2h

1

3

h

3 2

(1,ll

h 1f-1)

(3,1)

h

(7,1)

h

f-

(8,3)

C

(11,5)

Ph 2 f- 13 1

(14,2)

h

(15,1)

h

(15,4)

h

(14,6)

h Ph

3

0-

4

2 2

2

h

2d

a a

2 0

2

3

f-- V

f- A[ 8] f-

3h

a

4

2

0-

f-

a1

+1)0-

319

/1

15 14 13

/1

11

10 9 8

Vi

7 6 5 4

/

• •

•

•

3

1 •

-:

12

2 1 15

16

17

18

19

20

21

22

23

A[16J

(16,7)

Pc o

0:

(18,2)

h h

(19,9)

p

2h

»cr 18]

(22,4)

he

A[23]

(23,5)

h g

2

2 (24,11) p c o

24

25

26

27

28

29

30

Notation: (16,2)

h h

(17,9)

2h

1 4

p

1

(19,3)

e

(21,3)

h

(23,4)

he

(23,9)

h i o

2

A[19]

1

2h 2 4 4 0

2

1

3h

(25,13)

p

(27,13)

p 3h2

(30,2)

h

2 4

h d

(25,9)

h Pd

(29,8)

h 2j

2

0

=

C[ 18J

(18,4)

f

2

(32

(20,4)

g

2 1

vA[ 19J

(22,8)

Pd

vC[20J

2

1112

2

2

(33

(28,8)

Pg

A[8]C[20]

(29,7)

k

A[30]

(30,6)

r-

= h 0 Pe 0 hok

C[20]

Pe j

2

0

(25,8) (26,7)

2

0

1/2C [ 20]

0

(23,7)

v C[20]

= he o 0

2 0

2 4

h g

3

Multiplication by h (17,5)

e

(26,6)

0:

1

(17,4)

1111

0

2d (30,12) p

0

:

(20,5)

h 2eO

(26,8)

h2 i

= hog = hoj

(22,10)

h p 2h

(28,9)

h Pe

2

2

0

2

= h02Pd 0 =

hoPg

320

23 22 21 20 19 18

17

1

16 15 14 13 12 11 10

9 8 7

6

5 4

3 2

1

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

321

Notation: (31,1)

(31,5)

n

(31,11)

(32,2)

h h

f-

A[ 32, 1 ]

f-

1 5

(31,8)

A[31]

q

f-

A[32,2]

(32,7)

(32,15)

(33,4)

P

f-

vA(30)

(33,12)

(33,17)

(34,2)

hh

(34,8)

2 e o

(35,7)

m

A[14)C[20]

f-

(36,12) (37,8) (38,4)

eog h

2h

h

f-

035

B[38]

3d

(38,16)

p

(39,5)

h e

(39,12)

Pd e o 0

(40,4)

f

o f-

1 1

f-

1

A [39, 1)

A[40, 1)

(40,11) (41,10)

(34,3)

h h h

(34,11) (35,17)

p h

(37,3)

h

(37,11)

o

0

(32,4)

(32,6)

2 5

de

oc

f-

4

(34,6)

h n f - vA[31]

Pj

(35,5)

h d f - vA[32,3]

4

(36,6)

t

(37,5)

X f-

Pk

(38,2)

h h

(38,4)

e

(38,6)

v

(39,3)

h h h

(39,4)

h c

(39,7)

(39,9)

U f-

(39,15)

(39,17)

h p i f - "1

(40,8)

g2

(41,3)

C

(41,21)

pSh

f-

2

2h

A[37)

f-

2 5

1

Ph h

(40,19)

p

4c

A[40,2]

f-

1 5

A[39,3]

f-

135

(40,6)

(41,16 )

B[34)

f-

025

f - 11"1

o

4

3

p e

o

2

2 1

A[36]

f-

o-A[30]

3 5

5 0

o-A[32, 1]

f-

A[39,2]

2 2

o

f-

4

C[20]2

2

f - OC

1

5

(42,6)

(42,9)

v

(42,12)

Pe 2 o

(42,15)

(43,11)

