Introduction to Topology

Table of contents :
Cover Page
PREFACE
1 First Notions of Topology
1. WHAT lS TOPOLOGY?
2. GENERALIZATION Of THE CONCEPTS OF SPACE AND FUNCTION
3. FROM A METRIC TO TOPOLOGICAL SPACE
4. THE NOTION OF RIEMANN SURFACE
5. SOMETHING ABOUT KNOTS
FURTHER READING
2 General Topology
1. TOPOLOGiCAL SPACES AND CONTINUOUS MAPPINGS
2. TOPOLOGY AND CONTINUOUS MAPPINGS OF METRIC SPACES, Spaces R^n, S^(n-1) and D^n
3. FACTOR SPACE AND QUOTIENT TOPOLOGY
4. CLASSIFICATION Of SURFACES
5. ORBIT SPACES. PROJECTIVE AND LENS SPACES
6. OPERATIONS OVER SETS IN A TOPOLOOICAL SPACE
7. OPERATIONS OVER SETS IN METRIC SPACES. SPHERE AND BALLS. COMPLETENESS
B. PROPERTIES OF CONTINUOUS MAPPINGS
9. PRODUCTS OF TOPOLOGICAL SPACES
10. CONNECTEDNESS OF TOPOLOGICAL SPACES
11. COUNTABILITY AND SEPARATION AXIOMS
12. NORMAL SPACES AND FUNCTIONAL SEPARABILITY
13. COMPACT SPACES AND THEIR MAPPINGS
14. COMPACTIFICATIONS OF TOPOLOGICAL SPACES. METRIZATION
FURTHER READING
3 Homotopy Theory
1. MAPPING SPACES. HOMOTOPIES. RETRACTIONS, AND DEFORMATIONS
2. CATEGORY, FUNCTOR AND ALGEBRAIZATION OF TOPOLOGICAL PROBLEMS
3. FUNCTORS OF HOMOTOPY GROUPS
4. COMPUTING THE FUNDAMENTAL AND HOMOTOPY GROUPS OF SOME SPACES
FURTHER READING
4 ManifoldS and Fibre BUNDLES
1. BASIC NOTIONS OF DIFFERENTIAL CALCULUS IN N-DIMENSIONAL SPACE
2. SMOOTH SUBMANIFOLDS IN EUCLIDEAN SPACE
3. SMOOTH MANIFOLDS
4. SMOOTH FUNCTIONS IN A MANIFOLD AND SMOOTH PARTITION OF UNITY
5. MAPPINGS OF MANIFOLDS
6. TANGENT BUNDLE AND TANGENTIAL MAP
7. TANGENT VECTOR AS DlFFERENTlAL OPERATOR. DtFFERENTlAL OF FUNCTION AND COTANGENT BUNDLE
8. VECTOR FIELDS ON SMOOTH MANIFOLDS
9. FlBRE BUNDLES AND COVERINGS
10. SMOOTH FUNCTION ON MANIFOLD AND CELLULAR STRUCTURE OF MANIFOLD (EXAMPLE)
11. NONDEGENERATE CRITICAL POINT AND ITS INDEX
12. DESCRIBING HOMOTOPY TYPE OF MANIFOLD BY MEANS OF CRITICAL VALUES
FURTHER READING
5 HomologyTheory
1. PRELlMINARY NOTES
2. HOMOLOGY GROUPS OF CHAIN COMPLEXES
3. HOMOLOGY GROUPS OF SIMPLICIAL COMPLEXES
4. SINGULAR HOMOLOGY THEORY
5. HOMOLOGY THEORY AXIOMS
6. HOMOLOGY GROUPS OF SPHERES. DEGREE OF MAPPING
7. HOMOLOGY GROUPS OF CELL COMPLEXES
8. EULER CHARACTERISTIC AND LEFSCHETZ NUMBER
FURTHER READING
ILLUSTRATIONS
REFERENCES
NAME INDEX
SUBJECT INDEX

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

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