milnor - topology from the differ en ti able viewpoint

8/2/2019 Milnor - Topology From the Differ en Ti Able Viewpoint

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TPLGY


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FRM THE

DERENABLE

VEWPNT

By John W. lnor

Based on notes by

David W. Weaver

The Universit Press of Virginia

Char ottesville


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Copyright 1965

by the Rctor nd Visitors

of the iversity of Virgii

The Ui vrsity ress of Virginia

First published 1965

Librry of Cogress

Ctog Crd Nuber : 65-26874

Printed in the

Uted Sttes of Americ

Q


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ToHeinz

Hop!


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A

EE lecture ere eliere a h irit irgia i Decemer9 uer h rh h ageBar ecture Fuatihe ree me ic rm h egig lg, ceerig

au E J Bruer eii 9 the degree a maighe meh ue, her, ar h eretal lg, raherha he cmatrial meh Brer he cce regularvalue a he herem Sar a Br hch aer ha eermth mag ha regular ale la a cetral rle

imli he reetai, all mail are ake t e ieliereale a e exlicitl emee cliea ace mallamut ie lgy a ral arial hr i ake rgrate

ul lik her t exre m grai Dai Waer, himel eah ha aee u all i lle e tes mae thmaucri il

Princeton, New JerseyMarch 965

i

,


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CONTENTS

Preface VB

l Smooth manifos an smooth mas Tangent saces an eriaties 2eguar aues 7

Te funamenta theore of agebra 82 . The teorem of Sar an Brown 10

Manifos wit bounary 2The rouwer xe ont theorem 3

3 roof of Sars theore 164 . The egre mouo 2 of a mag 20

Smoot homotoy a sooth isotoy 20D riente manifos 26

The rouwer egre 27

6 ector es an t uer number 327 Fae cborism the Pntryagin construction 2

Te Hf terem 0

8 xercises 52

eix: assifying anfos 5

bgrahy 5

ne 63


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TOPOOGY FROM

THE DIFFERETIBE

VEPOIT


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SMOOH MANIFOLDS

AND SMOOTH MAP

I e us exin some of our erms Rk enos the -iensioneucin sce thus oint x Rk s n e x . X ore nuers

e U C Rk n VeRI e on ses ing fro U to Vwriten U V) is ce sm if of he ri eriisanf/axi ax exist re connuous

More genery e C Rk n Y C Rl e rity suses oeucien sces m : Y is ce smooth fo ech x here exis n oen se U C Rk coninng x n sooh mngF : U Rl th coincis wth hroughou

: Y n g : Y Z e smooh noe h the comosiong : Z is so smooth The iny o ny se is micy sooh

DEIII : Y is ce dieomorphism cis homeoohicy ono Y n f oth n sooth

e cn now nce oughy wh dretia topolo s o sying h ses hos oes o s R wch nnun ieomoh

e o no how wn o oo com i ss Th foowing nion snges o ic cie uscss

EIII sse Rk s c sooth maiold o dimesiom if ech x hs nighohoo W h s oohc o on ss U of he ecin sc R".

ny c eomos : U W s ce paramerizaio o h gon W The ins oohsmW U s c sysm o oodiates on W


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2 oot

Figur

Paramrization of a rgion in oeties we wi nee to ok a anifs of iension zero B

enition, s a anifo f iensin er if eac E has a neighborhoo n consisting o an

XMLE The uit shee S2 consisting of a , y z) E R3 withx2 y2 + is a sooth anifo of iension 2 n fac theieorhis

, y , y x2 _ y2,for 2 + y2 < I araetrizes the regin z > 0 of S2. y intchagingthe roes of y z, an hanging the signs of the ariabes, we otainsiiar aaetrizations of the regions > 0, y > 0, < 0, y < 0,an z < O Since these coer S2 i foows that S2 is a sooth anifo

More generay the shere sn C Rn consisting f a X , )with is a sooth aifo of iension For eae

SO C R is a anio consisting of just tw oints soewhat wier eae of a soth anif is gien y theset f a , y E R2 with 0 an y sin)

TGET SPCES D DERTIES

To ene the notion of evav fr a sooth a of sooth anifos, we rst associae with each E C Rk a inearsubsace T C Rk of iensio cae the tge pae o j at Then wi e a inea apping fro T o T, where y eents o the ecor sace T are caed tget veto to j at

tuitiey one thins of the iensiona hyeane in Rk wics aoiates na then T is the hyeane though h


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age spces

rigin ta is arael t tis mare Figures 1 an 2 Similrlyne ins of e nnoogeneous inear maing frm e angentyerlane at to te tangent yelane a y wic bes aroxiates ansating bot yerlanes t te rigin ne btains df

Before ging te actual enition, e ust stuy e secial casf aings beteen oen sets For any oen set U C R te agepae T is ene to be t entir ector sace R For any smootma : V te divaive

df R R

s ene by t formua

dfh l i (f( + h fO

f E h E R. learly dfh is a linear function f h In fac dfs us tat linear maing wic corresns t te X matrixf f rst artial eriaties, ealuate at

ere are o unamental roerties of te eriatie oeratin:

1 ain rue). If f : U V ad g : V ae smooh mas, wihf y, hed(g 0 ) dg 0 df

n oter wors t eery cmmutatie triange

of smot mas etween oen subsets of R\ Rl Rm tere corresonsa cmmutatie iange of inear mas

2 If I is he ey mp of U e d is e e ma o Roe gee C e oe ses ad

i U U


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mooh miolds

is he iclusio mp he g d i he ideiy mp o RkKoe aso:

3 L : Rk R is lie mppg he dL = LAs a sme acation of te to roeries one has te foon:

AEO f is deomophis bewee ope ses C Rk dRI he mus equ l d he lie mppi

d : Rk Rmus be osigul

The coosition 0 i e identity a of enced ) d is e denty ma ofR Smay dx 0 d ) is e deyma of R Tus d has a o-sded nerse and it foows at l

A arta conerse to ths asserion is d Lt : Rk asmoo ma i on in Rk

v Fu he eiv ive d Rk Rk is osiul e mps y sciely smll ope se bou x deophiclly oo ope se )

Se Aosto 2 44 or edon 2ot tat may ot e oo i e lr en f ery s

nonsr A instrcie exame s rodd y te xoeama of te comx ae io tsef

ow e us dee te e spce T for a arirry smoohmanfod C Rk oose a aramrzaton

g C Rk

of a norood g) of x in w g u x ere s a onst oR T of g as a ma fromo Rk so a e dae

d : R'Rk

s dend e T eql o he ie du R) o d (omar e must oe at ts cosrcton dos not dd o ar

tcuar co of arametzaton Lt h : C R e aneraramzaion o a eorood ) of x and t v h- )Ten h- g as soe neoood U o u doorca oo norood of v Te commtai daram of smoo m


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6 mooth maifods

smooth ma

ith y Te deivativedx Tx T

is dened as foos Snce is smooth there exist an oen set containn x and a smoot ma

F R

that concdes ith f on n Dee dx t be equa o Fx(vfo all Tx justy this dtin must re that dFv eons to T

and that t ds nt dnd on th articuar chc Fhos arametrzatins

g : U C Rk and h: V C Ror nehborhoods g(U) of and ey) of eacin U y a smar

set if necessary, e my assume that g(U C and tat mas g(U)nt (V. It foos that1 0 f og : U V

s a edened moot manonsidr the commutate daram

P. )R1

-' 0 log

r

U )Vof smooth mains beteen oen sets Tain dritis e otana commutati diaram of inar mans

ee u = g-1(X), = h1(y t foos mmdiatey tht d carries T = me dgu) int

T = ma h) urthermore t resutn ma x does need n te rticuar choice of P fr e can oan te sam inear


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Rgua au

transformaon on around the otom of the diaam That s:

d = dh 0 ho g 0 g1Ths comletes he oof that

d Tx Ts a -dened near man

A for th deriatie oeraton has to fundamentl roertie:

hain rule : and g : N P ae smooh with () = yhn g = g 0 d

2 is t identy m o he d i th intity map o Txoe geely with incusion map i, thn Tx C x withicsion m (omare Fiue 2

