Introduction to Topology

Citation preview

Introduction to

Topolo9Y

I ntroduction to

TO POLOQ Y

10 .

r . 6opKc o lJlt'f .

H. M.

E /nt 3111llJ:OIl ,

JI . A .

H 3pllHlIe DHOf.

T . H . ¢IoMeHkO

BBELlEHHE B T OOm a n-tlO

JiJ.a a Te..'1bC'T BO « B .. ICWWII W1ICoJla»

M """",

Introduction to

lf~p~~~~y YU. BORISQVICH. N. BUZNYAKOV.

VA, IZRA ILEVICH. T. FOMENKO

Translated rrom the Russian by Olea Efunov

MIR PUBLISHERS

• MOSCOW

First pu b tl. hed n u ~

o

from the 19Itl It...,.;.,. edHloa

" m~

..

8~

mKQM )O,

19«1

() EnalW> u ana1f,Ooa. "'Ill PIIblWlt rs 15lll'

, J'lRST NOTlOm OF TOPOl.OCiY 1 .W!laI Il~ 1. ~1Wotioft of ,be COflCqltJ

of ~I*'C IIIld '''''''ion

3. From I rIlftric to lQPOlosical watt 4. The notion of ItiaDaan _fad: 5. SomnbiD& about knot, FlIfIher; .-lin,

" ,"". ""

GENSRAL TOPOLOGY l. Topo lotleal lpoIftvu , liven '/Ii'ithoul proo f and saves as a basis for me IntrodlK1ioo o f the dqrcc of. nuoPPinl of sphcns an d the characteristic: of • vector r.eld (willi the Brou wer and fundamental theor em o f alarM' bcinl Qedueed); while in thc bo mo locY IlfOUP section (Cb . V). th e t echniq ue b U1 cl:Kkd to CU(t SC'QUCIJCaI. In p*rtil;Ular , t he lJou p H ,. (5" : Z) is com puted, and the Brollwcr an d Lefschcu. rl!led-poi nt theorems arc pro~ . Tn spite of havln! prepa red cY

- ,~ .. 0

red_,. into ' l

""ov, - 'v

.. ' I'

(6')

O. and'2 inlO 71 '" .. .

(l _ 1) . r "" J:' - 'l 1 - '2

lransro, ms, as can be n~l)' seen, alacbraic tqua tion (6 ' ) lero .he atsebr ail: (quallon ,,2 _ 7 = O. 1M COfTUPOIId1n& fIlIIPpinl . : C x C - C x C, {to (T, ,,). reduces 0 2 into n I .Dd is • homeolltOrpbiam. and the homromorphi$m n I onlO5 1 is pVUI by pro j«tiou (2): " = I' (t}, .here t ~ ('I', . ). Thus. we ha..., 1M commutat ive dlq:ram

w,-

") If tile In~oI

.-

", ,;w en OQ

,

.

-

IRIi:......ot:- rilG - rJd: . jR«, w)c4

n..lhea bonmnuJ ma ppiap of d/aaram P ) mIIbk us 10 tn.n d ona IlIIl'1tOtile ill. • j A(. I... )dP.

"

on lIIe ... sptu:n: Sl, wtme A io• nuand f\lAlCUOOlo. Th is _ I,u q.and by lhe fo""" EllICf I:IIbl.Iit utioll

• _ fr_

lJ for

the ra:.loaaJlUIlOll of

ee

-. .J§:

:........:..J. .

1 -

'J

We will come to I n essentially ne'" fcsulllf ....e cOnJidcr a polynomialp (t} oflhe thIrd dcu«. ThUll, corwd« an II1lu , we 111.10 them alone ,he cClfnll)Ol)di llS cu u : The constru ct ions arc similar 10 IboJc inIDealed in r IA 2S • ..o will produce a

_ ("1 ) , -1 1 +1 2 2

sphtrc Wllh

__ I

• _ _

. ,

,

. . -1 2

..

h .nllks. n ns IS the

R,cmann surfau of fUn 2 and Is an integer, and verify tlull il is .. -sheeted and 10POlopeal ly equivaknlto lbe sphere, Th e inVUliption of Don·algebraic lZlal)'lic functions in the ~·plane aJso ICllds 10 Rieman n surfaces QI1 wbid! the attal)'1ie fWlet ions all: OM-" . lued . Enrdfe 2-. Cemsid",. tbe Iopritbmic: fun I ....... bel- of buic ~ (iaoc:Iud.lnI 1-...;....,..;00 · 1 MId dl~1WlIl mlJlirolds) il carried: 0lIl. by Ef'ml~ in ~ f1/ DtrJW1t/aT7 AIlzt" -wtia, V. 5 ~'7. "~tif TopoIOu f2f 1 (pP. 476-'~ VlnlaI ~ 1(7) by Boll)'ln' sl111ld Etranoridlmay be Quite _rlJl rt>r I h~ bq;iDncr. lt upl""l the ide:u. basic notloflS ad I'"LI'lIlnas of IOPOloIY in I popuIlr mannl:\" . Abo r nl eo.w.plst>!ThpoJu lUI by Olinn WIdSlem.od Ib outd be noted . 1'bt: problems or ,luiq IOJfthcr lwo-dimcmional ,wraon 1ft' al s.o coveml ln l>O~r boIlkJ: Whlfl /$MQlhnntlt/ a1 [131 (Ctl . V) by Co urant and Rob blJlJ. AlfSdNlueM CAomrtrle , [19J (0) . VI) by Hilbert and Cohn·Vossen. New Mtll~t1«J/ DI"""ioIa f rom Sdtt,,1(fIc A/Iffrl('a" 1"'1, and ~ Unexpect ed 1I_,In, ilItd Other MtIIJre",.. ,kaI ~ (HI. etc . llcy cacruplalN bow M~ld .slrips Ir e llScd for l'indin• • 'Nhm ~ lnt~ tilt EllIer dwaetoislic. we abo ldcd _ tedu'liclues rro-! A " A~ I of tlte &.sic Itletu of Tf:l9OIoD by BtlhYIlIlkJ' IIld Efr cll\O\icll (I6I, ..... _ from Cm.dCr (141. The cl.aail'"atiCICI of l-.dimenslooal surracesl:s _ ed 1'ftY lill::lroq/lly in A ~ Topo./tIu : AM l " trotIuU_ Ij 21 (0.. I and ClI. 2) by M.wcy 1Ad.lso" !he ~_ T~ [7 11 (ClI. IQ by seirert ..,d Tbldf.P. MCIril: WlICa IIIId thdr _plftas an: dQk with ill 1,,/I'Odw6tM I t:> SeI 71fer;Iq aMI a-rfII T~ PI (Ct. 4) by A1allftdtOl' UIdill lbc ICIfl~ at rw.etionaI-l1IU. (461 (Q. 1491 (CII . lTd 1l'I. An e1M1m!At)' approach 10 1M idea or. copoIop:,al '~ fIlIY be found Ia !he 1_ boob _lolled II thc bqlrICIlaa; or tbiI wnoey. Note abo ia tllD _nealon · l.m~ ' 1Inll ' Hiltorical NoI.e' 10 A.. I rfOllll !be boot by Bourbati Topotjlfw ~ 11' 1 wllidI _ _ flI in Scc:. 1.

m.

n

"

IlIl:rod lltllon to Topoloa y

The "", o:oocepu of 00lft(lk:I wariablc f....etloo theory I h.1 we fc furilld 10 in Sec. . .... y be fOtDld . tor exUAplc., .. Inland dootftenl or \he notiont ioIrod llo

...

TH EOflEM 1. A (1)vuf" g !S"l nQlurally tCne'ltl~J Q tQJ)Ology on X , vh .• the cametlolt of n tr (v _ " 5..1, 1~/r~ r'I! K lJ on tub /In" )' finite $IIbstl from tal, lJ 0 lu:st

for lhe topology .

PaQOv. Verify Ihal the co.lltttion (1'1 $8\is fiea the cri terio n of a base . In fact, pUI V,. '" v. n V"for v.. " V".ObviDUdy. V,.'" IVl,and , l hertfore, the aiterio no fa base is flllrdled. • Th us. the co " mn. [S.l o f th e set X dacrmilKll 11 IOpoloU OD X whow ope n seu ' " aJJlhe posslbk union s U ( " S.) an d Ihe cm Pl )' set . DEFINITION l . The fami ly IS.. J IcntniCS.

...

iii

c:aIkd a s..bba5o> fOf t he toPOlogy whic h k

""""'=

S. ler. X . R I. Set.5of tbt formS.. '2 ~ :x < crJ, cr lJR l , and S, = Ir:i > ,/JJ, ,/Jfl R I , form a w bbLW ror th e lopolou of th e Dum bcF line R I. 6. La X _ R· be an II-dime nsional vector space. A base is a ccl lealon o f seu B ,. {Y• • ~ In R" . 'Il'hcn: v..... - Ir E R It : . / < f, < b/, I _ I• . . . • IIJ. f , is me j -(h

coonliftate:of the: vectot..t

-

(t l , h . .

. . ,(" );,, -

(01"

• . • o.J and b ,. (b ,•

• • • • b,, ) an vbitn.ry "ecton III R " , 0 / < b,. Sets lite: v.... arc c:aJItd optll ptl1'af~"p/pMJ In R -. •

£xvrirr 2". Pr O'le: rn.t the set of para llelepipedJ: described in bur rOf' Ihe topolo lY on R" .

~ample

6 fornu •

"

11 ls natural. fo r a topological spa«, 10 51:1«\ • base with lhe IcaM possibk numb« o f clementi. For examp le, sets II' .. ('t . in R I, whae '1 ' /2 arc ration al. form • base c:onsistina o f I w unabk set of clements. Similarly, mere is. countable base for R- cons u lin a of puallelepi~..-ith ra tional vcrtica or the fonn

tv

...,1'1 -

(x : ,~
N .

Conside.r an ope n ball D .VCxr}) in Y and den ot e 11 by V• . lIS inverse Image 10 the continu ilYo f! . mceecver Xo ' (V. ). The poinl ..-ob elonp to/ -I(v. ) tOj elhe r wilh some ball D.Cxol o ( rad ius

onR l : ( I) W~ J( .. (fl ' . •• • Ell)' Y '" ('" • . • , 11,,)

arc 110'0 .rbill'a1y vect o rs from R It•

Let IU veri fy that this il a mie Tie . Evklently. Properties I , II , III of a met ric (sec Sec. 2, 01. I) are fulfilled . Con sider Praputy IV. II is rcqu~d to prove the incq uality

for ar bitrary real numbers lemmata.

E,. "'_ fl'; '"

I• .• .

,n. The proof is broken into

LEMMA l (THE CAUCHY· BOUNIAKOWSKY INEQUAUTY). For any rtal ~I' ", i = I, . ..• n , th~/o/lfJ winB inrqUD/ily holds

PROOF. For an arb itrary real 1\. we have

E 1- I

E: + ~

1:• 1_ I

E:/l/ + >,Z

•

•

E (fl ,. ,

two

num~n

+ 1\11;)] ;;, 0 , whlffi~

E ,: ~ O. Consider t he left-han d ·side of the in1_

r

eq uall tt as I polynomial in 1\. It cannot have t wo different real roo ts. Therefo re , its d.Iscril:ninant is lIOI\opolltive. Hence , the inequalit y

E: ,-E, 'If· •

Ch. 2. C."",al TopolO!y

4?

LEMMA 2 (THE MINKOW SK I INF.QUA..L ITV) . For Ilrbitrory N!UI numb~n ~I' " I. 'h~foJlo wjnll ineq uality is valid

I '" I . . . . • n .

( i:

0 .. l

" ,+ "" )'" • (

i: 'i)'" + ( i: ,i)''',

I. •

• .. I

PlOOF. By using the Ca \lchy-Bo uniak ows lr.y ineq ual it y,

•

;

r ~

Q', + >1,)2 '" I

•

(~ f + 2l:,l1, + ,,~) ,r- ,

< ,. , e

r

'in r r r- "'r r

+2 ( ,.r ,

[et 'i

+ (

,.r , ,,1

,

,

+

r ,

,. , "

'

an d by tak ing the sq uare root of bolh sidrtofthls inequaHIY....eobtain the required inequalily . • We ca n no... com plete rbe verif>eatio n of P roperty IV of t he met,ic. Usins the Minlr.owski Inequality, we o btain

c

( ,~.,(~,_f,)l)'" + ( O ~.I

is a metri c o n R" . • (~y . . .. , ~~) be lhe ce ntre o r a ball D ~""ol, an d x .. (~ ; •. . . • ~,, ) its arbitrary poi nt, Then the coordina tes of a po int X sa lisfy t he ino:quaillY ( 2) I ~I _ ~Y 12 + .. + l ~ ~ _ ~~12 < r 2, Thus,

p

Ut x o _

A ball in R" is often denoted by

U;""ol an d called

an oJNn 'I_disC. A set of pointsx

whose coordinates 58tlsfy th e unsmcr inequal ity

I~ i -~~ I '+ .

+ t~n -~I ' ,;;; r2

(3)

is called a cloud ball (d OSlid n-dl$c) TY: V r;). The (n - l)-d imensio nal sp here S; - 1(.l'r;) wilh ra dius , and cent re at lhe paim xl) is d efined by the equ ality 1£1 _ ~Y1 2

+ .. . + I ~" _ f~12 = r 2,

( 4)

We will call It th e bo undQry of tht! disc 15'; or JY;. A metric on R" can be defined in other ways , for e: "" , )' ) =

(I ~, , . rna" " ."

'fII J.

'"

llltrodlldioo loTopoloty ~

I - _ Describe II ball in R" by means o fm~ (' ). Show th . Ihc E lJdid ear mel ric and metric: (S) ind uce Ibc same topology . ConsId ft" the complC'Jl .. -d immJionll1 space C" ;

c- .. l:t ; c ..

«I' . ..• c,,).

c,. .. x. + iy,.. x,.. J',. I! R I . k .. 1• . . . . Il lllIe ma.rk on II iI. introdl.ll::«l ln the same: 'II'lIy as in the real case ; p(t ' , z " ) .. ( Ici - :t:i ' 12 + . ' . + I:t:'; - :'; - 11)11lI. wher e.t · .. k ; • . . . • t~ ). e ' - (t j ' • . . . , 10POlosY is detmnined by the: marie p(t · .Z H ) .

mLll .. . I.

••

c~ ')

are: elemen ts o f C" . Th e -..mc

ICk - t k· 1.

We no w for mula le: a condition for the conl in uilY o f mappinp o f Euclidean spe.ees. A mappingf; R" - R'" a ssocia tes each poi nt (fl ' .. . • fIll wilh a « nain . so that we: a n wrile po int (" I ' ... , 'I,,> '" - f t U I" . . • f ,,),

(6)

...... f. (f l ' · · . . f,,> .

wheref••" " I•• _ . , m ia a n~ fuoction o f II variables . This fUnCiio n determinc:s a mappinzf,..: R " _ R l by the: n de 'Ii

- f,(f l' • • . • f .).

(7l

It if evidml that th e COIUinwty o f th e mllppill a /' is cqu ivalml to I ~ co ntinuit y of the nwtlCfical functio!t Jj U I' . . . • as iI is dcruted IJl analys.is. CaU mappinl (7) the i..fl\ component of the mapplnlf. The ma ppina l is det ermin ed by spc:cif)'in l aU itH » mponcn ts f ,. j - I • . . • • no. THEOREM). A nuzppilfZ f ; R " - R - is cot/';""ow if llnd 0 .. 1)o rpbisrCl (8 ) is also dcrlrled o n S'" - 1, and lh al

so:. -

r

II SO _ > -

IS" _ 1- Thus ,

0- -

I is homeomorphic 10

We now csu,blish llnOthrr impon an t

St'. - '.