Pm

(43,21)

pSh

(44,10)

d r

(44,16)

p

(32,8)

h k = h

(33,13)

h p 2d

(35,8)

he = h0 m 2

(36,13)

h p2 e = h p2 2 0 0 g

(38,6)

h

(38,8)

h m

(38,18)

h p h

(44,4)

g

f-

2

C[44)

Multiplication by h (31,9)

h

2Pg

(34,9)

h d e

(37,12)

hPj

2

= h0 e 0

= h 0 Pk

h (39,13 )

= h1eog 2hom 2 h p g = h Pd e

(41,17)

h p

(43,12)

2

2

0

3d 0

f-

2

5

3g

:

= hodoe o

2 0 0

2

2

o

l/2C[20]2

f-

0 0

= h p 0

3e 0

2

2d 2

h x

1

1

4

2

2 3d

2

3h

(40, 4)

h

(42,13)

h Pd e

2

0 0

3

h p e 2

= h p 0

h

5

o

e

0

0

2h

0

(40,12)

h

1 3 5

=

h Pe 0

= h p 0

3

g

2 0

(42,16)

2

2

0

= h p2 e

= h 02 y

0

0

322

•

29 28 27 26 25

24

23 22 21

20 19 18

17 16

15 14 13

,/

12

,-

,

,-

,-

11

10

,-

,

•

I

I

»

,•

.,

: I

I

•

9

,,

I

8

,

I

7

• I

6

5

•

4

I

.

,•

•

Ix

•

3 2 1

44

45

46

47

48

49

50

51

52

53

54

55

56

57

323 Notat ion:

(45,3)

(45,5)

(45,9) (46,7)

W f- A[45,2J

(45,12)

Peog

(45,15)

(46,8)

N f- 1)2C[44J

(46,11)

gj f- 1)A[45,2J

(46,14)

(46,20)

p 4d

(47,5)

h g

(47,8)

Ph e

(47,10)

(47,13)

Q'

(47,13)

Pu f- 1)2A[45, 2J

(47,16)

(47,20)

(48,4)

he

o

(48,5)

he

(48,12)

Pg f- 1)8[47J

2 2

f- vC[44J

f- A[47J

5 0

3 2

(48,6)

h h e f- vA[45,lJ

(48,7)

B

(48,15)

2 p t

(48,23)

p5 e

(49,5)

(49,11)

gk

(49,14)

pz

(49,20)

(49,25)

(50,4)

he

(50,10)

(50,13)

Pv

(50,19)

(51,5)

h g f- uC[44J

(51, 12)

e

(52,5)

D

(52,14)

Pd r

( 53, 7)

h C f- vA[50,2J

(50,6) (50,16)

5 0

2

C f- A[50,2J 2 2

p e

o

(51,8)

h 8 f- v

(51,15)

p m

2D[45J

050

f- a

6

f- vD[45J

2

o

5 1

f- 1).

5

f- A[50, 1]

5 3

(51,9)

gn f- 1)A[50,2J

(51,25)

p6h

(52,8)

(52,11)

gt f- A[52,2J

(52,20)

(53,5)

h h e f-

(53,9)

(53,10)

x' f- A[8JD[45J

(53,13)

Pw f- 1)A[52,2J

(53,16)

(53,19)

p3 k

(54,6)

G

(54,9)

h h i

(54,10)

R

(54,15)

Pgj

(55,11)

gm

(55,20)

p3 e d

(56,9)

Ph e

2 2 2

(54,8)

h i

(54,12)

e g f- A[54,2J

(54,24)

p d

(55,17)

2

5

2

o 5

2

f- (3

251

o

6

v A[ 50,

1]

f- A[54, 1]

5

-(54,17)

o 1

o

2

1

h p

2e

1

0

g = h

7R 0

1

(55,14)

Pe r f-1)A[54,2] o

(55,23)

p4i

(56,10)

gt f- v 2A[50,2]

(56,13)

(56,16)

p2

(56,19)

(56,27)

(57,7)

Q f- A[57J

(57,8)

(57,12)

(57,15)

(57,18)

(57,24)

(57,29)

p u

(55,25) (56,10)

R'

o

o

5 0

0

A[56J

g2

2

f- a

7

324

•

33

32 31 30 29 28 27

•

26 25 24 23 22 21 20 19 18 17 16 15

•

)'