T

Figur 2 h angn spac o a 8ubaniod

The roofs are stratforardAs fore these to roerts ead o the foon:

AEO : s eomohsm e d T TNs isomohsm o veco sces icl he eso o us be equ o t dmeso o

REGR ES

Let a smooh ma eeen maifolds of the samedmension* e say ta x is a egul o of if th derate

* T co movd


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is nnsinula. I this cse it flos from the inerse functioterem tat mas a neiorhood of in dieomricaly ontoa e et i N. The oint Y N is caed a egula vaue if y)contains ly reular oints

f is sinuar ten x is caled a citica point of and te imae( is caled a citicl value Tus ac Y N is eiter a critical aue or areula alue accordin as y) does or does nt cntain a critica oint

Obeve that i is compct an Y N is a egu vlue the y) a nie set possibly empty) or 'y) is in any case comact bein

a clsed subset f the cmact sace and 'y) is discrete since is nene in a neihrhood of eac 'y ) Fo a smoot : N it comact and a rua alu y N

we ee y) o e the numbe o poin s in ' y) Te rst obsertionto be made about #'y) is ta it s lcaly consant as a functon of y(here y ranes on trou reuar alues. Ie thee is a neigbohoo V N o y such th t #' y = o any y V Let x . Xkbe the ints of y, and choose airise disjoint eihorhoodsU" U f these hic are maed dieomorhic aly onto neihorhoods V . , V n N. We may ten tae

V = V n V n n V - U, - - Uk).]

THE FUNDMET THEOREM OF GEBR

As an aicati of these notions e roe th fundamental teremf aera: evey nonconstant comple poynomia P) must have a eo

or te rof it is st necessary to ass from te lane of comlexnumers to a comact manfod. onsider the uni ser C R3 ande stereoraic roection

h+ : - {O 0 } R X C R3from te "north ol 0 0, of . (ee iure 3 We il identifyR X ith te ane o comex numers The olynomia ma P fromR X itsf crresonds t a ma from to itself ere

x = h'h+ fo 0 0 O 0 , = 0, 0

t is ell knon that this resulting ma is smooth een in a neihbor


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Fnamena tho em o ab 00

Figur 3 Srographi projion

hood of the orth e To see ths we ntrduce te stereorahcroecio h from te south ole (0 0 ad set

ote by elemetary eometry tat

h+h = = z

ow if = a + a1 + . . . + a wth a 0 the a shorcmutation shs tatQ = ( + + . . . +

Thus Q is smoth in a eorood of 0 ad it folws hat = Qhis smooh i a eiorhod of (0 0 )

ext osere that has oy a nite mer of crtcal oits for Pfais to e a cal dieomrism ony at te zeros of the deriatieolyomal

P()

= a

-

ad tere are oly tey mayzers sice P s not ideticay zero e set of reuar aues of ein a shere with itey may oints remed is terefre coected Hece the ocay costat fuctio #y) must actualy be cstato ths set Since #y) cant be zero eerywhere we cclude thatit is zer owere hus is an onto mai ad the oynmial Pmst ae a ze


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TH THO

O SA A BOW

I GEEL, it s too much to hoe that the set of critical alues of asmooth ma e ite But s set ill e small i the sese idicatedy the ext theorem hich as oed y A Sard i 942 folloiearlier ork y A P Morse efereces 30 24

Theem Let U Rn be a smooth map, ene on an open setU Rm an let

= (x U I ak f < n } Then he image ) C Rn has Lebesgue measue eo

Sice a set of measure zero caot cotai ay oacuous oe sett olos tat the comlemet Rn ) must e eeryere desei R

The roof ll e ie i 3 It s essetial fo the roof that shouldhae may deriaties (omare Whitey 3)

We ill e maily iterested i the case m n I f m < n theclearly = U hece the theorem says simly that (U has measureero.

More grally coside a smooth ma f M , from a maifoldof dmesio to a maifold of dimesio n Le e he se of all

x uch thatf :

* o wods, gv ay 0, s possl o cov y a sqc ofcus n avg oal dmsoal volum lss a

Povd y Au B Bow 1935 Ts sul was dscovd Duovck 953 ad To 1954 Rfc 5 8 36


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u ues has rak ess tha n i e is ot oto The will e called the t

of citic pints ) he set of citic vues ad he comemtN te set of eu ves of This rees wih our reioudeiios i te case n) Sic M ca e coered y a coutalecollecto of eioroods each dieomoric to a oe suse ofm we hae:

Clay A B. Brow) The set o egu vues o sooth p : M N s veywee dense in N

I order to xloit tis corollry we wil eed th follwin:Lemma I N is sooth p between ni ods o dien

sion n, nd i y N is egu vue, then the set 1 y) C is soot niod o diension n

OOl Let x 1 y Sice y is a reuar alue te deriaie must ma T oo TN The ll sace C TM of d ill tereforee an -dimensiol ector sace

IfM C

choose a iar ma

L Rk Rm that is osiular

on this susace C M C Rk ow dee

y )

F :M-NXRm

), T deriatie d is clearly ie y he formulad) = d(),

Tus d is osiuar. Hence mas some eiorood U of xdeomoricall oo a eihorhood V of L oe ht 1 y)corresods uder , to the erae y X Rm I fact mas1 y) n U dieomorhically ot y X Rmn n V. Tis roes thaty) is a smooh maifold of imensio n

As a examle we ca ie a easy roof that the uit shere 8m1

is a smoot maifod onsider the fuctio : Rm R deed yfx = x + x + . . . + x

Ay y r 0 is a regular value, and the smooth maifold l is theuit shere.

If ' is a maifold hich is cotaied i M it has already eenoted that J is a susace of M for x M' The orthooal comlemet of i TM is the ector sace of dimesio called he spc of no vecos o ' M x .

I ricul le ' = 1 for eular alue y of M N.


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2 2 down hoe

Lemma 2. Th nu spc of df T TNy is pciey qul oth ngent spce TM C TM of he subnifo M' = 1 y) Hencedf ps h othogonl copeent o T isoophicy ono T

OOF From the dagram

i 1 fy Nl

e see that df mas the susace T C T to ero outigdmss e see tha df ps the spce o no vectos o isoophicy onto TNv

MIFODS ITH BODRY

The lemmas aoe ca e shareed so as to aly to a ma deedo a smooth mafold h oudary osder rst the closedhafsace

H = { X . . . , X r Rm X 0 .The bouny aHm is deed to e the hyerlae Rm1 X 0 C Rm.

DEFO A suset C R is caed a sooh nifod wihboundy if each X r has a eghorhood U n deomorhc toa oen suset V n Hof H. The boundy a s the set o a ots hch corresod to ot of uder such a deomorhsm

t is o hard o sho at a s a ell-ded mooth mafodof dmeso m Te inti a s smooth mafod ofdmeso m

The tage sace T s dd just as so that Tx s a ful-dmesioal ector sace ee f X s a oudary ot

Here s oe method for geerat exames Let a mafoldtout oudary ad et g R hae 0 as reguar alue

Lemma The set oX in with gx) 0 is sooh nod, w thboundy equ o g )

The roof s ust e the roof of Lemma


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ouwe xed pon heoe

E he un d Dm coitig of all x t m it

1 0 ,i smooth manfod th oundary equal to 8m

osie smooth ma f N rom an manifold ithoudary o a mfold here > n

Lemma 4. Iy N egul v lue, both o nd o the e tction a then (y C ooth nnod wth boundy.utheoe e bouny a(f(y pecey eu to the nteectionof l(y wth a

OOF Sce e hae to roe a local roerty it suces to cosderthe secal case of a ma f Hm n ith regular alue y t n Let t \y. If is a iteror ot the as efore l(y is a smoothmafold i the eighorhood of x

Suose tht x s a oudary oit hoose a smooth ma U nthat i deed throughout a eghohood of x i m and coiides ith

fo

nHm. elacg

Uy a smaller eghorhood if ecessary e

my assume that has crtal ots Hece l(y is smoothmaifold of dmeso n.