~(omorph ism .

THEORE..'Iot 4. T1l~ disc D '" .. II ortWomorpJrk to /II( space R"'. m ;;0 1.

PII,OOI' PUllinS m .. " -

I . we use t h( pr( vIou s constructl on . We tl'3nslace the

. pace R~- I, II "" 1, so Ihal ch( orls in of coordinate:s 10 l ).

DI!F1NrTlON • . A fam ily Q • (IQj]' hPi/ll,where IQIJ is a flnite set or disjo int plan e polYlons and I.. ill a finite K1 or &l uing homcomo rphi5ms of pairs o f edSel, each tdae bdni slu ed lO only o ne ed, c. is called a d~w:/QpmMI . Olu; nl the cdt. es or the SlIM polylOtl totUher is perm itted .

60

IOlr OOul;(;O/lI OT opoloay

,

,

, 'D ' ---------t t~~ 0

, I

,

-.... - --71' - --- ; ,

,

Fit!:. • • No te th ill if the the location of a polytlo n Q, on lilt: plan e is altered by I homeomo rphism ..,. then we act new homeomoill hisms (OJ'''u-at IJ t ha t s:Jue iu edges, and which we shall not distinl Uish hereafter fro m the hOmeomorp hisms (YOu!. In particular , thc famil y , we 3pCCify. ciralmnaviBation (i. e. , ori enta!;on) for eaeh polYlon. Each ro ge o f each polygon Q, will be denot ed by a letter accordi ng 10 th " follo wing rule: given a tIomcomorphl&m "'lI for a pair of edges. w" uenc te on" of th e C'd, es by II and cltcc:k wMthn Ihe Olienlatlon th at the homeomorp hism '1'/1 induces (eartiel; o ver) from II 01110tbe ~ond edse ari""ick:s with t heorieDlatlo n Mlh" laller . If thc )' do coi ncid " lbal lhc IoCtX)ftd ed se o f Ih" pa ir is also inS deooI ed the ed.&es of aD Ihe pol),lOm Q,• .... e obtain a sa o f wonls (wCQJ\. ....here ...CQ,) is a word dc nodna Iftc 'Vui.llS· ruIc: Of l he polygoo Q" In ldeli· don . tllle lcUef1l in Ibe won!. ...CQ,) arc ....ritl Cft In the onier in whicb ""' tM axrespondinl sides o f!be poInoo Q,.eeordlnl lo itl o rien tal ion . It i5 d ear that the Ia~ted sa of $YRlbo lic 'IOOrds !o-CQ.)I determ....es the "'elopmcnt Q . Two IIWn types of devc:lopmcn u can be t.in&kd ou t . OEFlNlTIO!'l 6. A T'ypr I auton icrJJ ~r is. dc'Iocloprnent I or poInon detntnincd by. word who« fonn

is . -

lI,b"" Ib, IlIP }!'i 'b i I ... lI.,b"p;'Ib;' I. ", >

consisti.Dt of ODe

o.

OEPINrTlON 7. A ~ It CilnoJt/all dewIop_1 is • ~dopmenl ~ o f llIlr pot)'lon with • word of the: form " I""'}!'! . . . ""pM' m > o.

We now formulate the basi c resul t. THEOREM I. A II)' dtwlopmMr Is ~{Wl/enr 10 " ~ /

()T

JJ Ctmr" ,icDI de...m,p.

mornlocclHdin, 10 ir" orienr"bility lH non~"'"bi/ily. PIIOOP T wo mt'IIIrks II nut. To bc&in with. it Is eaJ)' to see tbat by s1uilll, lite development C(ItTapondios to a triangulation K o f a w rf aec X ean be redllocd to , development a:msisl ml o f one pol)'go n . We shaD ther efor e consi der onl )' lb i.l kind of deve lopment. Sc. 2. Gcner.ll Ta polallY

a

iKlSon it according to the ru le ~ ...x .. (~Ia~ I' ~"·h ..... ~~~ .. I ). ThUs,group WI be identified Wilh th e un it circumference S l in th e complex plane C . He nce. S ' ICU on the coordinate {, e C, and the o rbit o f the poinl tj ln C is the clecumference of radius 1 ~ 1 1 if I~I I "" O. Therefore. Ihe o rbit Ox :::> (r'"xj (0 " (l> Ihat O~) n A .. 0 . Le t.e " II! O~) be an arbiU'lU'}' poud.

"

..

Then fo r any DeiahboUrhood v(r ' ) o f the poiDt)t ' such lbal 1'(1") C O(r). we kne V(r ' )nA _ 0 . thc:rerlX". x · is notalirniIJlOUlIOrA ~ O (lr) n A · '"' 0. 1'bIu. OU') C X '.CA U A " ) . and bec:aw.er is vbitrary, the 5C:t X '(A U ", ' ) is

.,..

Eumx 2". (I ) Vcri(y WI (A UB)' _ A ' U B ' ,

(A nSf CA ' nB '

and

(A 'S)' :::>A " ' 8 ' ,

Lei X .. la. b l be .. spa cc o f two c lemen t' cq uip pro wilh lhe topology eo ns;stins Qf lllc thte" sets: 0. X , la). Gi..." an ex.1lmple of a KtA C X fOI wh ich the ind usia n (A T CA ' is 001 valid . We shal l now prove .. buic st atem ent abou l lhe structure o f the:: closure of .. set.

(1)

TIlEOR.£M 3. X - A U A '

l or anyMt A C X .

8 y Theorem 2, ee KI A U A ' is ~ . Therefore. by the d efin ition of .. do$u rc, A C A U A ' . On t he other hand. it is ob viou.s thu an y clcna1 Jet contairlinl A also COIltaUu all limit poi nu o f .4 , and m a do!c oonlairu A ' . Heoce , A U J!' C A. ThIlS, X '" A VA ' , EnrfiM 3· , Let A W 11v XI oj ""tioll4lpoi1!u on 1M. ,"1$trai$"t liM R I . ~ PJOOf'.

l.!lacii - R I •

If. topolopeal lPKC X h., .. eounable su bset A ..mote do$ure urinddcs wit h X, Ihen It Is ~ to be sqxupble. (I is cu:r to verify that ~mty is a lopoloPea! P!VP"r1y. Extrf:ius, 4" , Sho w that the Spate R ~ . the dise D ft • an d the sphere S" j",

- I are separ able .

Verify t he followina properties o f the closu re o~fat ion :

m

'" A U

B,

A,~C A n B,A ' Be A'::B, 6", Let Y be a $Ub~ of a topoloJical space X and A • sulurt of Y. Denote t he doIure of the set A in the SIIbspaa: Y b:r A y . and Ihe c10lIIfC of A in X by A . Show

A"

that Al" -

A ny.

DEFlNlTION 3. A pol nt x ~A is oaid 10 be isolDlftI if there is . nci&bbowhood 0(%') of the point x JUdI that it docs not contain lilly points o f the Jet A olha than x ,

A poin t X 6 A is iloJated if and o l\ly If x e A ' A ". O£Ft1'4ITtON 4. A set A is said to be d i$r:,c,r If each of its points is isolated .

z. The

Interio r

of a

Set.

COll1idef t WO other lmponant no tio ns co nnc:cted

wilh that o f neighbourhood. OeFINITION ,. A poin t x e A IS call ed lUI iIl/mo, poiflf of a set A i f it hu • nci,hbouthood fl (x') sUU- Co lUider A _ [0, II, Ihe line ·sca,mc:nl of the l u i st rap

cur 10 sec lhat (nt

"

lnuodu o' ion Co T..."olo V

lliEOREM 4 . For 1m }' XI A ex. It't' /IQ ,'li': (l ) Inl A £~ (IJl ~/I .J1!r; O l lnt A is IIw Ja,.,ur 0Ptll st.1 COnlD;nM ill A ; (l) (A IS O/H n ) • (In cA .. A ) ; (4) I,l"G InIA) • (I' E A orfd}l is 110 /" J"n" polit I of X 'A) : (S)X ' ''' = X , In tA .

_....

PIl00f. Properties O ){l) au aImasl e'ricknl . We wit Y'tTif)', fOf cump~. Propm)' (I). Lei x E Inl A . Then Ihne is an ope n nclshboufbood U(.l') of Ihe pai nt x $uch that U (I') C A . The rcrOft:, IJlI.A is II OOBbbourhood of e~ of Its poinU and e eeee u

AI for prapmy(4). if x E Int A lMn , obviou, ty. }lE A andx'l"(.X' A )' . ConVUK . Iy. if Jl E A and Jl £ (.X , "' r then th~ il; :a nei,lIbou rtlood 0 ",") CA . ther efo re,

X l' lai A .

The vm fica1ion of propert y (S) is k ft to !.bl;: rc.clu. _ lbe ~ ltI1 (.X has to be c:on.sidc~ quite oflt:n II Is aflcd 1M u.tcrior of the lei A IUId dmotfd by u t A .

' A)

~l· .Showlhat A "

X 'edA .

3. The Bound ary or a

Set. The fo11owinS important conceP'S.llfC those 0' boulld ary po int an d the boundary of. Jet A . They ..-c ~ecnu.nd Iea,'c the ot hen as Q O. bein& contained ..molly in A: this folo\WS f, om the 6f:rmi(ion of th e marie topoJoay ,~;

"'.1

(b) the condition x EA ' is equivalent to the ctistencc of • sequence con\'l:l",ent to Jr . where a• • A , a.. ~ If . In fa ct . if Jr GA ' the n for anY " 1 > 0, there is an clement " I in A such that IIle D,. ,,"), III '" x . Le i 0 < "1 < p (.c, a ll . then alain there is an element

*'

"1 E D, (xl. a1 Jr , etc. Th us, the sequ ences ir..J and 1a..1C A (ire co nstructed such thtp';... Jr) < "., ". - 0, a. '" If, Le .• a.. - Jr. Conversely. lei there emt a seque nce a.. - s , where a. oil x, a.. Ii A . T hen for any nriahbou' bood 0(:0") o f Ihe point .... there W SI a b. 1I D.,,") C Q(r) ud N (e) SIKh th.l p {a•• x) < e fnf It ;l= N(e) . H mee G" € 0 ,,") w~ It ;l N (c) :l nd G. Jr,

*'

. hich comp letes lbc proof. Th e doerlD itioa o f . limit poinl in tmns o f KCI_ cnn~&Cft1 to it livu abo¥e is a1.... ys used in anal ysis as lbc de flnitioo nf a limit po int or. set; (c) th e condit iOn tha i a itt A is dowd Implies . just like lOt • to polngkal JpK e, lhaol A conta!ru lllJ its llmi l poi nls.. This cond ition n. eqllivak'ntlo the fac1 Ihal th e condition z. € A fol ln. . from the emlen c:e nf a scqucnoe \ll.l C A convusellt 10 .... III face. th e condition that A is dosed is equi valent . for eumtlle , t o lhe con di tion lb. l A • C A (ICe 5«: . $) which Is equiv alent to the previous $IDtemcnl ;

*

(d) llle con dilion ;It 11 llA is eq uivalent to Dr "" ) n A 0 an d D , (K) n (X 'A ) 'fJ 0 fo r any , > O. t.e.• an)' ball wilh cent re althe poin l'" will ' JCOOp' OUI the po inu of A I.fId X ' A . Th is st.tement is ob~iOlll . We ar e also liv inl an eq uival ent definition which Is n lt m used in anal ysis; (e) Ihe OOIlditiofl x E aA is cqui'flllenllO the uistrncc of a Jocquenr:e ~~J . X ' A (OIIYCf'1lCf1.t tox. and to t he exlSlenoc or. It:quencc 1a,,1C A "OIIVCIlIenl to x . In fact. su ppose X li! 'M. Then for . ny " > O. the ~n Dr,,"l 'KOOPS' poi nts OIlt of both A (i .e., tbc po int a, ) ~ X 'A (i.e.•• th e poinl G; ) . AswImin a!hlll. ' .. "•• ". _ O. we obt&in th.. sequences G ,. E .4, .~. C X ,A sudl tha i . '. _ x, G;. - x,

Introd uction co Topo\oJY

"

Con vtndy, ih. - x.Ia.1C A. and/l~ - z. I'r~ 1 C X , A., thmany ba UD, (.l") coa tains both the point " . U>dth e point ,,; fCll' a sv.rrociently IaraCIi .. " (r); lherera rt,

JUliA .

2 . Balls

an d Spheres in R" . We lba.Il invcstip

lc the sptlcrc S· , 1he opcn 4ise

1>" .. I an d the dosed disc1)- ... I in R" .. " TH.EOREM I . The/ allowin g

tqutllil~s are Wllid : U-

.. 1 .. u)i+I) .. (D" '" ' )

.

PROOF If llle 'r ay' [txoJ, 0 " t < +00 , I, co nsidered (i t emanat es fro m tb e cenIn o f the ball, the poun o , and passes t llro u&b the poinl Xotii b" .. 1' %0 .. 0), then

lbe poinlSxi .. Ie ;

1 xo Ofthis ray Imd loxOand lie ill D" " l (vuify th is by ou.inI

lhc metric on R" .. I ) , aDd the points h

'"

~ %0 also Jie ill D"

... I and I.md to :mo.

Therefore, (D" " ,)' ::l D" .. " On the ot her hand, (0" .. I) c b" .. I (JIere (L)i'TT) is lbc lopolocical doI ure of the ball D" " I). In fact . it Xl ii D" .. I . i,e., Ify e (D" .. tllen

x. - "

'r

p (Y, O) " p()' . x. ) .. p(x. , 0)

< p(Y ,.ll"... ) + J.

_ hmce by taklna: lnto accounlillal p (y . Xl> - 0 as Ie - _ . we ban p (jo. 0) " i, i.e.•Y E D"" I. After eo mbinina: the Indl.lslon malions Ihu we b vc obtained wilh Ihe eviclmt relation (d' .. 'r c ~ , we have

u+

I

c

wheGl% the I tt.teme nc o f tbe

w .. 'r c (D". theo~m

1Y'+ "

I)C

readily foUowI . •

THEOREM l . ~ $phtn Is Ih~ boundary Qj o boll : S - .. a(D"

PROOf. IA. .1'0 5 S" (S" ,p. 0l). Then

'-1

XI< .. -

.-

.1'0

+

' 1) .

E lY' '' I, and the tequem:e

~tJ

u.e., with radillS

0 . 2. o.ncnoJ T OPO!ou

r

> 0 and orllt.~ at Ih" po illt x~ by the eqlluun D,~~ ,., ll' IE M . /I (x. x~ " rl. S, tlr'ol ..

...

" 1M' e M. /Itlr', xol ..

Not" tl\1I1 D,tlr'ol. S,tlr'ol are eIosed XU ID M . l.ll faet , i ( p(E-.. ?} "

rJ. ll'.J E D,tlr'ol and ]C. - ?

Ptlr'o- :r.) + Ptlr'•• ?) " r + Ptlr'• • ?).

YoiImoc pl;co-?} " ' , Le.• ?eD,tlr'~; S, = M. SI (xol .. M ' lxol and (0 1(}CO» C D 1(}Co), Fu.:1.humOfC. (D .(X0l) ¢ D I,""ol. $1'""0> ~ ,)Dl(x~ " 0 .