14

/

•

13 12 11

/

/

/ /

• •

10 9 8

i

7 6 5 4 3 2 1 57

58

59

60

61

62

63

64

65

325 Notation:

(58,6)

(58,14)

(58,17)

(58,20)

(58,23)

(59,10)

(59,16)

(59,19)

(60,7)

(60,9)

B

(60,11)

(60,12)

(60,15)

Pgl f- ljA[59,2]

(60,18)

(60,24)

(61,4)

D

h (A+A' )f-A[61]

(59,13)

d W f- A[59,2]

(59,29)

p h

(61,6)

a 7

2

f- (3

7

4

3

(61,6)

A+A'

(61,7)

(61,9)

(61,11)

rn

(61,14)

(61,17)

(61,20)

(62,2)

(62,S)

H f- A[62,4]

(62,6)

h n f- A[ 62, 3 ]

(62,8)

C

(62,10)

R

(62,10)

(62,10)

PG

(62,13)

gv

(62,15)

(62,16)

Pe g (-

(62,22)

(62,28)

p 6 ct

(63,7)

(l-dC'+cX (-A[63]

(62,8)

(63,7)

A

E f- B[62] 1

cC'+(l-c)X

2

a

(61,23) 1

a f- A[62,2]

2

a

2 Tl B [ 60]

a 2

5

f-

(62,19) (63,1)

h

(63,8)

h B

6

2 3

(63,10)

(63,15)

(63,18)

p2 e r a

(63,21)

(63,24)

(63,26)

ha

2 5h 6

(64,S)

hD

(64,6)

A"

h A = h A'

(64,7)

h A"f-B[64,2]

(64,8)

gg

h Q

(64,10)

(64,14)

PQ

(64,14)

(64,15)

(64,17)

Pd v a

(64,20)

(64,23)

(64,31)

p

(65,3)

(65,6)

h H

(65,7)

h h n

(65,9)

he 2 a

(65,10)

B

(64,2)

h h

(64,7) (64,8)

(65,7)

1 6

f- B[64, 1]

2

3 2

hD

3 2

2

f--

A[64,2]

2 3

a

2

1

(65,12)

(65,13)

(65,13)

(65,16)

(65,19)

(65,22)

(65,28)

(65,33)

( - IX

8

2B[62]

7

f-

'3'7

f- A[64,3]

2 1

C

a

2 5 23

f-

TI'3'

7

326 Multiplication by h : 2

- ( 45, 7) (48,6) (52,21) (55,21)

(45,16)

(47,17)

h p

hhd = hhe 2S0 oso

(48,16)

(49,21)

h p4 d

4

4

4

3

h 2P g = hoP eod h h p

3k

2 0

(57,25)

-(53,20)

h 2P eo = hoP g

h pSd 2

0

=

= h PSe 0

S

(58,21) -(59,20)

(60,25)

h 2P eo = hoP g

(62,11)

h B = h B 2 21 0 22

(64,24)

h p k 2

0

S

4

h p t 0

h x' 2

= h 0R'

h 2hSi = hohsj

0 21

4

(56,11)

0

h R' = h B 2

o

3

h p e d 2

0 0

3

h p t 2

h D

2 2

(62,29) (65,11)

= h 0p

2

h B

2 22

3

e

0

h B

0 23

0

= h p 4e 0

= h0p

h p h

(56,20)

h p3 k = h p3 t

(57,11)

h R = h R'

(58,24)

h p4 i = h p4 j

2

2

2 2

0

0

2 Sd

6

0

0

1

1

2

2

2 6d

h p

2

0

h Q = h

0

2

0

2

(54,26)

m

= h A

7

h p h

= h 0p

3 2

= h 0 p 2e 0 d0

3g

h Ph h = h 2h d 22S OSO

2

oB3

(61,24)

h p4. = h p4 k 2 J 0

(63,25)

hPg=hPed

(65,29)

h p 6d

S

4

2 2

0

0

0 0

h p6 e 0

0

APPENDIX 7:

ELEMENT

REPRESENTING MAPS

REPRESENTATIVE B

M

Xl\Y

Xy

B

BOUNDARY

(X

" BYZ

) v

(B

li 1

1)

li 2

V