et l(y deote the coordate rojecto

We clam that has as a regular alue Fo the taet sace ofl(y at a ot x s equal to the ul sace of

dg = d nut the hyothess that f aHm is regular at x uaratees that thsull sace caot e comletely cotaed i m1 X 0

Therefore the set (y n Hm = ly n U cosstg o f all x gl(yth (x 0 is a mooth mafold y Lemma 3 th oudaryequal to ( Ths comletes the roo

THE BROUER FIXED POIT THEOREM

We ow aly ths result to roe the key lemm leadg to theclasscal Brouer xed ot theorem et e comact mafoldith ury


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14 2. Sard-Bown theormLemma Tee is n oo ap f a a lavs poin-

wise xed.OOF (foloi Hish Suose there ere such m

Let y r a e reuar alue for Sie ? is certaiy a reuar aluefor the idetity ma a also it foos that l(y is a smooth 1manifold, ith oudary cosisti of the sile oit

But (y is lso comact ad the oly comact -maifolds are ite

disjoit uios of circles ad semets,* so that a(y must cosist of een numer of oits This otradictio estishes the lemma

I rtcular the uit isk

D = x r x x is a comact maifold ouded y the uit shere sn Hence as aecial case e hae roed that the identity p n cnnt be exended st p Dn - .

Lemma Any sth p g Dn Dn hs xed pint (i e a oitx r Dn ith (x = x .

OOF Suose g has o xed oint or x D let (x sn ethe oit earer x o the le throuh x ad (x (See iure 4 The Dn sn is a smooth m ith (x) = x for x r sn\ hich isimossile y Lemma 5. T ee tht is smoth e mae the fooiexlicit comutatio: (x) = x u, here

x g(x = x g(x t = -x u+ VI - XX + (X'U)2,

the exression under the square root si eig stricty ositie eread susequetly x denotes the euclidea lenth x x )

Bruwe Fixed Pint Theem. A ny ntinuus unin G Dn Dnhs xed poin

OOF We reduce this theorem t the lemma y aroximati Gy moth maing Gien f > 0 accordi to the Weierstrassaroximatio theorem, there is a olyomial fuction R Rnith x G(x < f for x D Hoeer, may sed oits

* poof gv Appdx fo xamp Ddo 7 33


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Buwe xed point tee

)

Figur 4

of Dn ito ts outside of Dn To corect this e set

( (x/( The clearly mas Dn ito Dn and I P (x) - G(x)1 < 2for r Dn

Suo that ( x fo all x r Dn The the cotiuous fuctio Gx - must take o a mimum j > 0 o Dn hoosi

Dn Dnas aoe, ith x - ( < j for all x, e clearly hae x xThus is a smooth ma from D to itself ithout a xed oit Thiscotradicts Lemma 6 ad comletes the roof

The rocedure emloyed here ca frequetly e alied i moreeeral situatios: to roe a roositio aout cotiuous mais,e rst estalis the resut for smooth mas ad the try to use aaroximatio theorem to ass to the cotiuous case (omare 8Prolem 4


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OO O

SA'S TEO*

let us recall te statemet :

Theem f Sardo Let U R be sooth p with U open in Rnn et C be the set of ctc points tht s the set of x E U wth

radfx

< p.Then f(C) C RP h esue zero .

EK The cases here n p are comaratiely easy (omaree ham 29 . 0 We il hoeer ie a uied roof hicmakes these cases look just as ad as the others

The roof ill e y idctio o n ote that the statemet makessese for n 0 p 1 (By eitio RO cosists of a sile oitTo start the iductio the theorem is certaily true for n = 0.

Let C1 C C deote the set of all x E U suc that the rst deriatiedf is ero. More eerally let Ci eote the set of x suc that all artiaeriaties of f of order ais at x Thus e hae a descedisequece of closed sets

C C1 C2 C3 The roof ill e diided ito tree stes as follos:

E 1 The imae fC - C1 has measure zeroE 2 The imae f(Ci - Ci+ has measure zero for i .E The image f(Ck has measure ero for sucietly lare.EK If haes to e real aaytic the te itersectio of

* Ou poof s as on a gvn y Ponyagn 8 as a omwaa nc w assum a s nny na


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Step

the Ci is vacos unless is costant on a etre componen of U .ence in ths cas it is sciet to arry ot teps 1 and 2

OO O E This rst step is perhaps the hardest. W maysme that p 2 since C = CI wen p = 1 W will eed the welknown theorem of ini* which asserts that esuble set

A RP = R X RPust hve esue ze i it intesects ech hypeplne (constant X RP1in set f (p )diensinl esue ze.

For each x E C - C w i d a open neigorhood eRnso that ( n C) has measre ero ince C - C1 is covered y cotaymany of these neighorhoods this wi prove that fC - C has measrezero

Since x tere is some partia derivative say whichis not zero at x Consider the mp h U Rn dened y

hx) = x X2, ice d is onsinar, h maps some eighorhood of x dieomorhicay onto an on st '. h comosition g = h wi tma ' into RP ot that the st C of critica oints of g is recisey( C; hence th set (' of critica vaes of g is eqal o ( n C.

Fgr nstrctn th map * Fo a asy proof as wl as a alatv poof of Sad's orm) s Sn

g [35 pp 5152 Strg assums tat A s compact, u t gna casfoows sy fom s sca a


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3. rf Sar's heremFor each ( X . . . xn) { V' note that get X elongs to

the hyperplane X RP C RP: ths g caes hypeplanes into hypeplanes. et

g ( X Rn1 n V' t X W

denote the restriction of g ote that a point of Rn-I is citical fog if and only if it s critial fo g fo the matix of st deiatve o has the form

According to the indction hypothesis the set o ctical vals o ghas easre zero in X RP Thereore the set f critical vles o ntersect each hyperplane X RP in a se of ease zeo. Thisset gC s measuale since it can e expressed as a contae nonof compact susets. ene, Finis teorem the set

gC = V n Chas easre zero and tep is compete.

OOF OF E 2 For each { Ck - Ck+ there is some + derivative ak+1frax" ax'H wich is not zero. Ths the fncion

w(x = akt/axs, . axmvanishes at x t awax. does not. ppose or denitenes that s = lThen the map h : U Rn dened y

hex ((x) X .

xn)arries some neighohood V of x dieomorphically onto an open set Vote that h caes Ck n V into the hyperplane 0 X W gainwe consider

e

denote the estriction o g. By indction the set of crtical vales of gas ease zeo n RP t each point in hCk n V) is cetainy acitical point of g since all deriaties of ode vanish . Theefore

ghCk n V = tCk n V) has mease zeo.ince Ck - CH is coveed y contaly many sch ets V, it folowsthat teC Ck h mease zero


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Stp 3

OO O E 3 et e a cue wit edge f k s sucientlylarge k > np to e precise we will prove that t(Ck asmeasre zero. ince Ck can e covered y contaly many sch cuestis will prove that tCk has measure zero.

From Taylors theorem the compactness of and te dnitionof Ck we see that

were

1)

f(x + h) = t(x) + R(x h

I(x h) c Ilh Ikfor x C x + h . er c is a constant which depends onlyon and I Kow sudivide into ces of edge r et e a cueof the sdivision whic contains a point x of Ck Ten any point of Icn e written as x + h with2) Ilhl rFrom it folows that t( lies n a ce of edge + centeredaout t(x here = 2c 0/+1 is constant. ence t(k iscontained in a nion of at most ces aving total volme

f k > np then evidently V tends to 0 as O s f(Ck J"st have measre ero Tis completes the proof of ards teorem.