Fillll Uy. when r > l,we have

D,tlr'o> = D, ,""ol - M . S,u~ " 0:

mOfCO"et".m) '" l'J,Vol *" (D, 0, there is N(c} such lhat P ~,,+ m , x,, ) t:;II: . ll " N(c). m ;;a I.

(I )

However. Ihe co nven e i. nOI always true . DEfi NITION I . The space lM . p ) in which Cau chy' s criterio n hokl ' true Il.e., any fundam ent al sequenc e h M a limit) is eaIled a clImp/ere spare. E XMll"UlS

2. Let M = Q c R I be the set of rational num ber s in R I. Th is met ric space i . no t com plete since there W s! seq\lcnc~ of ralionalll\lmbc rs convergen t lo an irra tional number (i. c ., funda mental, but havina no limit in Q ). 3. The. spaJ of the point ff;t.o>. there Qisu a tlri&hbourbood OCKol of the poin t ..., SllCh t hat/lO(.x-oil C av{col. ~ 1-. Thl! fol1o'lrina propeny o f. 1TUI!t~/ : X - Y is cq WTalm t to the cmtinuity aI. poiM: the luJI inverse ima&er l'OVf;t.oI» of ...y nciahbourhood of Ibc point/{c~ is a nci&hbourhood o f 1M poml XO'

TKEORE.... l. A _~/ : X tcJt polll l XE X .

Y is COfJlIlIUOUS if lHtd onJ,

if Jr is conrinwou:l ll t

' lOOf'. U:t/: X - Y be cont inuous. ro e X

llII ",bitr&ry paim, llIId OVf;t.olJ an aTbilralY nei&bbo urhll(ld o f the pe int/{coil. Th en then u an ope n set V C Y $\1ch thai YcOVf;t.cll) and/f;t.oiI11 V. Put U - r l ( l'}. U I5 an opc n,Jet and xo E U . Thc:n I (U) c nV. ) , an d I I x;. the restricuon of 110 X"". The d iagram

.

~

.~

1/xa, '"' It

.., /'

Jf,,:,~

can be: nal ura lly completed 10 a commutative One by the product o f twO mapplnlP bee th e dotled arrow) . We letS. U n v "" 0 . LeI (.r(loYO> E' U . The sc t.l'o )( Y is hom~onorphic 10 Yand. l ha'efor c connect ed ; inlen«t lna U aI the point {.loo. ')10>. II lies 'lItboUy in U. wtticb followl from the COMK"tod" GS o f U. The leU x Y . ye Y. inlcncd ~ x Yand lhenforc U. H oweva. bdn& ronnected . they lie Mi oUy in U. Th us , U (X x JI) ,., X x Y C U . TherefOR , V .. 0 . The conl~dil:lion proves the

:x

,H

......... theomn . •

9- .

Pro~

II co n ncc leCl spac:es (II > 2). mnneetedneu of the TibG/lOY p .odud. n X .. .. Y of collMC1ed • ....

Thron:m 6 for the pTod o.K't o f

10·. P I"O¥C t he lipac es X" .

H iJIl: Collllcler I M .-t R of tbe l!9lnu or the prod lll:t !hill can be Joi ned 10 a CftUin point by conn~ oct!, aDd vaif)' that R .. Y.

J. Connected Compone nts. If a spece II d llCo n nec;lro Ih m it is natural 10 alt cmplln decompo se it inlO co nnected pleas. We describe th it decomposition. Let x e X be a point in a lopoloakal $pace X . Consider the lar,cst connect ed seiconlainin, t he point x ; Lit " UA.... WhCfC all ~Jf ~ co nn ed ed seu eoIltalniJl& tl\c: point x . ~ set Lx Is c1osrd.!in« the clo.w~ L",orthe con nected set L" Is ronn«ted Isee EllCrd_ 2) and hence L ., C L.. l.e.,' L~ '" Lr • DEf ll'(lTION 6. The X I L" is u Ued the COIIn« t«l f:tmIpofle'll of a poilU.., in I topo loPtal spaa: X . ~ x, y eX. x ~

y.

COnsidathc Ku L~,

L., Owill,l lathrir~nnedednas and:

~IY, there are t1WO pouibiliti£5: either (1) L~ .. 1:, or (2) L~ n l I A - 0 , '" 18 • I ond

Ifor ony x ex.

Pl!.OOF. 1...cI A an d B b.e two arbitrary closed sets in X , A n B .. 0. We assc ctete uc;h rati onal num ber of t he fo nn r .. k l 2n, wher e k .. 0 , I, ... . 2", with an open set G (r ) so that the following pl opert.ia are fu Um ed:

"

In ,,(x~ and ",(.x~ .l! x C U,,/.xol lhen X I;' G(rr). Therefore .. (~ " ro' Fu r-

Pu tUN(x# = G (rol 'G(ro

x E X' G(ro

112N)

ex,

G(ro - 1/ 2"' ),

Ch 2. Gcn .. al Topology

"

therefo re , o - 1I:r' .. • • 01", - o'v>a. O'S - b , o .. V'~ .b(xJ " b . )CEX.

where 0 , b (0 < b) are arb it rary real num bers. In fael , lf .. (x) is the Uryson fu nction th en me functi on V'a, b(xl .. (b - a )",(x) + 0 Is the one requ ired . THEOREM' (l'lETZE.URYSON).Fora/lY bou/ld~ mtrllnuov.r!uncr;on '" : A _ R I thfi1/td 01/ a clas«J .rub#t A % normal spoa X , there existsacontlnuous/UIIIClion t :X - Rl suchthot4> I", . ",and s~ 1.(.%)1 "" ~ 1",(%)1• ....OOF. We $hall co nst ru ct the fu ncti on'" u the limit o f a certai n sequence o f Iunc-

ucns. Put ...o = "" and

It is clear that t he seu A o• Bo ar e closed and d i. joinl. By the ma jor Uryson lemma. there exists a conlinuous (un ctio n

t (.%) =

o

' 0 ;

X - R I such that Igo(.%)1 "

(-1'01' 001'

if if

~ and )

xa AOo )C

6 8 0-

Now. we define the funct ion "'I on A by the equality ""1 = "'0 - 8"0' Th e fun ction "' I is

therefor e continuou s anda l ..

notation

s~f 1'1'11

"-iao'

Simila rly, b y in tro du cing th~

Introduction . " TOPOlDi)'

"

wh m

xeA,

when

J< e

B.

a

"" I (x) = "'o(x) = ""s~ of a H ousd orff SjX1« Y. Ihtn X is closed.

PROOF. Let Y E ~ X . Por any po int x e X , since Y is Ha~orfr. there arc open neighbourhoods U.. (y), u,1'(.1") of the po ;nrs Y. x such uree U..(y ) n UJ'(.1") .. 0 . Th e family lU... x fo rms a covering o f X . Because X is co mpact , there Is •

(.-n••

•

finite subon, ill not ce ntred . Th US,me su b$yn cm IX'U• .D . I h.as the empty inlcnection Coc iOftle a t. 0 1' whence IV..,: • I iI ar lll.ilc w boa vaill&o f t he covcri n. [U. ). Theref ore, the sp*oc X is compact. • wi:. now con.ndCl" !be p topen.)' o f ~l1'lpKInesa. II is iltllftStina: I/) exa.mim the rdation o f paracompaclnc:u: 10 me other propen iCi of \ o po!o Pcal lJlacel. Cotl · sider the l o-QI led loca !ty compact spaocs.

... ,a,_

DEFJNmON 6. A space X is said 10 be IQqlf~ r::olPlpDCt if it b Hauadarrr and eaUi X posses.ses • neilhbouT1tood U (.t} whose dO$Wc is eo mPKI .

point

% .

OIIe exam ple of a Iocalty compaa spaoc is lbe space R "; an Olhcr II •

I WOo

dUneruioMl mani fold (Icc Sec. " , ClI . II). THEOREM". U It top%fial/ spoct! X g 10NJJyr::olPlP «' th,,, i t is ~u/flr.

cx

"'-OOf . Let fI e X be an arbiull.~ poin t, and F « closed Itt not contalnins the point fl . 1'1Ien X , Fil open, and. Q E X ' F. Because thc spe ce X isloo;ally compact ,

"'"

Im r od uc ll o n 10 Top U (x) interxas aD N' E ; . Since V is u1)iltary, we dcdu ce thl tJt t' n N'(l nd therdore JtE n NY) • • N> • •

,." • •

H~

are some exampla which ~monst~ te how the eompattneu of .spaoc an quickly W detC'r'llliaed by lbe. Tibonov theo rem. EXA> O:ruell thaI flny ~t

In X of diameter less tha n.l lies wholly in a t:erlo;n elemelll of tile ro veing 1U] .

berci# 7· Let. metnc space X be: compaet,and / : X - Y a eontinuous mapping. Pro~ tbat for any eoveriol V '" IV.l of the space Y. the re exbls a Lebesgue number ~ ... 6(U) such that for any . ubset A in X , of diam eter Ie:» than ~,the image /(A ) is whoUy contained in some element of !.he d PonU1Qbl '1 e -,/roIfOou; (;ro,.ps r6oll\. "The to&loWt& bODb are - rul Utnl nwtriailO ~ Iopia IIl Ihit chapter : f'fnI Q/ CA__ Topo/lIkJo iJr Pro/MfM Md E:r:vriJa [7J bJ "'~ &lid P- . , -. Probk.oou ill ~tq [611 by NO'iikov n aI., ..." ProbJmw I" Di//6Vl1flIJ ~('7 tIItd TopoWU 1'91 by Misbcbut:o Cl Ill. lU rcpRls Indirid..al~, _ ~ &aU llIe follow!llt r~ Foo' Iho Itvdy or tllc -..u 01 ~ aDd meuk 'P"'C aDd I1Icircontill.. 0U5 -wIII» (Stu. I IlDlI 2), tJw cOllapCll:1lll.q: ~ or tlIe above tiIIes art ,ClC:CJInII\oOIl. 5«. J. The ~ of r~ or spKC 11 _ I IborouPl1 apooJfded in me boot; by JWlc)' ~)

eo-

1"'1 (Cl.tlf), IDCl lhat by BolirtIUi l it] (0..1. Sec. 1). sec • •• Tbl: dllSlifieation or doled Iwo-dirnen1iolllll Is pr.wlled well '" Scifet1 and Tbt cllrlll 1711(01 . VI). UId . morcmodun Ippm.tb by Ralr.c\man et al. ln Jnrrodwtlo" /0 Dl/f~mrlksl Geom#try 'r,. 1M ~ . [1)1 (~. 101.and by Massey in A. "~"'k TopoIou: AII1",rollrQ' rrotn thc QlqOfY of base point spaces 10 the C

"' .

c l~

1

From tbe diqronmatie POint of view, this homotopy is aplained quite simply (F'. .. :19).

Now: to sho.... that if n

>

I , then

We shlll verify l.hat the mappinSJ"

I'P J +

1,,1 .. ["'1 + I¥'J. remembe r tIll I

+ oJ. an d., +

- ( Y . y O> induces the group h om omorphism r(1". 'J~) {f) : r ,, (X. x O> - Tn I Y,yO> .

.

THE PROOF is left to the rea der. H im : Use the construction in Exercise 3· Th e h omom Ql"phism .. U~, ""l{f) Is denoted by In and called the ,, ·di mens ional ho mOlopy aroup homomof1Jhi5m indu~d by the colltinuous mllPP;"fl I . Thu s. the fu nctor r n • n > 1. acts fro m the cales ory o f base poi nt sp~ and thei r ccralnucus map pln8S 10 th e eategory o f Abelian sro ups and their hom omor phisms. Therefor e. if

I: (.X. xa) - ( Y,YO>,g : (Y. y ,)l - (Z. to> are continuous mappings th en fzf)n '" g"ln' wher e I n' If n • (gf)" are the co.re_ sponding ho momorphis ms o f n -dlmcnslona.l homotopy gro ups . 2 . The Fundamental G rou p. II will beinterestinS to co nsider sepa n tcly the ~,

TI (X. xo> = ... (1 .

al; X . x.)

ct ...

(5 I,

PO; X , xo>

which Is endowed with a grou p structur e in the sa me manner as " " . n > I. and is applie d in many problems. By general de rmltio n, ea is c:.Ilcd the fundatnUltal t roup o f a topological space X wlth a base point xo ' PROPOSITION . The ~l

of theproducl .

-.ex.xO> is II group und er the

d~ribed product

opera/ion

PROOf , Note tha I in the proo f o f Theo rem I. lhe condition" > 1 w at used. only ",h ite pro ving the commutat ivily o f the group _.. , where the second coo rdinate o f the spheroid was talrlng pan in the necessary homotopies. Therefore all the previou ' steps o f the proo f for Th e(WC:m I can be used fo r '0" 1(x , xIV without introducing any changes. In doinS so, the unil and inverse elemell\s in "I tx, Xci are defIncd uaclly in the ~ way. viz. , 6 ~ ["p,!. where " oU) .. Xo is a constant iooc : fo r each [..J e lI' t km of computiq b.i&het" borr\O(0 py P'O'Jpt is dlsclUKd. and thcir applic:alion to • ,",obft!Jl oooc:eminI the fi nd poUlU ~ .. c:ontinlK)\lS mappiq is Jivm (th e B.-ouwer lhrorem and the fundamental thCOfttD o f a1&dnll). I.

Line Paths on a Surface and Their Combina torial Homo-

clo$Jed S1UfaceX aiftn. 1.1 m Sec. ".01 . II, by in subdiviPoft. that. devdopmml n is Pm. and the $\Irt.oe X Is horneomoophklo t lw (.aor rpuc l1IR . wberc R b all equi...uenoe determined by uie homeoInorphlmu or the ckvdopmeal.. Denote the prod uct of !he ruiO\lS pcb in • triululaJ.ion K is homotopic: 10 II c;ombinlltoriai and OlI nlinllOus lil'lC padl . C'l tl. llI.JO .stud, the rdal ionsh ip bet _ ho mol optes. Heru fier. we co n5Oder onl y raed-md homOlop>cso f paths and klops . LEMM A 3. u r til triGnsu/II/io" K 0/11 P4r/ OCf' X "'" ,ivm . ur >.: I e K ~ til eo'" (I" !IOUS I" K, ),.(0) . ),,(1) ~j", rhe wrtK:es o/ the tritl" ,,,/,,tio". Tlun '''ere f:lCirIS II tiM PIIth /n K . wh kh I.t homotopic ro iI.

/Xl'"

PItOOl'. Subdivide the line-uJIllem ( ..

to.