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4. THE EGEE MOUO

O A MAG

COE a smooth map : S sn. f y is a reglar vale, recall that#1 y denotes the nmer of sotions x to the eqation x = yWe wil proe that the esidu l ss dl 2 y) des nt depend nthe hice th gul lue y This reside class is called the od 2degree of . More geeraly thi same denitio works for any smootmap

f :MN

where is compact withot oudary, N is onected, ad othmaifolds hae the same dimension. We ay as well assme also thatNis compact ithot ondary, sice otherwis the mod 2 degree woldecessarily e zero For the proof we introdce tw ew cocepts

SMOOTH HOMOTOPY AND SMOOTH ISOTOPY

Gie C k, let [0, 1] deote the sset* of Rk consistingof all x, t with x E ad 0 : t : 1 . Two mappig

f,g

are alled sthly htp (areiated f r g if there exists a

* If M s a s anf w nay n M 0 ] s a smanfd d y w "cs f M Bnay pns f M w gv s c s f M 0


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tpy n istpy

smooth map X 0 1] Y with(x , = f(x , (x , = (x

for all x Tis map is called a sth htpy etween f and g ote that the relation of smooth homotopy is an eqivalence relation

To see that it is transitive we se the existence of a smooth fnction< [0 1 [0 with

(t = for 0 t

(t = 1 f i:t (For exampe et < (t = (t /At + A t wher A = for 7 and A7 = exp( 7 for 7 > 0) Given a smooth homotopy Fetween f ad , the formula G(x, t = (x, < t dees a smoothhomotopy G with

G(x, t = f(x fo t G(x, t = (x for i:t 1

ow if f " and " , then with the aid of this construction it ieasy to proe that f " h.

If f an g apen to e dieomorphisms from to Y we can aodene th concept of a "smooth sotopy etween f and g. This alsoil e an eqivaence relation.

DEFO The dieomorphism f is stly istpc t g if therxists a smooth homotopy F : X [0 1] Y from f to g so thatfor eac t [0 1] the correspondence

x (x, t

maps dieomorphicay onto Y wil turn ut that the mod 2 dgree of a map deends only on its

smooth homotopy class:

Hmtpy Lemma. et j

g M N be ty htpic pbetwen ifld te s diensin, weeM is cpct nd witutbundy. f y N is egul v lue f bt f nd g, ten

OO et X [0 N e a smooth homotopy etween nd g. First suppose tha is also a regla val for . Then (y


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Dg mdul

i a compact I-maod with odry qa torl(y) n (N X 0 X ) = (y) X 0 (y) X 1.

Th the ota nmer o odary poi o rl(y) is eqa to#(y + #g-(.

we rec fom tht a compac mniod way ha a evennmer o odary poits Th #r) + #gIy i eve adterefo

(

Fgur 6 Th nbr bnary pnts n th t s cngrnt t th numbr n thrght ml 2

ow sppoe that s not a regar ale of F eca from that (y' ad g(y' are ocay costat nction o y' as logs stay away from criical aes . Ths there is a eihorhood

V C N of y, cossting o reglar ales of f so tha

fr a y' { V ad there i a aalogos eighorhood V C N so that

for al V Choose a regar vale z of F within V n V The

which competes th proof.W i aso d the folowg:

Hmgeneity Lemma. Le y nd z be bi inei pints test, cnnc nfld N Then ee exis s diepis h N N is ml c idntit nd ies int z


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py andstpy 3

For the special case N sn the proof s easy: sply choose h to

e the rotation whch carries y into and leaves xed all vectors orthogonal to the pane throgh y ad z )The roof in general proceeds as follows: We w rst constrct a

sooth sotopy fro n to tself wch

1) eaves al ponts otsde o the nt l xed and sdes te orign to ay desired point of the open nt all.

/ "c/' .V

Figr 7 Drming th unit ball

et Rn e a smooth fncton whch satses'(x) > 0 I xll


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. odu eefre eac s a dieomorpism from R onto itself etting

ary we se tat eac s smootly sotopic t the dentity nder asotpy wic leaves all points otside of the nit all xed t clearlywt suitale coice of c and t, te dieomorphsm will carry torgi t any desired pont in te ope unit all

consider a connected manifold N. Call two points of Nisotopicf tere exsts a smoot isotopy carrying one to the oter Tis isclearly a eqialece relation f y is a nterior point te t as aeirod dieomorpc to R\ hece the aove argment sowstat ery poit sciently close t y is isotopic to y

.In other words

eac stpy as of ponts n te nterior of N s a open set andte ter N is parttied nto disjoint open sotopy classesBu te terior f N is connected ence there can e only one schsotopy la Ths compltes the proof

W ca ow prove the main resu of this section Assume that Mis mpact ad odarylss that Ns conected and that J N smt

y nd egul v lue of f he#1 (y) #1 ) (modulo 2)

This coon sidu clss which is clld h mod degre o f, ependonly o the sooth hootopy clss of

OOF Gien reglar vales y and z let h e a deomorpismfrom N t N whch s isotopi to te identity and whic carries y to z Ten

s a reglar value f the compositon

h ince

h is omotopi

t te omotopy ema aserts that

B

o ta

Terefre

a eqired

# (h f(Z) #z mod 2)

Call tis commo resde clas deg) ow sppose that is smoothlyomotopic to . By ards teorem there exist a element y N


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tpy nd istpy

which is a reguar vaue for oth f and g. he conrence

deg #y #g-l(y de g

5

md no shows that deg f is a smooth homotopy invariant, and competethe proof

XMLE constant map c M has even mod deree.he identity map I of M has odd degree. ence the identity p f cpct bundy less nild is nt htpic t cnstnt

In the case M = sn this rest impies the assertion that no smothmap Dn+! sn eaves the sphere pointise xed. (I.e. the phereis nt a smooth " retract of the is. Compare , emma 5.) For ucha map f woud give rise to a smooth hmotopy

(x, t = (x ,etween a constant map and the identity.


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ORNT NFOLS

OE to dene the degree as a integer (rather than an integermodo we mst introdce orientations

DEO An orientatio or a nite dimensiona rea vectorspace is an eqivaence cass of ordered ases as foows: the orderedasis b, , b) determines the se ien t n a the asis b, , b)i

b=L

b with det( > 0 . t determnes the

ppsite ient tini det ( < 0 Ths each positive dimesiona vector space has preciseytwo orientations. The vector space Rn has a stndd orentation corresponding to the asis 1 , 0, . , 0 , (0, 1 , 0, , 0 , , (0, , 0, 1 .

In e cas of the ero dimensiona vector spac t i covenint todene an "orientation as the symo + or -

An iented smooth manod consists of a maniod togetherwith a choice o orientation or each tangent space Tx. 1these are required to t together as foows or each point o there

shod exist a neighorhood U C and a dieomorphism h mapping Uonto an ope sset of Rm or Hm whic is ienttin pesevin, in theense that for each x U te isomorpism dhx carries th speciedorietatio or TJ nto the standard orentatio fo Rm

f is connectd and orieta the i has preise twoorieta tions.

f as a undary we an distigish tre kinds of eor inhe tangen space TM" a a oudary poi

there are t veors taet t the odary, rming a 1 dimesiona sspace T(a C Tx

there are he "otward vectors, orming an ope haf spacounded y T(

3 h ar h "iwar vrs frmin a cmpmenar afp


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The Bwe degee

ac orientation for determines an orientation for a as foows

For x a coose a positivy oriened asis (V 2 , vm for Tin sch a way tha V2 m ar tnget to the ondary (assmgtha m and that V is an "otward vector. e (V2 . mdetermines th reqird orientation fo M a

f the dimension of is the eac ondary point x is assignedthe orietation or + 1 according as postivey oriented vectorat x poins iwr o otwrd. ee Figr .

Fgur

H rn bunr

As a exampe the n sphere sm

- C R can e orientd as hondary of te disk Dm

THE BROUWER DEGRE

ow e M and N e orieted dimensioa mnifds wihotoudary ad l

e a smooth map. f is compact and Nis connected, then the degreeof f is deed as foows

et x e a regar poi of , s that d TM TN is ainear isomorpism etwee oriented vector spaces. Dee the sgof f to e + 1 or 1 according as df preserves or reveres orientatio.For ny regar vae Nden

deg( = sign f y)

As in 1, this integer deg(; is oca constan fnctio of y isene dense open sse of N


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8 Oeted mids

Theem The ege deg y) des t deped he hie

egu vue y t wi e caed the degee f f (dnoted de .

Theem B f is smthy hmpi t g, he de f = de g he poof w e essentiay the same as tha in t is ony ncessary

to keep crefu contro of orientationsirst onsider he oowin situatio: ppose that is the ondary

of a compact orieted maifodand tha M is oriented s he ndary

of .Lemma M N exeds sh p N, the

e( y) = ery egu vue y.OO irst sppose that y is a rear vae for , as we as for

= F M he compact -maifod - y) is a nite nion of arcs andcrces, wih ony the odary points of the arcs yi on M = et A C y) e on of hse arcs, wit A = {} {}. We wishow hat

i d + sin df = 0and hence (smmin over a sch ars tha de( ; y 0

0,Fir H rin F(y)he rietations for ad N determine a orietatio for A as

foows: Give x [ A et e a positivey orieted asis

for T it VI taent to A. The VI etemies the equie etatif TA if d y d caes V + it psitively itedbsis j TN

et x) denote the psitivey oriented it vector tane to A at x.Ceary is a smooth fnctio, and x) points outward at e oudarypoint say ad iar at the other odary pit


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Th Bouwe ee

t follow immediatel that

= -,

9

= + 1wth um ero Add up over all suc ar A, we have prove thatde( y = 0

More geerally, sppose tat Yo s a reular vale or , ut ot for The fucto deg( y is constant with ome eghorhood U of Yoece, as i 4 we ca choose a relar vale y for F wthn U andoserve tha

de; o) = de y = 0 Ths proves mma

ow conider a smooth homotopytwo mappigs = FO

F : [0 1] X N = FI, x

etween

Lemma 2. The egee deg( y is equal to deg( y o any omoneua al y

OO he aifold 0 ] X ca e orented as a prodctand ll the have ondary consstn of 1 X wit the correctorentatio ad X (wt the wrog oretation Ts the dereeof a[O ] X at a rela vale y is eqal to the dierenc

eg; y deg ; y

Acord to emma ths dierece must e er.

The remaider of the proof of Teorems A and B is completelyanalogou to te argment i 4. If y and z are ot reglar valueso N, choose a dieomorphism h N Nthat carries y to zad s iotopi to the dentity. The h wll reserve orietato and

dg(; y = de f; hyy isecto Bt s homoto to 0 hece

deg( f z) = de; z mma 2 Terefore e(; y = de( z) whch completes the proof.

MLE The omplex fuctio z Zk, Z 0 maps te it irloto itself wth deree . (ere ma e postve, eatve or zeThe degeerate ma

oa N


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Oeed mioshas dg zo A diomophism N as dg o -1ccodg as psvs o vss oiaio Ths n oenonevesng eoophsm o compc ond ess mnod s nosmooh homoopc o he den

xamp o a oiaio vsig diomophism i poviddb h cio i sn s wh

h aipoda map o sn has dg as w ca s b oigh h ipoda map is h composiio o n cios

Th n s even, he npod mp o sn s no smooh homoop ohe den, a ac o dcd y h dg modo

As a appicaio oowig Bow w show ha sn ms smooh ed o nonzeo ngen vecos nd on n s odd Compa

ige 0 ad

oFigure1 (above). A nonzero vector eld on the I-sphere

Fgure11 below). Attempts for n 2

DEFO A smooh ngent veco ed o M C is a smoohmap M sc ha vx 1 TMx o ach x 1 M. I h case ohe sph sn C n his is cay eiva o he codiio xx = 0 fo x 1


43/80

h Brouwr dgr

ing h idn inn od .

vex) i nonzo o x, hn w my w so 2 X = o x snFo i y e ex) = v(x)/ l v x 1 wod b vo which disy hi condiio. we cn hi o s ooh ioo sn o ie.

o dne smooh ooop

F sn X [0 J sy om F(x 0 = x cos 0 vx) si Compion show h

F(x O F(x 0 = 1nd h

F(x = x F(x = x h h ipod p o sn i s homooi o he idiy B o nv w v n h

mpossib.

n hnd i = 2 , he expii om

dns nozo ngn vo d o . hi comps h poo. oos idny h h ipod mp o n s homopic

o idniy o odd A mos hom o Hinz Hop sssh wo mppigs om od -miod o h sphe smooh homoopic

and ony hy hv h s dg. I 7

wi pov o g es hich mpie ops eom


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6. VCTOR LS N

TH ULR NUBR

a h appcaio o h cocp o dgee dy vc ds hmaifods.

Coside a op C Rm ad a smooh veco d

ih a isoad ze a h poi z t U he cio

x = x x mas a sma sph ced a z o he i sphee* he dgeo his mppig i cad he in L o a h o z

ome examps ih idcs 0 2 ae isad i Fge 2 imay associad ih ae he cve age o hich aeobaid by ovig e dia eaios dx d = V X . . . i he cves hich ae acay sched i Fig 2)

e h abiay de ca be baid s oos I he pao cmp be he poyoia dee a sooh veco edih e ie he igi a he cio e avec e h e id

We m pove ha his ccp i i ivaia de dieomhi exai a hi ea e cie he me

geea siai a map M N h vec ed ech

m

DEFO he vc ds M ad V o N ospon e i he eivav df caie x) o fx)) o each X t M.

* E ee i e iened e und he cending dik


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h n 33

{ = Q

+

+ Figure 12 Examples of plane vetor elds

I s domophim, cay i uy dmd by he oao

i = o l

l b sd

Lemma Sppos that th cto ld on U ospons toi = d v o r l

on U und a domophsm U U. Thn th ndx o v at an soatdzo z s qual to th nx o Vi at z


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3 6.

Assmig Lmm w a d h op o idx o co

d w a abiay maiod M as oows I g U M paamzaio o a ighbohood o z i M, h h in L ow a z is de o be a o he dx o h coespodig eo d g W 0 go U a h o g Z I ay wi oow om Lmm ha L isw dd.

h poo Lmma 1 wi b basd h p o a i dis

Lemma

Any ointation psvng oophis o Rm is

smoothly isotopi to th intity(I oas ma vas o m h xiss a oiao p

ig diomphism o he sph sm whi is smohy isooco h idiy S [20 p 404

OO W ma assm ha () 0 Si h divaiv a a b dd by

= i ttO

i is aa o d isoopy

b he oa

t = tt o < t , 0) =

To pove ha is mooh v a t 0 w wi i e om*

wh g g ae siab smooh cios ad oe hat

vas o hs / i isoopic o ia mapig ih ca isooo e idiy i povs Lmm 2.

OO O LE W may assm ha z ez 0 ad ha Uis cov I psvs iaio h podig xay a abov

* f xaml 22, ]


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h n s

w cosc a o-paam amiy mbddig

U Rm

wih o = idiy ad () = 0 o a t do hvco d v f 1 o ( U which spods o v o U hsvco ds a a dd ad ozo a suciy sa sphd a Hc h id o v Vo a 0 ms b a hdx o v ' = VI a hs pvs mma oiaio psvigdiomophisms.

o cosid diomophism which vs oiaio i s i

o cosid h spia ase o a cio hV ' = p O p ,

so h associad cio v'x) = v ' x/ v ' x 1 o h -sphe sav' p v P

vidly h dg v' a h dg v whch cmps hpoo o mma 1.

e i sdy h oowig cassica s L M be compacmaod ad w a smooh co d o M ih isoad zos. Mhas a bonday then w s eed o point otwad at a bonday pon

Pinca-f Teem h sm o the indces at he zeos sch a veco ed s ea o he e nmbe

m

(M = ( _ I i a Hi (M i = Q

n patca ths inde m s a topoogca nvaiant M it dos notdepend on he patca choce o veco d

dimsioal vio o hs m was povd b Poia 188. h hom ws povd by Hop 14 i 192 a alipaia ss Bow ad Hadamad

W wi pov pa o his hom ad sch a poo o h stFis cosid h spcia cas o a ompa domai i Rm

X C R' b a compac -maiod wih bday. h Gamapping

g ax 8m-

assigs ac x 1 aX h owad i oma veco a x

* Hr H ns h hmy rup f M s wll ur r as fnc lgy ry


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3 6. Vo lds

Lemma (p) I X m is a s ec eld wi isaed

zers and if poins out of X along bounday en t index su L is equal te degree of the Gauss apping fro ax t m. In paricularL does not depend on te choce of .