II wilh a finite nllml:>c1" o f polntt (ttl~_ o

Va .. 0,'. .. 1) inl D-tr\denlly m1Iline_:a.cpncnts so Ibat for cadi int erval Vc _ "

ru l ). k _ I , .. , .II - 1. mer e may be a vertex A c _ K w eb that tile imllll' ),.tt'_ I.'h ,) o f tU imen21 may lie . holly m tbe star S lAt ). the' lIn>onof the open tnan&ks and cdp of the trillllJU1a.tioo K .c1jaeatl to II eertain "",rtex A. and the ven"" A . uxIt. SinceSl,Ac ) is an opeD sa in X . and),. is a con lmuOU$ :n l ppl l!&. Ihis can a1 W11)'1 be Kltic-o'cd (see Ex. 7, Sec. I). Ct . II) . Now. we associat e each po int ' t e I Witb lbe venes A ~ E K. Nou. mor eover, thai for

an, . .. 1. .. . . 11 -

I,

),.(('k' I••

,»

C SVt c) n S (At. I)' where S CAt) n S Vt t .. , ) obviously con lain l the tria ngle whkh is adjaCCI\\ to botlt A t l\IId At . " TherefDre , If A . '" A , . 1 tlten tltey ar e jo il\cd in K by an cdtc wlticlt ...e will denote by 'c ' Let A; ; 11.1" 'k . ;l be an eleme ntl ry path which is th" 6_ Ictl$lon of the indill:atcd eormpotodcnoc of tbe Ve"fca IIId pointS ' k' If A ~ "" A t ~ , lben OR comidn to be equal to uro. The produa of elemen \llry

'tc

>.t

'.1'. "

01 l Homotopy Thoo. y

'"

paths 11 k determi nes a line path ),' : I - K caUed a lme opproximal/(m of Ihe porh. Th e paW), and ), ' ace homo topic; 10 one anolher. In fact. ill virtue of tilt structc re of the path), ' , fo r an y point If: I , t he images ), (I) and), ' (I ) lie in the same: closed topological t riallile from K. Therefor e, they can be joined by a ' Hne-segmen t, lhe home omorphic image of a line-segment III a lr,ang le of Ihe development; CQn&c:qutntly, i' is natuta lrc give a linear defcrrrsuion of the po,nt JIll) int Q th e point ), '(1) which determin es the req uired homotop y, Not e, mor eo ver, thet poim MI ) does netleave that elO$C'd triangle, edge or vertex, in whiclJ it init iaUy was In the course of the homoto py, • II is necessary 10 di ~tl nauish betwC"n line loops which an: homoto pic to a con. slant one '" ' he to polog ical cr ccrnbmatorial sense. We will call .3 \oQp whsch is homotopic IQ a constant one contraCfible or comb inalorially contfQetib ~ loop , rupcctivdy LEMMA 4. A COfIlraclible lirl e loop), Ul a Iriantllllal' On K t$ comb/natorialfy COrltrocl . (bl e in K. PROOf'. U1 a line loo p ), be given by a mapping o f a Iine-segmem

F : It

)C

t : I t _ K. Let

' 2 - K be Ihe ~n tTllction of the loop to a vertCl< Xo E K , i .e"

Fl/, " !J) .. .", Fl1, ,,UI .. Co : I t

-

XOE K.

It is d ear that F l lOl >t/ l : /1 - Xo and FI [ll " l l : 11 - XoSince. F is a contraction keep ing the ends of th e loop fIXed , th e ediles A B , CD and BD (Fig. 61) are mapped. into one poi nt "'o- We mack those points onAB whose images are the vcnices of K , and dra w vutk:al Jtraight lines tllrouilt them . Then, by drawi ng additionally other vertical and bor;wntallincs an d diqonais (fig. 61), we will ob tain a &uf rLCicntly floe triangu lat ion 1: of the squau A BCD for lhe image of the star S(V) of th e t riangulation 1:: under the mappmg F to lie in the star SeW) o f a ecr1ain vertex of the t riangulation K (tltif follo ws teem Ex. 1,~ . 13 , CII. II). We no w associate the vertex V with t ilt vertex Wand per form a similar ope rat ion over aU the vertices of the triangulatio n t , Then we extend this mapping to the edges of the triangulation I: In pr~ y the same mann er as we did In the proo f for thc lem m a on a line app ro!timation o f a Jllllh . The mapping whl~h we ob tai n, i.e ., F L : I: l - K , where I: l is the union of th e edges o f lbe triangulation 1:, transforms the subdivided side AB into a certain line loop;; in K . We now sho w that ~ is ~ombinatorially deformable Into ),. In tact, dur ing a line ap prox lmatjon , no po int o f a path leavcs th e rrien gle, edge or venu.iIl whlch it was

Fig 61

'"

lOlroduClJon (II To pololY

~~~--e-~

COCO

COCO

Fi• • 68

posjlioned . Th erefore, the loop;; con sists o f Cbe same d~ntary pal bs as). (if the null pat hs ar c neaJcetcd). Howeve r, Imerally spc:aldna. some edges CiIIl be: r un scverllJ times in differml dirmiolU. Th us , we can make a transfer frem ;; to "). by Type I com binatori al

Ie ", (Xo' pl.

~.

4° . U.lns the van Kun pm thWTcm , de , ;ve 111 ... ,bp if X has ee type Mp Of by 01' 01' ... ,Oq i{ X has tbe type N q. We denote th is group by G . We shaD now co usider the embeddin g mappinlJ i : X l - X and the hom omo rphism of the fundamental &To ups,which is inductd by Lt. viz.•

ex

I. : '"'1(Xl'''-ol - :l"ICK.xol·

.,(X.

We will calculate th e grn up xo1 as follow 5. First, we prov e thlll I. is an epimorphi!>RI . Then. using th e theo rem co ncerning epimoillhisms , we nbtain :l" 1(X' x ) '" :r ICK, ..-) /Ker i . .. G /Ker; , . Th e ealcuhuion of the kernel ~rl . will complete the proof o r the theorem, We first pro~ that i . is an epimorphism . Let as :l"l (X, "-01 and K some trlangmat:ion of \he sur face X . The n. by Lemma 3 concerning une appro~lmatlom , there is a line loop)o, (in the subdivision X ) in the homotopy c1aS$ of II. K ma y be assumed to be obtaine J tdaticm

or• c. •

canonical dctck'Ipm enr wtlose ward Is IlOUP hnin. one IflICTllto r at and the ddlnin&

jl'CJSK$Sft •

4't4' I' the~ron: (X. x~ is. cydic

,.,

0 , J, HOtDOIoprTheory

COROLLARY 2. ~ /N1td1l_"tIl16roup 01 till! tonu .. ,(T 2, x~ is II l'ft A Ntill" 6rollP willi 111'0 ~ton, f' 1t0 0P. Th~ tonlS r» poacaa: I canonical d.cvclopmcnl wllh the wor d IIbt1-'6- ', and , COI\MIQumtI)', __ obtain that the IJOUP " l (r , .~ b aenerated by II , b . The rdUion IIbtJ- l b - 1 '"' I! provloda I CiOftdi1ioo foc iu convnutl1iYity, viz" fib "" N , . Ocomttrica1Iy, to tlIe &eeera1Of II , o f the tIlDdammla1 ~ of th~ projea.lve ~,tbefe ......u e:spouds its ~lllte (lee 1M moddsof RP in Su. • , 01.. II). To

the aeneraton III ' 6 1 of the flll'dazDe:ntaJ PtNpoft be tonu r 2 , lh«e eo n espond iu !"aBllel and meridian , the IWO principal DODcorlltlCtlbk loops 01\ Ihe tOt\ll . ~ 7- , FInd ali t wl\al reomelric mamin. th e ,e Denllors of !he tundunmlal IJOIlP' hi ve fOt th~ ~Ilr rac:a M ,Nt. The fundamental crou p orJe knot c:omplemc:nt pla)'J IIlI imponant pan in knot ela.ul rlClliOll,

_.

EJttrc/# . - , Prove that th~ triv ial knot is not eqlliv alent 10 either lite trefon or rl6\lre-of-eiabt knots. HI" t : Show thal !be rundame Jllli Voups of the c:ompltmt:tlll lD Il J 10 thcsc bwu ....e

JIO(

5, The To pological Invariance of the Euler Characteristic of a

Sur face. Lcl X aad x ' be IWO bomoomorpbk elosied I\,q"facr:s with somesubdivl· sions n, n ' ; let x(ll) and x (II ' ) be Ihe.- EWer eharaetcristi is . rcprcsmtl lin o f its iIna&e MII I.. I. The sptlennd '" Is ' attathcd' to • point Xo- Md lhe sp hero id f.I..f)p to the point WX'tt~ =:0 :1:0- Ihe fo nner brio. homOiopic to the Lw cr In virtue o f rI - l x - Sup. pose lhat the poUlI Xo sbifts 10 • painl :1:0 under di is hom oto py, desrnoln, • path ...( , ) ill doina 10 (FI, _ 7 1).

l..u w(l } i.lIduce an isomorphk mappina ... (X, tol!! ".,, (X. xoHiC'eTheo~ ~. Set. 3}. Th e holllO«)p)' of lh e 'r. heroids ,. &nd CtJ)p geDc:rIleI' lhe homoloPYo f the l.. I. Therefore, ' J ill,,] • 1IJ"'1 .. la l _ S ::- l C",l , spheroids U I" M da from which me ans th ai the foUowlna dlagr.un is comm lill tlve:

s:-

Ch. J. HomolOllyTht " . Though rcq ulrin • • development of a special method. the fanner Is dancntuy m oqh. We list the followiJ\l reJiulu witho ut proof- :

_ I(S" ) ..

-2(S~)

= ... = "-,,. •(S"l '" O,1r,,(5") .. Zf1l ;;. I).

Hence. It follows. in particular, ttlat the sphere S" is noncontlllclible 10 a"y o f lu poinU .

Th e lOCondease has not been fully investiga ted , an d the diHicultlCl increase wilh 0 111 and Ie - n , Here are lom e of the limplffi rnulls:

the il'oWlh

1rl (S; .. Z,

-.($;

= Z2 . ... '''" .-+ I(S'') '' Z 2f1l ;:,. l ).

This refutes lhe intuit ive assumption lhal Jr.t(S " } • 0 whClIt > n. Tb Ui. WIlen n - 1. 1.. .. . the sroups ",,,(S" ) are free Abdi:m Itoups with o ne , eneralOT v, -r" beinJ th e homotopy class o f the identi ty lD.Ippinj Is- , S" - S". The mullipM: d assa I . 't" can be imagined all the homotopy dasscs o f ma pplnp .. :S~ - S " UK:h thai 'twisl' the sphere S" onto itsdf I times . In addftion , if I > O. then t~ onent.hon of the spbe re under the lIUl PP;01 " is said to be praervcd. wlWst it I < O. the orieatatiorl as said to be cban&ed (cf. t"e bomotopy dasses fro m 1rICS I» . £xociu I I " . Ltt S" be: a sphcf e with the 0Cll1re at tbc: onJin o f th e spatt R " " I. SIlo.... thallhe m.lppi.tI. o ( S " into iudf p n .. b)' the WflQpcln deJlC'C "'1.Xl . ... • x"

I-

( - X I' Xl '

1).

determines alsomolopy eIas£ eq~ to ( - -r..). lt illqulle simple lO proYe tJw 1r, ,(X.... ~ l . if w JP&CCX is contraelrio;a!lP\lfo.=h . The: d ..mmll tn holIIOIopy lh

.J"'(xl"

"

. x,,».

to besmOO/1l (or d if/t rtfll fob le) of

I . . . . • m , halIall con tln uo us paniai

$" -

3, On

U foreveryorOc:r \lp tos '" r

inclusive:. Smoo th mappinp / o f ctass C' are also called C"-moppings and written as

rec:

If aUthe fundion s/ t possesscontin uOlls pan ial derivatives of My order the n the ma ppinaj is said to be i'l/initd y smooth (I e C"" ). Continuous mappings Il.U called CO-ffUtppill&S. Il ls obvious thai the following relation s an': valid CO :> C l :> . . . :> C':> . . . :> C-. In ¢Ue au \he runcl.ions /~ are anal ytic (a function il said lo be 1l/IQ171k if ilt T aylor noonverlU to il in 1M ncipboluhood of eadl point), the mappln.. /ls said to be IlnllfYfk if" C"). ~ foUowlna SCI incllU;on Is val id C'" ::J C".

paI\5";oo

DEFINITION 2. Th e matrix

o f the fInt derivatives of t he mapping I .calculated at a point Xo is called Iht JecobiDn m atrix of t ht I/'IIlpplng/ at Xo and denoted by

('£\ I iht )

"0

Th e Ja.cobian mat rix detenni na a linear mappinl R " _ R"':

..

,

t~ ·~ ~I~ ,~ ~·L.~ + ~ii~~

whkh II called tbe dtri"'llli"O/rM ~J'iIl8I'"the poinl XOand denot~ b y Dx!. deri vati ve is a 'l.incariwlon' of the ma ppi ll&/ . 1..1: . , th e .rrllle mlppinJ ICX,} -t (D"f!(x - Xd comdde. wi th f(x) up 10 inrmltcsimab o f • higher order tban b - xo' . More prec:lsdy, Dx/it . unique Iina.. meppins of R " lo Ii. ' ' fot

~

which

"fVoD, is siYal 011 V as a fllnetioJt of t he . &Ddard roonliDatcs o f • pOOlt )I. th en it QID be ~ as I (un.ctloo of tbc sundatd coonfulat uQf a polnt . , l.e., as a func:tion o f lh e eUl'Yillneu tolmtinates of tbe point r - In analyslll. such an operation Is lermed a m OIl,e of WlriQbIa . In Olha wor ds , we replace t be coo r·

din'tea In ee evese imqe apace o flhe funa ion , . which ;s equiYll!ent to consider· iq the fu nc:lion V ln$1e4d o f , . It goes wilh ou t ..yinl lhat m appiogs an d s1m iltr eltal1gea o f variables ean abo b e eonliidered . Th est m anleS ar e al so mad e III th e space o f Q1app.!na images, l.e. , lf , : W - V is I mappina of, set w e Rift th en in$lead of the a udard coord ina tes of t he point , (rl , t: e W . the curvilinear coerdin lles ot itic: point , (d determi ned by th e honteomOl'llhlsmJ I.Te eonsidem1. Sudt • ehM&t' Is equivalen t 10 COflIideriq the mappinJ r 1, ins tead of . mappina , . Not e th al the rank o f . smooth mappill, is Wlalt ered USldn • sm ooUl dtan,e of yariable:a . , . A Theorem on Rectifyi ng . Tbu ,tllfdtrnf emMddill6of R - into R · .. tis • mappin, R - - R" .. . spedfted by th e l;:'Clr1UpOndence (x l' ••

ta"1' .•• • x• • 0.

, Jr.. ) -

.. . • 01.

Th e stalldtud proJ«tiolt of R - + " 0111.0 R · js tbe mlppirl, R · ... " - R · det er -

mined byt~ ~c:e (rl "

"

,x.'x. ..

I' "

. x. .. Ie)

- (xl . · · · , • .,).

C h . 4. M a nif olds a nd

F, bt~

Buodles

'"

,"

v

... ,

Fig. 73

THEOREM>3 (O N REcnF YINO A MAPPING IN THE NEIGHBOU RHOOD O F A REGULAR POI NT). L~t U C R~ be On open Rt.j : U - Rift 0 C '· mop pin l, r .. I

ond Xo . In fllct, since the mappingF'i l is bijec tlve and F1(x ,0) .. !{x). ....ehave(Ff)lx) co F(f{x» .. fl lif{x» " (x,0) . In case lhe determinant of th e first" rows of the h cobian matri;c ( :: )

I"", i,

equal to zero, Ir ls finl necesswy ( 0 renumber lhe coo rdinates in Rift (in othe r word" 10 male e a , pecial change of coordinates in R'" by means of th e C"' -diffeomorph ism It ; R'" - R'" ' 0 thaI the delermlnant made up o f tile fill t n

. matrIX . rOW5 o f th e Jacoblllll

('~ - 'f» a"

I

may be difforent from zero). We con-

"0

struct the C'-diffcomorp hism F for lhe mapplnu '- 'J lIS above. Th uifg - I will be the required C' -dIffeomorphism for the mapping£. (B) We represent the dem ~nt:l o f ure space R~ .. Rift X R~ - '" in th e form {X,y ),

wnere x "" {xl ' • .

,

z.. ) e

R"', y

fl) .Fromlhedata.wehaveranlc:

..

o., . ..

, y~ _ "' ) 6 R II - "' . Let.~o ..

,c;.!=m,or ran k UO .n

(..!!.....) 1(,II.A a (x,y)

(x'l,

= m. ,

Assuming at IIut thal. the d~lenninllnt made up o f the fust m columm of the Jaco-

(-'1....) "".r'>

1 is different from zero. consider the m.pping a ~.y) F : U _ R'" x R II - '" given by the ro le F(x , y) _ «(:C, y ), y). Th e Jacobian

bian mat rix

matrix o f the mappins F at the poin t ~,yOj is of the form

::: I ~.~, · ..,~~ L.

'I,

'I,

'1.1

'I.

1 1 I ' J", "'--r' . ' ._J>.." " ';, ~-_'" - \P •• , r• ,

1oJ, _'" •• . r.J>.,: . . . • ~ 1 ;;-, "'.... ·'r' _

J

Itt

...

o

o o

Int,od"" limI co TopoIoay

'"

By tho:" ",", plion , lb onnminanl in tho: up per lefH :lMd coma' is diffen n t from zero. thcteJ on: rank (A4. ~ '" n . By lhc: inyenunappia. thcorem.Jhere c::ilit OpeD qhbowtloods V(r'.',;u) Mel W(F~. ~I o f th e poipu tr". y"') &lid F~. /1>, repectivdy, sud! that the m appina

W(f~. Y">. yO)

F I I' ,- ,

and,

,

l at ~ +

PIlOO. , Put

, /(.r) •

l(.r -

xO)clI ,

"'By appl ym, the dmlentary IratUformalions tAo..... from an alysis, _ /(.r) -

m omlYN ,

!.

If/?· W

-L , oo

I

+:

dt ..

- .10,>

,

(.r, -

xf>

r::,~

J

1(tl

(.r/ -

+ t (.r - x"})dI .

xf) ::,

.

L , oo .

~+ (.rj -

obl.ain t(.r -

;Il)

)dt

~'l(.r)· •

b~ .... Let

f be a fun etion of dus C' • 1, , :> o. Ji veo On a CQlIYU nci&hbourhood "~of a po ltll xO 1lI JtA . Sho....that fl,r) • f W?

+

~ ~j

,- ,

-

z't):, ~ +

r •

'.J- 1

(1"/ -

.¢f.rJ

-

~¥I,rl.

2. SMOOT H SUBMAN IFO LDS IN EUCLIDEAN SP ACE I. The Notion of Smooth Subman ifold in R " . in the courses o f analysis and analytica.l .ftlmetr)', Inl()()(h surfaces in the tlltcc -aimcn5ional Eucli· dean 5PK'!' lb M are IPvm by an equation: • fl,r . y), whCftf is a smooth fllna;on o f l W'O varia bles dd'"med in a rqion Doflne plaM (r.y). ...e Q)llsidcred . Mon:c:o mplCll IlItfac:es(e .... closed) which are aiven on their plUtkuw regions (l .e .• kalIy) by cee of ttle foDowing equ ations t .. P I.r. y • q(x, ;z:). ¥ .. r(y, Q are abo «msidned. Th e limpkst eumple o f slldt a swfao:: b the $phni S1. Other objects It udi ed in anal ysis met anal )'tical loomdly arc 5mOOlh Cllrva Jiven Ioca1Iy by oar; of the l}'Items or equtiom,;

»,

¥ ..... (d. { ]I -

\Io u ):

{Jr: •

=: ... lJ').

j{y):

{y.. . . (.-), : ..

~(r).

AD tbC$C obj«ts are mlbrlCC'd by the sin gle Mlion of smooth 5IIbnanifoid in a EIlC1.ideaa IP'O=. We oow c:onrider a subset M in R N 1.1 • lopo1opeal sp4CC equipped with Ihc lopo101)' lndllOCd fro m R N . Let ¥ be. poiDl o f M. and Ul,rj ils opc:n nciahbourhood (U1M ).

DEFINITION I. If a homeomo'llhl, rn " ; R~ - U (r) , n " N , u Ilsfyinll the /;Ond i· lions (i) '" Ii C', ' ~ l ,u.ml ppi n. rrom R~lo R/Y,(ii)ran k ,,,, '" n for any poin l , . R ~ is linn, t hai Ole pair (U(,l'). ,/,) is caIlcd a elrrv1 ill lire polflt ¥ on M of ( 11;1$$ or a C'polor1cal strur:fl;1't:I; C"S1 ructurcs (r ,. 1• .. .• OIl ) Ill: calIcd smoofh (or dlfjtrtfft iIJf) st ruclUres.

"YeO

lati vct..

DEJ' INITION , . f'r, topolOSicai space M with 11 C'''$Cru et ure liven o n it is Qll1ed a C". mQlli/oid (or . " umi/old olrloss C'). and the d illltftSlon o f the space RIO from wh ich Ihe hOPlCOmOlllhi$ms of dlans let is ca1Ied the dj~ of lhe C"·manifold .

Similarly 10 C'"-SlfUI:$ ures. c:'-manifolds are said to be topo/06icrz/1lIId C· manifolds (r :a I• • .• • OIl) pn()Olh. SomelilM$ (for bl'CYity) C'"-manifo lds will be sim ply reJo:ned to as manVoJds• .... d C"'-at1ues u lllliut$. If in conditioD (Ii) o f Definition I. Ihe homeomorphillTlS .;,- 1.... .. - ~ u,: analytic INppina:s (". _ 1,., ..._ C"') . theD. the charts tv, ...). tv,~) on Mare said 10 be CW-(;OmpIltiblr. C'-atlasu. C'4nIc1ura and C"'-manifoltls are defined t\;Ilurally. C"'-SUUetllres lIlDd CW·manifolds arc ealkd "',""ylie strucr..1U a nd. IIIUl!yllt numijolds, rnpedivdy. To indicat e the dimension of . mani fold , we win wri te M". and abo d im M • " .

t+..

l\IOTIl llMo c1illWRsioII of a CO-m.;ul lrold Is ill i,,.-ariant. I.e., ;rldcpmdenl or 1M dloou o r an

li t.. In facl. ir M . d",itt«l .,Ia04 [(U~ . ¥'. l](...~ :

R" - U.). [W, . +, JI(~, : R'" - V,l