Fo examp, i a vc d he dis Dm pois ward aogh bdary, h L = +. Cmpa Figre 3)

Figur 3 n xampl with inx sum 1

POO. emvig a ba ad each zeo we oba ewmaod ih bdary. The cio vex) = (x ( x maps hismaod i sm . ece he he dgees v esricted hvaios boda cmpes is ze. B v ax s homoopic o gad he degee h he bodary compos add p -L he mi sig ccrs sic each ma sph ges he rog oiai.) Theee

dg() - L = 0as eed.

. The degee g s s as he "cva iega ax, ice i ca be eprssed a ca imes he ga verax he Gassa cvae. hi egr is o cse ea o he mbe o X Fom dd i is a ha he mb o ax.

xedig hi e t he maiod m moe primaes a eded.

is aa o y cmp he dx o a ec ed a a ze z


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The u 7

i ems f he divaivs o v a z Cosid s a vo d v pe s U C m ad hik f v as a mappig U m so hav m is dd

DEFO The vor d v is nondgn a z if he arasfomaio dv is osiga

t oows ha z is a isoad zo

Lemma 4. T dx a a nondgna zo z s accodng as t dminant of dv is posiv o ngatv

OOF Thik of v as a dimps fom som ov igbho Uo of z i m. We ay ssme ha z I v psevs oieaio e hae see ha Uo a b dd soy i heidiy iho idg ay s S as , Heh id is iy eua .

f v vses oiai h siiay v a b domd io rcio hee L - l

o graly cosid a

zf a o d

w mail

M e . Thk of w as a map fom M o k so ha divaivew : T R is dd

Lmma T dvatv dw acual cas T no t sbsacT C \ and nc can b considd as a lna tansfomaon fomT o tslf If ts lna nsfomaton as tnan D 0 tnz is an isoatd zo of w wt indx qu o + o accoing as D isposv o ngav

OOF L U M b a paamrizaio of som ghbohooof z L d he h basis o o m ad

s ha h vos o a basis fo h a space T u W m om h ima t t (u) d he ia asfoaio Fis ha

1)

L v I b h vo d o Uwh orspods he e w By diio v dW s ha

w((u dv I hrfo


50/80

8 6 Vco ld

Combg 1 ad 2 , ad h valag a h zro - (z) o v oba h ormul3)

hs dw maps o l ad h dma D o hi arasormao T T s al o h drman o h max(av;/aUi) ' ghr wih Lmma 4 hs oms h roo.

osdr a compac budarys maiod C Rk. L N

oe h clsd -eghborhod o ( e h s o a x k w x y or som y ) Fo scly sma o ca shoha N a mooh manod h bouday 8 Problm 1 1

Therem or any vecor el v on wi only nonegenerae zero indx su L is qual o egre o e au mappng

g aN S In pariular is sum does no depend on e coice o vecor ed

POO. For x N lt rx) o h coss po o (Compa8 Poblm 2 o a h vor x r x) s ppdiar o hag spac o a rx) or ohrws rx) would o b h cosspo o I s suy smal h h uo rx) s smooh adl d.

Figue Th nghbohd f

* A n npan f s g as n gvn y Anrf anFnc: g f g can xpss as h nga v f a suab cuvau scaa, us yng an -mnsna vsn f cassca Gauss-Bnm (Rfncs 1J, 9 S a Cn 6)


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he e sum W wil also coid h sard dic cio

() = r ) A asy compaio shows h h gadi o is giv b

grd ( (

Hc o ach poit o h lv srac aN = h wardi oma vcor s giv by

g) = grad /1 I

grad

= ( (

d v o a vco d w o h ghboho N by sig

w) r + v r h w poit wad alog h boda, sic h i pdtw() g ) is al o O ha w ca vaish o a h zoso v i M his is car sic h wo smmads r ad v (r ar may orhogoal Compig drivaiv o w zo

z t w s haW (h) = d(h) or a h TMWh) = h or h TM

Ths h demia o w is al o h drmia o dv Hch id o w a h zo z is a o h id o v a z

ow accodig o mma 3 h id sm I s a o dgro g Thi provs hom 1

XMLE h sphr thr s a co d v hichpois "oh a vy po* A h soh po h vcos adiaowad hc h id is + 1 A h orh po h vcos covrgwad hc h id is ( 1 . Th h ivaia I is al o 0or accordig as is odd or v. his givs a w proo ha vrvco d o a v sphr as a zro

For a odddimsioa bodarlss mafod e ivaia I iszro Fo h vcor d v is pacd b v h ach id ismlipd by ( _ 1 ) ad ai

I = ( 1 I ,o odd impi hat I = O

* Fr xap, ca dd y frua v(x) ( ' x)x, wr (S Fgu )


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4 6 V ld

EM If L = on a cnnected manifd thn a theoremf opf assrts that thr xits a vcor d n with o zros at al .

I rdr t btn th f strngt f th Pincaropf thoemthe frthr tps ae ndd

E Idntication of te invariant L wi h r nb () It cent t constrct jst oe xam of ndgerate vectod n it L al t ( . Th mot laant way f ngtis is the follwig: Accding to Marston Morse, it i always pssble

t nd a e aued fuctin on hose " gradient is a ndegetvectr ed Furtermore, Mrse swd tht t sm f ndesasscatd ith s a gadent ld s qa to th r nmbe f etals of this agmnt te radr is rfrred to Mino [, ,36

E oving t to o a ecto ld wi dgnerat zConsidr rst a vctor ed an ope set U t a isoatd zeoat z If

A U 0

taks th vae sma ghborhod N f z and te va tsde a ighty argr brood N, and if y is a sitymal glar vale f the te vctr ld

x = vx (Ys ondegnrt* withi N. he of the ndic at th zs withn Nca be evauatd a th dere f the map

N and h dos no ange rng ths atrtio.

Mor gnraly onsd veco lds on a omat maniod yng s agnt localy w se tat any vcto eld wit isoatdzos can b placd by a nondgnrate vcto ld witout ating tnge L

E 3 aniolds wit bounday If C R has a bnary, thn vctor v which onts outward ang an gn b extnddver th nighborhod N s as to poit otwad aong N owvr,thre is some dicty it smoothnss ond t boundry of Thus N is ot a smooth (ie dirntib of ass manifd,

* Cearly v s degeerae wn N Bu f Y s sucey sma e vw ave n zers a al w N N


53/80

in m

l a -ifold he exsi w if ded s bfo b

w( = v (r()) + r ) will l be a oos vc ele M h agm nohelss b rrid o ihr b showingha or srog dieriabliy assmpios ar no rlly eessayo by o hos.


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7 A OBOS

TH OYAG OSTTO

E degre f a mapping M M' is dned ony when th manifoldsM an M' are oriented and have the same dimension. W i studya generaliation du to Pntryagin hich is dened fr a smoth map

M S Pfrm an arbitrary cmpact bundaryless manif t a sphr. Firstme dnitins.

Let N and N' b cmpact dimensional ubmaniflds of M ithaN N' aM . he ierenc f imension m n is caledthe codimension f the ubmaniflds.