a nd " '# "' . tben chue would be fOlS

U~.

v, I\I~ Ihat U. n v. ". 0 +i I(U. n V,l

an d til . IXiI " I. I .. I, . . . • nl. iDdud lll il from Ibe sphfn

SO' •

I.

11- . Show IM I lbe mapping

r 3)

(.l"t.,1'l',1'~ - ~,.xt. xf.X""l' x r J. x is . ho meom orphism of the projective pla1lcRpJ e r ne I subxt in R~. To lnduce the SI TU CI U ~ of the smooth manifold Rpl by th is homeomorphiam mean s (0 realize thereby Rpl as a subset In R'. 12". ConJlnlet the realiza tion of RP) in R'o.

• . Matrix Manifolds . WcUldow the: 5Ct M (m . n) or all m x elements from R I with i :

If

matrlen

the lopoioo' induced by the: nat ural

wil h

mapping

R- - M (m , II ) :

::~~

~,: ) .

(

(.l"1• • • • , x_) -

x(Ioo _ 1)10 .. 1

_

Then the hOllleomorphlSlTl I indu eeJ on M(m. n ) the stru~ urc: o f a c-· mani fQld of dlmeTlloion 1M . Denoting the subspace o r ma trices DC a fIXed ra nkle in M {m, Il) by M (m, " ; k ). wespcrify theSltudureof.C--manJ foldof dimmlion k(m + If - k ) onM(m , n ; k ). Nou-. befoulwld, th' t if YE M (pr. III and rank Y OJ k, then by mtuehanJina the I'OWS and eolwmu, the malm Y _ , be transfo rmed 10 lhe fOnll

, I B,). (A c; J Dr

wbne A r is • IlQn-slnjular $quare m8tri,x o r order k . In Gtlter words , there U1Sl IIOnsinaulaJ square 1NI1ricu P r E M (m, m) , Qr e M (n , II) weh tha I

P l'Q

r

r

• (A T

I

c, I

Br ) . Dr

We sh ow lhat rank Y .... k if lUld only if Dr ., C 0 the c:qul1lty

YID r

III fact, it follow. from

( _ cjAyI IJ~O_ )(~I~) · (~ I -C~ Y~~r + D)

"'"

",

Ch. 4. Miln,folds and F1 bre Ilulldkf

It C:1llI be seen from rhe lau e!" equality Ihat rank Y . k if and only If D y = C yty lB r No w let X Oli M (m , "; k ) and X I" vbitnvy matrlll from M(m.". k). DenOt~

Px.XQx, _ wh~

Ax . .-.. n l

Jq~

(Axx.. xo IDx.x.J Bx. x!\ . C

o

Xo

ma lrill o f o rder k . ColUid er an

V( XO> .. rX e M(m.,,) : cktA x .

of the ma trill X o in M (m . n ). Th en U(,XOJ - V(XOI ueilhbovrbood of X o in M {m . n; k) and the mapplnt

"x,:U O. /Mn UlSIs I

, 'T) _ [ 0 •

• • Fi, _I I

/I

C-·fune:tion ' t : R" - 10,

...·hen when

c; DsIl(O), X E" R"'D, (O)_ JI

•

JI &fI,('h

IJtq!

'"

ln1: M and rc:a l fllne:tions I,II' . . . ,I. d e:f.., ed on M . We will SoIy rhat / C'..smooI },f? d~purtb on IlI lIlutI(. l /olu f l , I) if there exists • C'"-full(t kl n U( 11, It ) o f , ui varil blo , I" defined on R· such rhat

.1,,(' ""

'1' .. .

J ()l) ..

UCNiC), . .. '/.(;/Jy ;;I - Vl (y), . ..• f' (,»), y e V(.lr) is a homeom or · phi$lll o f V(I-) onto tbe spa cc R ~, (ii) for allY po int y e vI;Ofllhism for any ... and (J. In facl . srnce I is a C"-d itrCOfT1Clfphism . its rcptC5r - ,.,.. is • C"-cm beddi.,. OD • cert ain AelSh bourtlood of eadI pomt x E AI" (th is fo llows fl om th e rheoretll on l':'CtifyioK. ma ppinz. see 50:. I).

O . 4. Mal'li(oIds and fibr~ 9ou>dIes

~ / ii; ·z

'"

ExAMI'LE 6. Amappiq j : R" - R~. H ~ n , o f dus C",r ~ I, det erm ines an immeniou in RN If ran k

( ~)

l,. "

(3)

at an y point Y e R" . Thus,jpoueJSCs no nonregular points an d , by t he t heorem on rect lryill.g a ma ppin g, is a loc al homcomorphtim between R ~ andf(R") . If, in ad d,tion. / is a ho meo ll"tOTllhism o f R" on to / (R ") then f is a C"oflIIbcddinl. Eztrc~ 51" , Veri fy IIa.I th e notion o f eh an on a C"-a.bnwtifold in R'" (see Sec. 2) is eq uivalent to the C"-embcddina o f R" iJl R N. Vert o ftm lI\&Difolds lie in ot her am bimt manifo ld$. It will be too Jmen.! to call ...y ~ manifo ld a subm ani fold o rlh e am bient manifold j ust like a su bset I:Ildo...• cd, in a lopoloJical space, with all arb itrary lopoloJ)' wiD not be: tcnned. a subspacc . It is n~ to Im~ rca",n.able restricti ons ir "'e ~lIire th.altherc WSI a $impt,. td ation bet ween the st ru ct ures of an embedded an d amblei'll manifo!ds . Meanwhile. th e notion o r em beddi ng proves useru t.

DE FtNITION 7. We ca ll any SII bspace M 1 in N'" whsch is the lmaae o f a certain C"M " - N'" with Ihe C"·strueture induced by the h omcomorph.isrn/ , a subman ifold of the C'·ma niJold H"'.

~boddilJ8f :

The su bml.niloW and mani fo ld i tnKlUrcs happen to be retalm in the fo UoWlng li mp lc fashion : ror a cenain atlas I(U... p.)J or a mani fol d ~, Ihe in tersc 0, Let xobC' an ar bitflry po int in M r Sincc N1'b aUlbt here u isu a cha n ( p", ~ ) . ff~OJ e P) from the mlll ima! a lias for the C'st ru cture liven on N'" such lhat t he pa ir (r n N t. '" IIll) is a chan of the muimal I tlas for Ihe C"ll ruttllre on ~ , Let IU, ,..), (.rol!! U) be a chl tt of the maximalallu few Ihe C'-~C1 ure pyen in M" sUoeh that flU) C 'P. Th en, (rOfO the d l t. a lYen. ,.. - Ilxo) is I L"t'gIllac point of l~ fWIppinl " .. II,.. ; ,..- I( U) - R - and , by Ihe Iheo rem on reaif)'irla: I lMppi.na , there t:ltist an open nc'-&hbovrhood m~ n ifol d ,

«:

·,,, YC,.- I:0>1 - W , such UQl llw:mappln, .F- 1 on the Sd W is the standard project ion o f R" onto R- . Note t ha' ,,(V(,.- l)) IS an opm nciahbowhood in M" o f the point .Ko an d the pai r (,, (11(..",F - 1) is . chart or t he maximal alias for the C"-S\ruct ure ,iven on M". Since flP- 1 is the stan dar d proj«: liOll, and thc set

'(xJ»,

or I/ (",(V(", -I (>:o>)) n M 1} C

R'"

consists o f polnll o f t he fo rm (>:1' .. . • x r O• . . . , 0), Ihc SC1 F ,.-I (,,( V(,.- I (.x'Ol» n " 9 C R" Q)llJists o f points ofthc fOC"m (1'

,x•. o

a,x,. .. "

than (.. (V{,.-' (.l'ol». )) n M"

R" - ,. .. It G> IK E R" : X• .. I .. . . . . . x'" .. 0].) Sudlach art can be ( o nSU\let ed for an y po int X o& M I " Th is pro ves tha t M[ is .. su bman lfol d ;n /of" o f dimension n - m + k . • EXAMPLE 8. It (OllOWI, in part icular , from Theorem 4 that the Inverse imaae o f a regular val ue o r th e ma pp ina f : M" - N'" Is either em pt y or & submanifo ld in M" of dimension II - lit . TIle follo win. fulHlamcntal fact Is pven witbo ut proof. (HCI'(

THEOREM j (W Hl TN'EY) A ll )' C'-mtUlijofd M" N il W cy..mb«/drd ;1I 'Ire £=/i_ . " " sptta RlI'«tor wh ich b JU!lab lc fOl" .,. arbiuv)o manifold. Let Jot" be . maniIokI of class cr. r- ;> I . F.... an ar bilrat')' point X E Ar and consieler tbue! T ohll triples (..1'. CU. 10) . II). wh en ( U. 1")lu d1art at th e poin t x . lIDd II • >'«lOr Ortbe~ R '", We d dine an ~ce re~lion on th e >el T u roUowl: (..1'. (U. !O) . II) - (..1'. (V. ~). &,) • II - D *-I(>1(!O - I.. ) CI') , ~

1*. Verify I h.l this relation .. an equivlllence rewio" , The equivalence d :us (,r . (U. 1") . h ) is call ed the tu,,~t I1«t or- at t he poinl JI, an d' l he tri ple (..1', (U. !O ) . II) from the equivale nc e d Ull a ~preRft{Ulil'e of tlw (u" sen{ vector In lhe ch art ' tU. !O)' Moreove r. we will cal l t be veeeer " the vector componen l of 1M ~PfW('f1{Uli... (..1'. (U, !O). II) and de note it b y II; , We n e~ l consideT the set o f all t anllent vec tors at a point JI, Denotin ll it by T;/01~, we Ii" a chart (U, ,oJ. x e U . an d co nstruct the ma pp inll

r.. :

T..-M~ - R~

(2)

l ha t . ssocIates each. I an,;~t ..«t or with t he o;om ponc:l'Il It of iu rcp r~tative in Ill" chatt (U , ..) , It is obYiollS that ,,, " a bij«tiOll and th cn rore the st ruC!. ur e o f lhe n-di mcNional ..«tor space R" h IUIolunJ1y transfened to lhe SCI T,.M". A more de tailed namIn.lw.. o r cJlis fac:a shows Wiehe a1s dlnlic ~ral.ions o f IIddi cion and nwlliplk:abon by . n um ber are inlnxll.lC'ed On T/of" in Icnns Of lhe COI'TaJ>OU4lnll operatioou ovu lhe Vt'CtOr eomponenel of Ill" repruen w iva of t he tanaent ..ectors

• e-a&, ( I ) cItmooutrat.. how iI>t.

"*"" .... ' h

a~

..rtbc.dlal t,

o..~.

M. " jfolds and Fibre Bundh:s

'"

in tile chosen chart (U, .,.) , If t he representa tive. of the ta ngent vector s are given in differe nt cha rH Illen Ihey should be repla cCO\llIt thai the fust aDd th e lui: funetiollAls are equal to

•

,OS)

•

~,

(I ~

Icspcttivd y, we obWn, jusr like iII ded ucina ( 104). by equ alirinl (11) to ( 19) and put tina • • J'j' thai

.

wl _

~

t.J

I'J

a(". -I~)f

ax

/ ~ ,

"

·

1, ... • m.

.

('20)

J

.rftnns

Hence, ma pp lns (17) Is of class C-. F nnn uia (20) the r. ct esublishedQtli"" th at th e vecto r componellt o f a ta.ngent vect o r Is t ransfo rmed by means o f lhe IinQt transfo rma tio n D_-1r.".loj -''''), A. T he Differential o f a Function and a Cotang ent Bundle . Consider the nel io n of a YKI01" Jr> E T..oM" o n th e funa'toaf ./ E d'(:cO). U lhe fUndioft f is nxcrl men there arisa a 1lDear f u.nct lo nal on the IJlK'Ii' T p.r : 1:.1> - 1:.I>f/) . This funetlOn:al l'l ~oted by the 5YflIbo i tcVl... and 'j(l'). Ihr:Il

( 2 1)

11I1('Dd O,lCllo o 10 T opoIOS)'

(I5IJ - 0 ....h en f ot

J.

loIopMs in X . In aa:ord~ ..uh DerIlliOOD 2. the path I : / - E ill c:a1Icd • lift of tM: J*h 1 : J - B (wtIid1 11

enabl Q

cowriltllhe ~ I ) ifpf ""

.

UM.\olA I. Let P : E - B be a eoverina: ma p . Thm th e followifta S1atemel\l.S an: t rue; (i) uy ~tb l' if) B ~In.'" a point btiE 8 possesses l unique c:ovcrint; pat h 7 inE wtIK:h stllU aI M Y poin t "oEP- I{boJ: 00 i f l' .. 1'1 ' l'1 b the product Or paths 1'1 aod 1'1 in B then the eoverin.S path i& ;: .. y\ . Y1' wh ere: l , :;1 ecver 'l't Uld 1'l.respc.ctively; (iii) if l' .. 'l'i I il the path in vene of 1'\, then:; = 'Y;-I.

:r

PRooP. Let l pat h 'r be give n by a mappilli l' ; 1- 8 . 1 - to, I ), g (O) .. boo e ach point of t he path ,.(t) bel 0lll' t o lOllIe o;:oordinate nd&hbourh ood U" and. Iher e b I Cl)nQ«ted ndshboUfbood (i.e. • an Interval) 0 , o f . point t e I .ueb thllt 1'10,) C U,. We pick . finite coverinl OUt of an o pen coverinJ lO,l o f t he line-segment I . Let oS ~ the ~ number o f th e covcrina Let UI bnak th e !ine-$Cll1Ienl / int o sepleutJ 4 , ... Ct, _ I,t4 0flCl\lth h:ssthaolbydi.... lon poinu t...l _ 0 , . .. • N , t il - O. t N .. I. Th eD -.(4,) lies in • certain coonlinale Mi&hbo urbood U,. 1 _ 1• .. , , N . Thcn'forr, eac:b portion "" o f the pa th ,. livm by th e m. ppi.... 1', :4, - B admiu atift 'i'", to the .h~ W. , Ii.-m by t he mappinll.!/ "" P;;' 11'1 : 4 , _ here p .. : W. _ II a homeomorphism O DIO the (:OOf"Iftnllle oei&hbOurhood U" We ~ . th eet co nlainilll th e painl,.Oe p - I (b'> as W'. and lift 1M: port ioll o f the path .,." Th en 7 1 s1iUtl1 at Ute point " 0' w hee ~ hll aJrady bun chosen and the portion oft he path 1', _ , lifted , we c:hoo5e u W .., lhe &beet COAlain ill& the Itmlinal poinl/, _ ICt, _ I) o f the panion o f th e pa1h 7, _ /• ....h.kh lic:I over Ihe point/, _ I ( i , _ I)' Then t he por1ion of Ihe pluh 7/ oria.in 'lQ a l t he polnl Ji _ I ( f i _ I)' We Iifl t hu, al l tile po rtions 1'/, I - I • . . . , N, o f t he path 1" Since Ihe mappings Jj : ~ _ E , I _ I••..• N are rompatlble 0\'1 Ih e co mmon Cft ds o f Ihe ad jacent inte ..... 1$4/. Iht)' Clll be combin nt Inlo t he m appinll.! ' 1 - E, 1(' ) -I,. 11", (.8. bO> be fu.ndamcntal JfOUpI ,and P . : :rl (P. xlii- 11"1 fIJ. "0> • homomgrpltism indu~ by the pro jecUonp : E - B . QlD.dd er me iDvenc: imIl,e P; I(~) o f the unil eleme nt of the group "1 (8' , bO>o II SUff1cellO show thatp; I (If) _ e ' , wnere e ' is the unit elem~t o h h e y oup 1l" 1(E .xO>. If (a ] E P ;I (/!) th eDa coven thepath l:l _ po. ....hich is homotopic to aco~anl pa th In B . Accordins to sta tement (iii), Q iii abo homotopic to (I COl\$t&Ilt ptUh (in E). and thnefore (Go) :> ~ ' • • Th us , iT; 101l01l'li from Th eorem 3 that the gro up ... ($) Is lsomorpl\ic: to • A1bpOUp of !be IP'OUP " t(8 ) (viz.. the SlIbvoup G • p. (1I"1 ($»). CollSkler the cosn.I (e.a.-. rlalu) o f the aubpo"" G of th e Ifoup " , (8 ) . The followiq: lmportalll th COfUll is valid . THl!O IU!M4. For tm,""~""_P P : E - B.tJre~ p -I(bO> irilfb(j~'iw~­ ~nce witll l1le/_iJy of CO#t$ of 1M IJlb8rf1UP G 0/ III~ .1T1IIp 11" 1(8 ) ·

paOOP Ld: \U as:soc:iale th e bocno&opy ~ lIJJ E r l (B . b~ with a polnU jJEP - I (bol by lhe foUowiDt;rme: _ lift. the paJ.h ~ to the palh a iD E with the oriain at tbe polnt ~o (\emtDa on liftin, a path) an d pllt ~jJ '=' a(l); by Lenma 2 {sutement (Ii». Ibe CDd o f the peth .. doeI" 1lQ!. dqlmd GO the d10lcc o f a repcaeutative ~ IS !PI. therefo«. the mapp{nJI " , (8, b~ - p - l{bO> . lISl - ~, Is def\ned. If 1IJ 1I. !P11 bdol1l 10 the the n 1#11 . lIJ 2 I E p . ("1ith t beorl,lin a lbo II homotop le 10 a certain loopJIa. wbere e iI a loop in E with the orlain at to' Den ote the Un. 01 the loo p fl I . fli ' with the origin at t o by a ' an d nOle that tbe l~ (r ' and (r ar e homotopIc with the ends nxcd (statement (ii) of l..anma 2), therefore (r' lIa dosed loo p co verinS the loop PI . Pi I_ But by Lemm a 1 (statem enu (Ii) and (ill». G ' "" 8, . Pi . The d05Cdmlill 01 the path gI . IJ; I llnpliCli Ihe winddenc e o f the ori&in of th e' path PI wilh th at o f the palh 12 , and a1Ioo f their ends. Th erelore. e~ "" t (l • Th Ull, the mawina IIJI - ~, Is con· stant 011 t he wse1 1i. M UII... hile,«l clIffhcm ~ there I;l)rt't$pOnd different im . qcl . In lact. if we »lUmc the eoou&f)'. theD there ' " 1#11 and t.B:!l f rom d iffer Cfll coseu. but '=' ~, . ... hidl mc:am that the e:ncb (alld tbe ori &iru) o f th e liflS of 4 1_ 1 eoilJOde. h.erd~re. 11 • 6 ; 1 \.s a loop in £ witb me orl&in at the point ~f1' P i$ 1 · Pi l ) .. p • . Ili l is: a loop (wUh lhe orip at the poinl bO>o wtI~ the hort)Ol:op)' dus {PI . IJ; II - lIJlI • 1IJ1r I o f this kIop bdonp. to P . ( I"I(E. ~o)). t.e.• IIJ I], 16:1are from th e _ coset . which is c.ontrary 10 th e aswmption . Fi nally, it larWru to Iho. thzt any poinl' e p - l(b.J is t he imqe e, for a «flam lISl . Con. sider a path .. .ioitlina in E the point eo 10 the poiDllo (us.ina !.be cxmd.ilion that £ Is

_e in !he fibre p-' 0O> - F. Let /PI €" '1" ,, and e" GP - ' by the ruJe t" _ t ... (or the 1l11ppi na "I : F - F by the rule a .. a ' ). II is CU)' to $CC t hat lhc mlppin, " I coven the I PBl;e P - 1(bo)' The folJowm. obviou s eq ualities: " fI . I - "I " I '''1 .. 11' if fJ E e (the ide nti· deHv~ flom~:nml I on liftin. path H i.nl tY ty elCl1lCYIt o f '1",(.8. boll, thllt the co rrespo ndence" : IPI - "} ~. reprcsmtati on o f the irouP ",(8, bo> by ' homeomorphisms', i.e .• ' pcrmu latJOrU' of the dUeret e space p - I (bo> (or F ). Th is representation tI is called I m OlltJdromy qf tfft co'Hrin6 , and th e set o f peml utat ions 10,1. ~ e .. ,(8, bo> the lI'IOItodrotfly . rollp of e e cov erill• . Thus. the mon odromy tI b • homomorphism o f the aroup '" (8, bo> 10 the grou p of all pam utltlons o f Ute film. It foUows from Theorem l !b ll tlae poin tc. Ep- I (belis flud for thosc an d only those pcm1utations tI , for wtIkh IfJ l e p .("I(E, . ,.)l. Thus. P . (" ltE. c. » is I sta bOit)' subpouf i)f Ihe point c. m th ego,ap " I (8 . Ito> Ktlnaon the fi1Ke p - ' (bO>. M OI"CO'+'CI" • .... ~.,j = tI, . {t.) If and on ly if IP ' ) E lP. ("l lE, c.» J 1fJ), i.e.• to tbe coset cont&iBin.!be clc¥Dcnt Il'J ( whence 1ltCOf'cm .. follo ws immediately). For dif· rcrCQ1 points c• • c• . • lh e P1blt0llpl P. (" , tE , c.». P . (" ,lE. , • . I.l"e collj uaate wi1h rcspca 10 tbat demenl IfJJ e ",(8. b~ for which ...,, ~.) .. t • . . In fae:t. if ' is the eo rrcspondiD i c:ovcrlni patb t~'" the OOIT~peodenoc ., _ -r" '" i -, . ., . 8. where bf e 7 1lE, c..). cstl blW\CS IJl i50morp lUun betw een " ltE. c. l and ... '.E, ',,' ) transformed by the monomorphismp. into the bomorllhlsm

",_I • ,,; '

»

»- ffJr'p. (" 1(£' 0'» [8]

P . (II,'.E . ...

- P . (11, (£.

l! .. .

»).

Lo:t us caleu lal e lhe monoll ro my group {O"~ I for the coverin . map P : £ - E I G '" B .encratcd by 1 propc1'"ly mscon tinu oWl trlnlromwion ItOl,lP G . LE.'otMA S. nc mOllodromy , rowp of I/Ie col'trln, _P P : Ii - E l a '" B fC/fU1lfed by IIJHOpUIy d i.sl:oit,jtl"OlU ~omultWtl I roNl' a of IIpatlt-«1nnrctftl spa« E is /soIIfO'Ph ic to

a.

P l OOf'. ~ 'o € E , b o - p(t~ bebasc porolS. We b.¥cp - ' (b~ = o~ , whe reO is tbc ol'bit paIolo!tta !ltloup the poin l t'oof lb ll: poupG, i.II:., let o f p&nb ~o>i~ • E G . Let IfJl " " 1(8 . b~ and . , Ihll: correspo Ddill. lI\OfIodlom, transformatOon. ,,~~. SIna: Ihe pdt 5 from "0 10 '.~~ i5 eanicd by !he bomco_orpblsat, E G 10 tbe rrom to, tI/J' alId Ihe patb 61 cov en l he path fJ • • ,,1#0> - 6 W~~ • ' ("e'~.

ee

n eD

-

path,6

Po

co>.

r.

Introdl1Cl.ion tD TopolOlY The

~

_, _ " determines a homomorphism o f Ihc moDOdromy iJ Ibc SIlpcfJ)OSition of - ' I aDd D'l ' Ihen

-'J. _', • 1I.,,r.~~ Tb~fOle -'I. -'I- "1 . " 1' Funbennore. the pcnn l.lw ioI'I -i' .. 0'0_ 1 correspond. 10 I i I ~ th e idmtK )'

pou p inlo the IJOUP a . 1n fatt. lf

(.,t,,'Jeo ....j1JII..I~~ .. "

I~"'~

pcrml.llation _, co 10 Ee) 10 '0 .. ~. thc icklItlty clcmmt of thc UOUP G . We Ulow IIW the bomomorpttism _0 - I , iJ ;I lQOrIomorphism o f the monodronlr ItOUP in lO 1M aroup a . In face. if I • .. so then ..lJIt o .. "~o - "0 for an)' 1 6 C , nd thcrcfOl'c ", b the idend l)' mappin, o f Ihc fibft: 0 . The I\Iljceu¥ity of th c bomomorpbi$m 0', - II foQo

IrOlD t hc

~.

CODJllOl:l.c rrlaflw 10 1M nomllli subvoup P . ("-Ile, ' 0))' PWO> - b. Is isomorphic 10 flit ,roup O. Paoof'. Coesl4cT the botnoIDorpblsm S ; "'llB. bO> - G given by th e OOIIl positio n of th e homomorphism o f l be 1l0\lp '11'1(8 , bol into thc 1lI0nodrom)' &Jllup of the OD'VCfiD, l.II.d bomofJlhism o f the trlOllOdromy aroup to the lJfOUP G. I.e.• the homomtJrphism siven by the corn:spondeocc 161 - .., Th e in!.a&c s - t{to ) COIIlUts of those daaa f6 J for whkbl, z .0.1..... ., is thc idmtitymt.ppin, o f the fibre p - IfJJO> . Th Cfcfore, S - "ltol • P . (..-,(£. to» aDd the fKl:or boIllOt!lOfllhism ~; "-1(8. boll,. (..-\(E. ~a» G is. ~bUm. COItOUARY. If . ro¥t:ria.a l!llP P : E - E 1 0 .. B is Lmlvusa1 U1Cl the Po -;cOo o( - ;col a .K/Io Le., .. pcnnUIO the po\rIts oftbe fibre . The amerad nl dement of the IfOllP "-1(RP", be> cqnespondinj to lbe dement _ is fanned b)' th e homot op)' dau of the pat h"..,. where,. is a path on S~ joinioll the poin ts xo and

I,.

wYC'St:

-

..

-s.

The UIIi venal ooverinJ p ; s ao. • I - L (4'; 1:1, .. . • k..) is Imcratcd b)', prop. u l)' discorttillllOUSaction of the a:roup Z",wil:hthe lencrlt.Or t1 : S :lo .. 1 _ S2Io • I. TherefOR ..-, (l..) _ Z",. the lrIOIlOllrotrly IJ OIIP is abo Z. and ac:IS" on th e fibre ; its ptleraar COI:fCSPOn4J to the 1ftlUIl10< f'l'1 E "-l (l..). wbc:rc,. it th e iXojo:\iod. of the PI th in s)o .. I ioinina the poiot;co to the paine "'(%0> ( rtod Q, flP'O> lISinl Sec. " Item 3, 0 . 111-

C b• • • M .... ralds and Fibre 8l1ndLu

Fig. 91

The u.niYUUll:Ovennl P : R ~ - 1'" Is, eneralcd by l pro per ly discontinuous.clion ot the &rOUp Z~ With th e Itneratou Q j ac:tinl by the nlle ~I ' ... ,Jel _ I' )C/. Je' . I' • •• • )C,,) _ ~I "

"

,XI _ I,Jr, + I, Jr, . l"

"

. Jr. ). ; '" J• • • _ , II.

Therdore, " 1(1"') - z " and th e lata, lon h j ) , ; = 1. 2, .. .• " ot Ihe ItOUp " 1(7"' ) conla1n the loops ,, / otIu.iDcd by !be projectioop ot the pWu inR" joonina lh epolntO 10 the points _i(O). Th e mo D04romy ar o up aets OG thef"IbnF - Z·, its aencrvan 0 . /, i '" I. . . .• " Ktin. an int qral VCdon (rom Z" by th e l'llIe: (A'1' . . . , .t:. ) _ (.t •••. • ,.t: j _ I' k, + l , .t:, .. , • • . . • k,,>. To It\tdy lllIivenaI CO'I"r:1'in$S, il is nea'Sar)' (0 im~ Iotronger Rquil'emc nu Iban t he patb~ on t he base space of tbe eov~1 OUlNlTION I I. A lOPOloaicai spaeeXis Wd to be: I«altyP4th-eoNfeCW it for My poInl s e x , then exists • bale or open pa th.colllla:ted DCi&bbourltoods, If ndahbourboods of , bale POS$e$l, in adwlXxt. Ihe property o f l-oonn ee:te dneu. then !he , pace is said to be: 10000J(y J""",lI«t«l. Exantples o f loxa!ly path-eonn«tcd and loc:a11y l -eonnctd) 'lil'1lQK IpKe mel bUc may be -.ida-cd lriarcWabk. T1IIac lriIn.avWlao:Illlay be cbosa sufflCimtly riDeand COlllflaIible 10 lila! the MI iIr«nc iaA&cs or a 'tUtu, &II od,p aDd I. tri&q1e fram ee bue ~ ""U of " vertlccs, edps Illlli triaqIQ. mpea.vdy. ThcufOK, lilt. tqIleI.i(y holdl:

ll tM' '\.

U

, U(I") a "ll W"' U, V rptl ) ).

0'

Le. lbe. full invencimaaep - ltpt:o!» colUiau of 1ft polml.l" ' • • . • •,;.... 1ben Ille full in· verse .."...,- ' ( Y(PW») ~ o f 1ft cllia U p/·). SiJla: the ~ ilU tol' j ls mapped 01110 u '(PerJ)j locally bolDeom orphkall, willi dqroc k / . s . I • ••• • 1ft . Ih. let p- ll) n CODIisu of prflCisdy 1* _I poUw for evel)' jJoim,. 4' ilVrptrl». TburlflfC, for every IIQl>W pcMi ..-'. lUld po(a u:rf 1;p - l (p W» . we hue

auWo)

•

,, - 1:

(l kJ.I-lj ... m.

("

, •.1

We !tOW Slue the 4ixs u(r4) IyIna

0V'Cl'

the disc

VfpuJ» to lbe qlal:C Ah

U

, U(I"l.

Dalote the obtained If*ll: by Al·. SilIoe, lie Euler dlaraaaistic of lllc di$c equals 1 and lUI

or lu bcModaty is O. _

ol;Jt.aia

•

1:

,. ,

(I .t,.' - I ) .

(5)

Ch. .. o Man' fo ldJ.and

Fibr~

'"

8 \>no.l l"

Ol~ oa& by OM new disca pI.KC'd o wr t ill: lunainina JlQInl.p(l"·) Ii M . i.e.• the proj«!ions of the silrplar points 11 .. T C Itt. we oblala:

x(.'1 j - x (v ,- ~ Ut>"»"'" 111 - ~ ....t.tte/ ls the rl...oa of different

Ilnases ptrl> of lbe

x ( \1'- L! UV» an

rdat~

and

( lk j '

_

(6)

I).

anaular poirruJl. It£tIIomber l.IIu

X(M'- l!Ver\o(e t he point of ccn tacr of the torus and plane by P . an d th e po int s of th e 10rus Ih at lie over p on th e pcrpcnd kul8r to the pl ane by q ,' and $ in order of In -

ctctUl.ng heiaht. Whlle inVI:$!i&ating fun etion s on the m anl fold . we wUl need tl'lc notions o f t he u beJ,lIe ret ('I' "c) .. ll'e X: .,, ~) " cJ of a fun dion ¥' : X _ R and thel~~1 ~I ,e l,e, 1,U

all exotlIp le Of a funCilon ,. posseslln l tbe indIcat ed propc nlll> , $pCCify me mtooth fun 0, i.e. , g.;s the coset 01, ~e naJl1 elemenl '" E Kcr l . rcs PQCliYe to lhc Rlbl!.rCup 1m I • • ,. In t U"1l,'" E Ca u d can I'C consio:kfcd as the: ~ 0 ( :;11 ~n.m elvncnt respective to t he wbarou p II foUows from 0 Ihat iJ.d e e~ 1 and fhat iJ.d E Xer "/t. _ 1 c ell. _ , from •• _ la. _ o . To describe Ihe

_

co,,~ ruction

de e.

e:.

I.", ."

£nrc:isrr J •. Show Ih.at Ihe ~el Iat d f o( an c1eme nl • • d in Hfr _ I (e~) docs not dePoCnd on Ih", d . oic", of the elemen ts '" and d ( rom tM corr rs poDding OSCU_ We auoc:iatcd ucll dm1cn1 .. (rom H . (C . ) willi lhe denlcnl "',td f' ft om H. _ I (C!) th ereb y specify;1t& a ~& wltidl _ ....ul denote by

O: C,,(l. Oj; G) -KCTao -O. H tftI%, _

oblam the bomoIcJ&y yOUP5 H o(. "; G ) _ 0

whCTI k > 0:.

H o(, O:G} _ O .

(l)

Before. ealt:ulatKoJ H o(' · : G) when /I > 0, _ 5Ol¥c. mo~ I.encr;t.l problem. COI15OdCT • Wriplic:ial compa X tyrn & ill the l\ypc.l'l)laliC n '" c R '" • I IUld1I point " e R- • n - . Wc will (lI11 the. CODKlio" o f $implaa c~ of Umf.lcxcs 'It e X. lbe. simrCll g and si~lua of lhe form (01 . ' ,'). Le.. ti mplcn s (01. iI t, . .. . g '! ) such th at ' l _ (a l. . .. . , iI i ) m. c:crlai n simplCll in K , m et'oM oK O'tIW I~ COM· plu K wllh ,~ ,"f1u iI .

I"

~I.U

'"

6 - . Show IlIl t " X is a Ilimp bc ,a1 complu..

PROPOSITION 2.. LdoK be a cone wilh

I. ...erto:

II ov er a loirnpl.ia.l o:ompkx K. T1'Icn

....he ll k > 0 ; H o(oK ; G ) .. G

H t IpK , G ) - 0

(4) ~ ' l "tI '

PROOF. Ce nllida an arbitl"llr)" G-dinwnsional cha in ' "4 + Co(PK; 0) ..

from

Ker ao' we have

E W,' a' -

.

K,O).

Du e to lhe equalit y

E Cll 'Qi •

an arl:Iilrary l;)'C:1c I ' . +

.. f!

+

I:, ~.

J;: "") ' 0 _hick U

DO(

el'

It" ) -

al a : .."" •

from Ker a o is ho molo&OIU to the cydc:, "(1

homoloaous to zero III Ihe IfOUP C o(4K : 0 1 when

"" O. We obuoin tbc bomorplllsm H ofpK ; G l

"

tI" ),

=

G.

Consider no.... an armerat)' k-d;me~ eyde in Ct (lIX; OJ

. , 't_1, 8/. h, E G an d [1',11.1. (g , or! tt -

when: / e III. . i E We ha~-c

E ' i' [rt l.;. E hJ ·l" . ..: - l J (I "er~t .

E 1/' 141 - E (rj ' lr!) "

Thcnfore th e cyck

:t ..

~

itt ... (r, ' Ia.

rlm = E 1/ "lo"J - I ). ,

is botnolOSlMls lo the l;ytlc

E, Jr; -Io·,l - l) - E, IIj'" 1l',,)o{Q.,/- ').

The eoc fflCieol of t ile sirnplQ

trf -

J

I in lhe sum i t

beln a o nly one w ch SImpl ex! ). Therefore

II; "

,] dell ' c oriented simplexes.

( [

, hi ·lo. ,J - ' I) ish; (Ihe re

L hj ' 10" , ,,}

,

- I ] il a cyd e If 8l1d o nly if

Oforeachj.

""\15. we h.a'·e ellab lished that in C . (aK: G ), when k

Ker ajo Is. homoiOSow 10 zero 1r > 0 _

In

> O. any C)"tle from

C..(/IK: GJ . Th erefore. Iljo(fJK . 0) '" 0 when

Note (h Ill the eOm(l1elt I" ~ l «>nnpond in.to l he $im p lex T ~ . . ",,0. o ~ ) is a 1I° IT~ - 'l ...ilh the: YfflO: ,, 0 OVer me f;ompkx p" - Ilwhlc:h f;orrap0n4sl o tlM:l im plex T" - I _ "" . . ... , o ~ ). Therefore. from equalities () a nd (">.owe obtain !he hornok>u aroup.l of a ll " . dimensional simplex: u

I;()QC

• •

I D. '( - u(_ I,m(l .,t ) ""

13:~

- a{1* - Df_ la:,r,

Extendins Df by linea rity to C;lX: 0 1, ,",'co ob tain the required homomorph ism

Df··

rDf l

We stress t he po int thai 1M construction o f is func lor ial, i.e. , ror ;IInr w nlinua u, mappin g ", : X - Y , th~ fa Uowing d iagr am is commutativc

C:IX;ClJ

V;

C:" IX.I;ClJ

!1'l'>t 1.J.. ,

1-' c:Il':GJ e: THE PIIOOF OF THEOIIEJ,l I. Let F : X x 1 We define: the chain homot opy

lDt

: C: (X: OJ -

C: .. (l'~I;OI

Yb 1. 10· , Let tbe cmbeddlnSi: X o - X be IIhomolopy equlvllcnce. Sho w thl t H; CX,

Xo ;O I c> O for ea ch k .

'"

Inlroouctioo

{\I

Topplosy

NoIc thai , enerall y ,pa.kln, ,......sen.ion tha t tl>n o r Ihc .. ·dilC (KIlO the 1)ounduy sphere ....cur bIIscd on t he fun ctori al propert y o r homotopy , roups and on Ihe re-Iu ll ....h lch

'"

has DOl been prov ed : ... ($ " ) • Z . Now . on the baliI of the cst ..bIidtcd '-not. pbis.m H:lCtlq the ~ f,

'x, "

aDd".

on the

bul.s o r l he lsomorphi$m or

5". LetA : R" + 1 _ R" + I be a nonsiDgLlilt Uncar opICftt or . We define thIC map. pio, A : S n _ S· b y the Co nnula .

A ~) _ " ;:~I ' e e s- . Prove tha t ra r the operator A _ - I: R n + I -

R" +

I.

the eq uality dell A _

- ( - I )"· 1 ho lds .

6". Prove l hat for an ubitTal'Y, nOIJ5in,gu)ar U.near opc r1llar A : R n ... I _ R " ... I. th.. equUlly ~A