DEFO N is cobodant to N' within M if the ubsetN X [0 ) N X (1 1

f M X 0 can be extended t a cmpact manifld M X [0

tat

a = N X N X 1 an tat nt ntrsect M X M X 1 excpt at the pintf a

Clarly cbrim i an equivalnce relation. ee Figur 15.)DEFO A raig f the ubmanifold N M i a smth

untn ich aign t eac x t N a basis

( = (v x) , , v x te p TN T f nrmal ctr t N n M at (e


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The ragin cnstructin( O Figure 1 Pasting together two obordisms within

gur 16) Te pair N, ) is called a ramed submanold f M T

frame submanifd N, ) and N', are ramed cobordant f tereexis a cbrdism C M X [0 1 beteen N ad N' an a fragu f s that

u' x , t (V i 0 fr (x , t N X 0 E

ui (x , t W (x 0) fr (x , t N X ( E

gain tis is an equivaence relatin cider a smh ap M and a reguar aue y P

The map induces a framing f the anifld r l y) as f Cea pitivey riented basis v vP) fr the tanget space T SP) r eac x (y) recal frm page 12 tha

aps te ubpace y) t zer and maps its rtgna cplementTrly) isrphicaly t P Hece tere is a unique vectr

wi (x Tr\y C M

at maps it

under It i be veient t se the ntati * fr te resuting framig wl ) f y)

DO This framed manild (f- l (y) , f*b) wi be caled theonryagn manod assiated ih

f curse f as many Pntryagin maifls crespnding t diferet chices f y and bu they all belg t a sigle framed cbrdismcass

Therem I y is another reglar vale and is a psitivelyoriented basis or TSP) " then he raed maniold - y) *) iramed cobordant to y) *)

Therem B Two appngs rom M to SP are smothly hootopii an onl i he associated onragn maniols are raed cobordan


56/80

44 7 Framed coborsm

N

[O, IJFige 6 Famed sbmaiolds ad a amed cobodism

eem n comac aed smaniod N ) o codimension pin M ocs as onan aniod o soe so mappin SP

Ts t ooo as o as a in onon ospondnit t ad obo: s o aniods

Te poo o Tho b v sa to the agnts in and . t i be bad a

Lemma 0 and ' are to dierent posiivel oriened bases at en he Pontryain mani f*o is aed cobordant to 1 ,t*' .

O Coo a soo a o 0 to 0 in t spac o a posivlyoind bas o TSP y Tis is possbe sin this pace of base


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h Ponryagn onsrcn 45

can b idnid ih h space G p R of maris ih posiivdrinan nd hc is conecd. ch a pah givs rise o hrird framing o h cbordis 1 y X 0 ].

By abs of oain e i fn de referc o * ad spasipy of "he framd manifd 1 y

Lemma 2. If y s a egula alue of and ucely close o y he z s ramed cobordan o y

OO Since he se ) of criica vaes is comac can choose

0 so ha he neigbrh f y conais ny rgar vas.Giv z ih z y < chse a smoh neparamer famiyof roaions i. an isopy) r SD S so ha r y and so ha

r is h ideniy for 0 < ') r as r1 for ' < and3 ach r z lie on he gra circle from y z and hc is

rglar vae of

Dn h homopy

F M X 0 S F t r Fr each oe ha is a rgr vae f hecoposiion

N SP folos a foriori ha z is a rgar vale for h mapping F Hene

z C X 0 ]

is a frad anifod and rovids a framd cobordsm bn heframed mnifods z and r Z r

z y Thispros Lemma

Lemma I f and g are smoohly omotopc and y s a egular aluefOT both ten y s amed cobordant to g- y

OO Chos a homoopy F ih

F t F t g < < .

Chos a regar va for F hich is cose nogh y so ha ' is framed obordan o y and so ha g- Z is framed cobordao g- ' y Th F-z is a framd manifold and provids a framecobordis b z ad g- Z Thi prvs Lmm 3


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7 ad obords

ROOF OF THEOREM ive an t reglar values y an fr ,e a hse rtatin

s that r s the identit and r (y) hus is hmtopic t r hece z is framed cobrdant t

r f Z r ) r1y) .his cpletes the prf f herem

he prf f herem ill be based the folloing Let N C Mbe a framed submaifld f cdimesi p it framing o Assume taN is cmpact ad that aN aM

Prduct Neigbd Therem. Soe neghborhood N n seomorphc to th product N X RP urthermor the deomorphsmcan b chosen so that each x E N corresponds to (x 0) E N X RP andso that ach norma rame o x corresponds to the standar bass or RP

EMR Prduct eighbrhds t exist fr aritrar submanilds mpare igure 7 )

MFigur 7 An unrmabl submaniol

ROO First suppse that M is the euclidean space Rn+ onsiderthe mappng g N X M, eed b

gx i t t = x + tIV X + . . . + tvPx early dg ex ; , " " O is nnsingular; hece g maps some neigborhoodf x 0) E N X RP diemrphicall nt a pen set.We ill pre that g is ee n the etire neighborhod N X Uf N X 0, prviding that 0 is sucietly small where U dtesthe neighbrhood of 0 in RP or therwise there would exist pairsx u) x u) n N X RP with u an u arbitrarily small and ith

x u x u


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Th Ponain contucion

i N is cmact, cud chse a sequece f such airs ith

cergig, say t Xo ith cverging t X6 and with u 0 adu The clearly Xo Xb , ad e have ctradited the statemetthat g is nene in a neighborhood f o, 0)

Thus g mas N X U imrhicaly nt a e set. But U,is dieomorphic to the full eucidea space R undr t e rrespoece

u u/(1 - I lu WN) .ince g (x, 0) , a sinc dgx o es hat is expcted f it, thisrvs the Product eighbrhd Therem fr the scia ase

M n+pFr the gera case it is necessar t relae straight is i n+v

by gesics M re recisely et g t . . tp) be the enpintf th gedsi sgmet f egth ) + + pP i M hichstarts at ith the initial ity vectr

t1 v\x) + . . tpvP(x)/ tlV(X) + + tvP(x) The reader h S familiar ith gedesics ill have dicult i

chcig thatg : N X U , M

is el ened and soth, fr E sucientl small. Th rmainder fthe roof rocds exactl as befor.

POO O TEOE C Let N C be a cpact bundarlessfraed suanifold. Choose a rduct rersentatin

g N X P V C M

for a nighbrhood V of N as abov and de the roection V P

by 7(g (x, y y. e Figure 8 Clarly 0 is a regular value, and7- O) is precisely N ith its give framing.

.

Fiur Construting mp wit gen Pntryagin mnifol


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8 ram cbrism

ow chse a sooth map < RP SP which maps every with

to a bae pit So and map the open unit ball i RP dephically oto SP so Dee

by

) < () for ( V

) So for V

learly is ooth, and the point


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h nya nsuc 49

V o ot contan te antpo y o Ientiyng V it N X Ran entiying ith R, we obtain corrsponng apping

th

an wih

dFe o = dG o = (ro cton to R)for all x N

We wil rst n a consnt o tht

F u) ' u 0 G u ' for x 1 N an 0 < l u l < c hat is te point F u an G ubelong o h sae open hafspace n R So the hooopy

1 t)F u) + tG ubetween F an wil not p any new point nto 0 at east for I l u l < c

By Tayors theorem

ence

an

I F u) - u l : c, 1 1 1 1 2 , for u l :

F u u u : c lu

F(x , u) u 2 I u l "- c, lu l >

for < l u < Min ) th a siilar nequality for GTo avo oving istant ponts e select a ooth ap " R R

wih

ow the hootopy

u = for I lu I I : c/2

"(u) = 0 for I lu l l 2 c ,

F u) = [ ut]F u) + "(utGx ueors F Fo into a appng F that () concie ith G thregon I l u l < c2 2 coice ith F for l u l l c an 3 a nonew zeros Maing a corrsponing eforaton he rgnal appingf, this clearly coplete te rf of Lea .