= sl.&n

IA I Is valid .

HIlI" ~ dUll '" the dass o t ~ Iincar 1IpfttlOfS , A II bcxruxopic: 10 an operat;Ot A ' wboK matri:II: Is dlqoaaI aDd wtIoW diqom.I CQual "" I. and contt.nla • oiDrplicial partiliorr or the spbac owbIdt II ia¥ar:iaru; willi rcapccr. to IIw uanslonaallOa A ••

'"'"-nu

Con~idca' a ma.ppirla. : U _ R IO" '. wlH:re U is a YICSliptinJ t he soIotions of t he oquation . ~)

_ o.

usion ill

R IO •

I.

WlIiIc: In_

il IScUSlomary to cal l Ih~ m.pp;ng 4> th~ vedaI' f~ld On U (a pO;nt;ll; is usoeia ted with the vecto r 4> (K)), and the sotcuons o f equ at ion (6) singuIQr points af the venQ r f~td

4> .

In praciiee , the mapp inll 4> is not al ways continuo us. If ;t has isolated po ints o f disco ntinuity (or points o f ;n dCl~nninacy of Yallle), then these poi nu ar c also ca.11cd singular ~nts . Most subsequenl statements.are also valid for such vecto r r~lds . LCI x In: lUI isolated sintilular point of a vector field It, i.e.• 4> (K0) = 0, and let Iher~ be DO ot her sol utions of equation (6) in a neighbourhood of the poi ntxO. Th en for a su fficiently small R , when 0 < I' < R , the degree of the mappina 4>, ; s" _ S~ given by the eCl,u alily (7)

Is derUlcd, and doc s n ot depend. on Ihe cho ice o f I' (compare wilh E>tercise 3) . DE FINITIl?N 2. The d~lree deg. {or of the maptngs f, (for wfflc\ently s mall r) is called the mdo: of the 'Soialedsmgular poin t x of the vecto r fIoeld .; we wi ll den ot e it by ind (K 0, It).

Lei a field 4> have no sin gular points on t he bound:uy S: (;r°) of the ball D: .. I (;r0) with radius r and centr e at the poin l xO (it is not a.s~umcd now t ha i x ~ls a singulu point an d I' sma ll). lt ts evident that in this case also, formula (1) de fines the mapping 011, : 5 " - S" .

DEFlNm ON 3. Th o degree deg f, of a ma pping of, is eo.llcd the chtlr«lu istir: of the vector f~/d 4> on th e bo uodary of the ba.11 D: +- I (K0). Wr: will denole th e characteristic by

x(., 5:",°».

Along with tIM: tcnn 'characlr:risticof a vector field ' , the term ' rotauon or e veetor field' i~ o ften used . which Is similv 10 th e 2-dimen sional cue, where r.,.. y>: 5 1 _ st. lite degree deaf' is lbe ilIgebraic nu mber of rotations o f the veao r f' tr l wh en X ranges over t he circu mfcrenoc 5 1 (in 'the positive direction). TH EOIlEM 1. U t tlf~/a 4> ha~ no slngU/Qr points I" a c/OMf1 boll D/ + I(;r O), then

:d . , 5:(;t°» .. O.

PltooP. The mapping polDI

liven.

~ e S~, I lI>x-.

of, is hom oto pic to the cons lant mOlpping 11>" of

the degree

5~

into th e

or 11>0 beina:

zero . T he corresponding IIomotopy is

+ xo ) + ;11;° )1

o c ec

e.a:.. by the fonnula lI> (t rx

4> (1 x), - IfI(tr,tr

I,

x eS"• •

COROLLARY. If ]((II> . S; ~ (}l) '$' 0 Ih"n th" f"' /d 4> hIlStit I""st on" si1fl,u/sJr/XIlnl in

tM lNllI D,~ .. l(;t0 ).

'"

Inttoclucllon 10 Topoloty

No te that the charlKtaUtk x (to, S;'(c D» is dermed even if the field. is given only OJ! U.C boundary S:(%O) of th e b all D: • l (r 0)",

The foUowin J theorem II a direct corollary to Theorem 2. Q ~Id. ~ siwn on the sphne 5:(r°) (1M Ailwe li D sj~ulU points. If x{+, 5:"° » .. 0 fh~" • alnnot iN ext ended to Ihe txlll"D,." + "," 0) without M,~ku poinls.

THEOREM J. ILl

The COf\VtT5e to Theorem:) is also valid; it follows from the abovt-melltloned Hopf Ih ooren> .

The d l.raacrlstic of \I VC(:IOf flCld • can be defined on Ihe boundary of an y resto n l) c R ~ + I whicll is I compact polyhedron provid«!lhat . (x ) ¢ 0 on all:. Th e roUowlq theorem which we give wilhl,)ll! proof relata the global characteristic x (. , an) ora vector"field4owith the Jocal characteristics, viz., indices ind to) of the qular poinu of the fldd •.

... s p c I>omu LosY aroup homomorp hism

By ddinit ion, we put 11./ _

E 1- I)"Sp!9!)!. l.,J.

(0)

•••

E ( _ I)" Splel;l / ,l _ ,I:_e( ,.e

I)" Sp [9 !' / . ). , 1

(10)

and that If AI " 0, tben there exist simplucs '" E J(ltl and ...p E K. SIIch W I T' C

,II' . Now , we considcr an C1U1p1ewhen! _

,.1'

and!b'> -

I.. : IX I - tX I is lhe idenUly milppull of th e pol ybcd tOll I K I . DcnOk thc: diIM nsion o f the veeee Sf)lIC hav etli = A . Therefore th e Lefschetz num ber of th e mapp ing / : X - X equals that of its simplicial apPToximalion/~ ; I K V} I - IK I. where K is a tn an gu la lio n of X The Lefschetz nu mber o f continuous ma pp ing could be defined a s that o f its simplicial ap proxima tio n withou t the U K of smgu lar· ho molo gy theo ry. Th e Icllowmg th eorem is qu ite use ful for vario us applicat ions . In il$ proof.we shall use th e unrqueness theo rem of ho mology theo ry. THE OREM 4 (l EFSC HETZ). Uit / : X - X be a continuous mupping 0/ a co mpact polyhedron X = IL I into iI~l/, and 11'1 O. Thf!1l there exists Q f IXf!d poinf 0/ the mapping ! . i.e. Q po inr x e X such fhat ! Vc ) = x.

*"

PROOf . Assume tha t J bas no flXed point s. Th en there Is (3 > 0 suc n th at p (/(;:), x ) ;lo (J for eac h x e X . Ut.., '= min (P , (see Ex. S). COnsider a triangulatlon X of finen ess or/ 3 an d a slmplicial 1'l3.afproxlma llo n f .,1l o f the map ping/. Fo r arbitra ry po ints x,y o f any simplex r ' e K V , we ha ve the inequaJ ilies

"'(X»

p(/., I]Vc).)' );;' p(/(x), x ) - PVc. y} - p(/, IJ(;:),fVc ) ;lJ ..,13.

This mean s th at the relation , o. a sufrJci enll y small o :> 0, any XCi M", Ind;n the Rieman nian metric. the ineq ualily (3 ~ < X~ ) . X~) :> ., Q ho ldS. He nce, 1liiY point X e M" is unfailingly shifted by the diffeo morphism V, lIonl the intq;ral curve o rthe polnlX for Isu fficiutl) y YNJI t :> 0; this can be ehtckcd by eon sidc:rifl8 Ihe inleya! cu....., in th... chan at the po int x, The 'last sta te me nt is colilrAry 10 t he eJd5l"'noe o f a fIXed point for the diffeorJLOJ))hism V, . • CO ROl.l.AIlY. (f illS '"'" tllt lf thue Is If01 rI mqk WJc:t8)' Theory

",

singular points on 0 co mp oct,smooth man ifold does n o/ defHnd on Ihe choice ojthe vu:/or fie ld.

We give th e pr oo f of chis lem ma io a nuts he ll. let M~ be a co nnetted mani fold embedded in R "' , m > n + 1. We select a su ffic ien tly sma ll 'tubular ' neigh bourh ood of th e ma nifold M~ in R"', -l.e.. a neigh bo urhood. U(JvI~ ) whk:h is the total space of a loc ally trivial fib re sp ace with the base space M~ an d a fib re homeo mo rp hic to the disc D '" - " . More~r, the pr.ojectl on r of this fibre bundle is a smoot h reuaetlon, and the ma nif old M~ is a $trong d ef ormation retract of t he space U tM'~) . intuitively, the t ub ular neighbourhood of the manifold M" ca n be Imagin ed 10 co nsist of di5C$ D ;' - ~ "') over ea ch pol nt ;r EO M~ Ihal lie in (m - n )dime llsion al planes orthogonal to rhe tan gcm planes o f the mani fo ld M" . Thc set U(JlI~) is a compacl po lyhedro n. It is not co mp licated to s how that H:" _ 1(a u eM"); z ) ... Z . The gener ator of this grou p is a cycle bO~lOding U( A1") . T here fore any mapping V' ; aU f) v('X) + x - r(x) . The sum o f the indices of singular poin ls o f t he field w co incides with the sum o f t he in dices o f singular po ints of the tan gwt flCld v (by mean s o f t he Sard the orem, the gen eral case may be red uced to tne study of smooth field s with nondelilenente singular points, and the app licat ion of the result o f Exercis e 7, Sec. 6). Th e field w o n aUW~ ) Is ho mo topic , wit hout singular poi n ts, 10 the vector field t tK ) "" x - rtK ) . Hence, fo r the no rmed mappings 1Oi, t , wc o blain deg ~ '" deg t an d the refore deg ~ doc s no t de pend on th e flCld v. Lemm ata 3 an d 4 lead to the followin &theo rem. TH EOREM 6. TIle su m of rhe indices of singwlQr po ints of Q vector field ....Ith iw/atu/ singular po/n ls on a compect, smooth numlfold equals the Eu ler characteristic of Ih e moni/Q/d.

Exer cise 8". Let M~ be 3 co mpact. smooth manifold , an d fjP(MIf) ~ dimcH;(M ~ ; G) '" O. Show th " l an y Mo rse Function on Ihe m" nifQld M" has nor less than fj" (frf~)

crilkal po ints of ind elC p (Morsc ineq ualilies). FU RT HER READING

In the last decade, th ere a ppeared $CVer al mon ographs prov iding " s)'Stemalic app roach to homol ogy theor y and ;15 applications . We indic a te, fitst of all , A lgebraic Topo log,y 1731by Spa nier Lec tures On Algeb raic To pology (271 by Cold, Hom ology and Cohomology . Theo ry [53) by Massey as th ose mo st co rrespond ing to thc deman ds of l oo ay . Recommendin g lh em fOT 11 pro fo und a nd systematic study

I n lf odu ~on 10 T opoJOI)'

of

bomoIol)'~,

'fie emplw.lze. however, tllat inlrinsically lhey aK ra lheT a u· IlI:XIbooks eofIemu:&t cd on spedal COUfK:l. SK. t. White siudyina tllor Oll&hly sqlat'ate 10pic:s 10Ul:1Kd u poa in th e prnenl dlap:a-, II ....m be undoubled1y inlaesl.in8 for the reader to ,..,.. hlsatle snion to the folknrin. Hleral ar e: the noIion of loornokIsY was lntrod uc:qS &IlI1 elaborat ed 10 lhe dassialA Nllysis sitOs and tbe live COIIlplc:llWIlLS 10 it by PoiDcari: (see the mlh YoWme o f Otvyns« H~fIri PoittaIrt 163J). sec. 2. T o lI udy dWn oompkxes an c1 lhcir bomo'olY &fOllPS, 1M ruda is ad · vised to I« 01. 11 of Homolou [$1) by M~L:mc . Sec. 1 . SlmplOal homoJocy lhco ry is wmpaclly and Iho routJtl y upounded in Oulfine o/CombinaforiGl TopoJOD (6S1 by PonlTYqin . Qui le u seful is also the ac q uainlanc:c "",itb Introdwc:r/o'l 10 Homological J)im,nslo'l T1wory alld Com' binalorlal TopoJav 121 (Ois. 1·11) by Aluand ro" and Homology Throry 1401 lOu . (-III) by Hilron and Wylie. Sec. 4 . A brief and Ilco melry-dtn.ble '\tentian ls p"ell lbere 10 th e"kdmlell de tai ls of the theof}' . In the PlllOf of the th rort'f7l on bomciirnofpbisms inducec1 by homotopy ma pp iqs. we (o Uo'" MacLanc I'l l (CII. II . Sec. I) and M U5C)' I'll. since this method enahks 1# not to introduce: CCIum WIICCPlS wed in II:XINsi¥C" CO\1IICS. The Iudti" eaII Sludy the rdation bc lwoen bomolos7 IIftd bomot09Y JrOUpi in the abo~ofJICIllIoncd boo h by TclCIIIaII 1791 (01 . IV, Sea. J, 7), Hihom and Wylie {~ (Sec . 1.1) and Fudu C1: aI. m! (Sec. U ). and also in Homotopy ~ (4l1 (01. II, Sec. Ii and 01. V. Sec . 4) by Hu S.-T. Sec. S. The uio ma lk . PPtoadl 10 bomoJoay theor)' is .;"en in FollJttkJtiOM 01 AI~1c Topolov flOI by Eilcnbcrt and Steenrod . A diteet proo f of tbe cqul\'akncc o f siJDplicW atld Jinaular thcc:lries on rhe QtqOT')' o f polyl\edu is &Iym . c•• .• in the book by Hilto n an d W ylie J40J (Sec . 1.6). The rca ckr ma y nn d AJeM.fldroY-Cech hornoJoay t heory in lhe book by T eleman [79J (Ch . n , See. 18) and, In arca.tcr d elu. In the boo h by A1cxandl OY 1'21. Ill. See:. 6. The homo;llo sy 8' o up, of spher es ar e c.scul"ed in all the co urses o f h(lm o loD the ory. We fo llowed th e book by Fu eh s et al. [111 (Secs . 12- 13). In the lIbQ"e· me ntJo nni Com blnatonill Topof()IY [ I] (Or. XV I) by Alcundroy. the llleofla of 1M:degree of .. mappill! . chara.cr eriSlk (If , vector flCld and inda o f a sinJUIar poinl are giyCl> c;ltmslyely an d on t he bw o f sim plicial homoiolY lheo ry . Sec . 7. As Icprds ce U hocnolo.y tllcory. we f«ammcnc1 lhe lect lll Cill by BollyaIl' sk y on IJtI.* CR &d'. Pans. 1m . 10 . Roll1ill. Y. A. &lld f\lt1lI. D. B.• FIn, e - tJl Topo/t)U. ~trlc ~. M_.lm rlrlR~).

11. Seiferl . H. IIld ThnUfon, W" lA/lrlh/ropo:flIC1 topolo"eal. 12, 4S. 59. 85. III o f mnlic .paces, Ie of lopoloP:al opac:a, 70 Rdinnnent , 9O,!in. 91,100, 177 Iklna. IU. I II, ,.IUOOA ddormaticD. 116. 117, 145. 1M.

,.,

'"Uk,

,4.4. 214. m

116

II '

Sard theorem , 117, Ill. 2-19. 2951 SccOfld _ 'o.biIity Wom. 19·9 2, 16' ~&lJ0tl WDnu. , .

"'-

COn"i'rjlftll, 17

fuet, 2' . , 156. zs -. 263. 2&5. 274, 279,

m. ""

lIomoJoe;r . 274, 281 , 2114

of . pal. , 263. 16S. 283,:wI lUn4amcntal. 7' of palnu of . I PKC. 17 op«l,a1 . 2j.4

""

rio$Cd. 4 1, 7 1. 1 4, 92. '3 . 91 coonpaa. 100, un, 1(Jol., 115-' 18. ... _ eel, 1'1 . - I.. .... u. n. TI l

............. da>>'rd. 1O dosa'ele. 71

......'"

_ .41 ,47 , 74. 11 .91. 9l, 116, \76 partidy «do red, 4 1

_ N . ... . 92 scq uaW.allr CompaQ , I OJ sloriati.... St. 117 Se'win&. 117, 1St

.-rldd, on

Simplu. :tW, 2:17-2S9. 262 . 264, 269. 2'77, 214, 19S·2;97 barycenlfe o r. 266, 276 QI,viUneu , 162, 26S, 267, 268, 210 o rimled, :U1l-26 1, .26) . 264. 2U ' inJl1l:ll , 169-274, 276. '2'77. 28J lI andto' d . :U1. :U8 , 269 , 216 Sim plIcial aN!l'onlma hon theo rem , '296 paniUon. 266. 2U SkclctoJI , 289. 290

s.-

base . 2I S·2 17 , 2 1t, 221. m, i l l. 2)4,

'"

b&Ic-polnl . I ll. 121. 271

..cak ,II S

Itclt Ktioa . Il l, 115. II"

"'rona ddonnalioa.

rlM>CtJoOf\au, K pa . . ...... !n LcbQaue. 74 . m . :w.

lII nllilc. 9I

eeee, l IS

COO\pacI. 91. COO\pIcl • • 16

105. Il l.

In.

114 . 214

......pIotdy .q ular, 106 _r~u..l.lon . 17 1, In. III. 119, Its, 211.21) ..........tcd.,. , ... II I , 115. 130 , 2.\0. 2) ) or oontia UOl, 114

boroOlOP7 " ," ole. III Ian. 69. 10) , 161 acnuabuod. 69. UJ . 224 liocv, 119 1ocIIII:r OOOlIIpaQ , 99 . 100, l OS. 112, 113.

'"

me..i