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5 F cbmPROOF OF THEOEM B. I smt mt t

mm 3 ssts tt t tg mfs (y (y m bt s, g m bsm , bt (y (y gumt cmlt gus t thp om cnstuts mtp

F X [0 1 - sps tgn mf F y F*) s pisl qul t , ttng F) F t tt t mps xtl thsm tg mifl b mm ; similF w mpts t pf f m B

EMARKS. ms , B, sil b glizd s ast p t mnf wt bu sstl i is tos l mpgs whi t bu t bs point mtp sss f suc mppgs

n spondnc wit t cbdism classs of faedsubmfls

N C Int(

f imnsi p If p m + 1, tn ths st of mtop classes b g th stuctu of an bln gup alld t -th co homoopyoup (, composition optn in (, cospondst t ui pti f isjnt fam submifds f Intio (

(mp 8, blm 17

TE OPF TEOREM

s xmpl, lt b cnt intd mnifold of dmn

s m p m submnifl f dmnsi p is just a nit stf pnts t pf bsis t ch t sgn( qul + 1 1g s th pfd bsis dtmis th ight wg intt sg( is ll qu t t dg f th asscatdmp - s But t is t diult t s that th famd cobodsmlss f t mifl s unqul tmi b this itg sgn( us te lg


63/80

Hp 5Therem f Hpf I M i nnee iene, an unae

ten tw ap M sm ae moohl mopi i an nl i aveh am eee

th t hand suppos that M s t tbl gv bass for TM w c sld x roud M ls lp s as t trnsformth gv bass t f ppost rtt sy gumnttn pvs t foowg

Therem I M i onnece u nonoientabe en wo map M sm

a moopi i an nl i te have the ae mo eee.e theory of framed cobordsm was troduced by Potyagn n

orde to stud homotop casss of mapgs

wth m p. F xmpl m p + , t sl twhomtopy casss f mappgs s SP. Potryg pv ts sutby classfyg ramd mfods sm Wt csb m

dult ws bl t shw that th just tw mtp lsssaso th cas p + , usg framd flds wvf p ths pproc to th pbem us t mfl cults

It as sc tur out to b as t umt hmtp clssesof mappgs by qut drt mo algba mthods* Potygscstuctn s howv oubdg to It t ny ws us ttslt fmt bout maflds t motp ty t vrsl bls us t tst fmto but mtp tmf th m f th dpst wk n md tpg scome from the terpay of ths two theores e hos wok ocobordsm s an mortat xamp of ths efereces 3] ]

* f xam S.-T Hmtpy Thry


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8. EXERCISES

IN CNLUSN here re soe rbles or the reader

nBLEM 1 Shw that the egree o a coposition g o is equl tthe ruct egree egree f) .

PRLM2. w tht every cplex olyni egree

n

givsrise t st r te Guss sphere 2 t itsel egree n.

nBLM 3 I tw ps f nd ro X to P satisy I l - g 1 1 < 2r l prve tht f is omtpic to the hootpy being st i fn g re smt

RBLM 4. IX is cpact show tht every continuus ap X- Pcn be unirly pprxite by st ap I tw sth psX -

P ar continuosl hootopic, sho tha he ar smoothlhomotopic

RBLM 5. I r < p, sw tht every p Mf - P is tpi

t nstant

ROBLEM 6 (Bruwer) Sow that an ap sn - sn w degreeieret fr (- 1+ st ve xed pint

ROBL 7 Sw that n sn - sn d egree us rry

se pir ntipdl pints int pir ntipd points

PROBLM 8 Given st nilds M C Rk n N C R , swtt te tngent spe T(M X N)cx , y is equal t TMx X TNy.

PROBLM 9. Te grp r st p f : M- N is ene t bee e f ll (x, y) 1 M X N wit f(x) = y. w h r is mt


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ci

nold nd he ngen space

Tr y C TM X TNys equa o he gaph o he lnea ap

5

POE 10 Gven M C Rk show ha he tangent bundle spae

TM { x v c X Rk I v c T }

s ls soh anold. Show ha any sooh ap M - Ngves se o sooh ap

he

: TM TN

deny deny ( (dg d POE 11 Slaly show ha he nomal bundle spac

{ x v c X Rk v TM}

s anold. I M s copact nd boundayless, show hahe cesndence

( f + R aps e -neghbohood o M X 0 n doophcallyono e -neghbhood N o M n Rk. opae e Podc eghbohod heoe n 7 .

OE 1 Dene N M by x + x. Sho hat x + vs cse + han any he pon o M sng hs eaco pve he naloge o Poble n whc he phee SP s eplaceb anold M

POE 13 Gven dsont anolds M N C Rk+ \ he lnng pA M X N Sk

s ene by A (X y x x I M n N e op,ene n bundayless, h al ensn + henhe egee o A s alled e lining numb l(M N Pve

l N M _ l M N

I M buns n ened nfl don om N pove l(M N O . Dene e lnn nube fo dsnt nlds nhe see +


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8. se

BLEM 4 THE NVARNT. If Y z are reglar vale for am S- 1 - S the the mifods 'y 'z c be orietedas in ; hece the iig umber l 'y z is deed

a Prove that this lig mber is ocaly costat s a fnctioof y

b) If y and z are regar vales of g als hre

fr l x rove tht (x) g(x < Y z

(y z) = l g (y ' (z = l(g- y g z rve tht '(y 'z depeds ly te mtopy clss

of des t depend o the chice f y ad zTis iteger () l ' y ' z) is caled the opt invaiant f

eferece ]

BLEM If the dimesio p is odd prve that () O. Fr comosition

rve tht (g ) s eql t ) mltiplied by the square f theegree f g

Te o bation S - S is ee b

(x X h- 1 (x + iX(x + ixere etes steregraphic rojeti t the cmplex lae Prvett l

POBLEM 6 . Tw submifols N N' f Mre sid to intescttansesally f fr eac N ( N' te subspces TNx d TNtogeter geerte TMx. (If n n < tis meas tat N ( N' If N is frme submifold prove tht it ca be deformed slightys s to iterset give N' trasversly. Prove tht the resultigtersecti s smooth mifol.

PBLEM . et V(M ete the set f frmed brdsm

csses f codimesio p i se the trasverse iterseti peratit ee rrespdee

If p ! + use te isjit uio eratio t me P M ita abelia rp mare 5


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EX

AS S Y - OS

W LL v t fllwing sult, whi s b ssum i ttxt A bif isssi f th clssiatin bm f igh imnsionl mnifls will lso b gvn

Therem. Any cnnece iensina ani is iemophic it to cic 8 som inta a numbs

An ineva is cnnt sbst f R wic s not oint It mb nit innit; losd, n, hlfn )

Sin ny ntvl is dimic t to [0 , (0 , (0 ) ,it fllows that t nly fu istint cnnt manifls

Th oof i mak us of th conct o "angth t b aintva

DFNN A m I M is paamizatin by acnth f ms I mhilly nt n n subst f M n if t " litvct T s unt lngt, f I

An givn lol amtiatin I c b tnsfmd nt amtiatin by lngth b stightfwd ng f ibs

Lemma. e f I M an J b paaetizaion b anThn )

J) has at mos tw cmpnents it a ny n cmpnn

ten can b xen o a paamtization by aceng unin) g J . I i as two components ten M mus be iemophic 81

* For example, use a dieomophism of he fom

J(t) anh (t) b. Thus can have ounday poins only if M has ounday poins


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OOF Clealy g- aps soe elativey open sbset of I deoophically onto a eatively open sbset of J. Ftheoe the deivativeof g is eqa to eveywhee

Cnside the gaph I X J consstng o al ( t wth = g(t .Then r is a cosed sbset o I X J ade p of ne segents of slope l Snce is coed and g ocay a deoophs these nesegents cannot end n he inteio of I X J bt st extend to thebunday Since g is onene and singe vaed thee can be aost ne of these segents ending on each o the fo edges of the

ectagle I X J. enc r has at ost to coponets (Se Fge 9Fteoe f thee e two copoents te t s ave the sop

8 -y _ L-

b c Figr Tr f pibiii fr

f r is connected then g extends to a lnea p R - R d g L piece ogehe t yield te eqied extesion

F I LJ ) gJ as two coponents with slope say + tey must be anged

in te lefthad ectange o Figue 9 Tnstig the intevalJ if necesay we ay ssue that c d = so tat

a < b c < a <

settng = 27t/(a a the equied deophsh S1 is dened by te foul

h cs i ) fo a < t < = g(t) for c < t { .


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asyg manods 57

e ige h(81 , being copact and open in , ust be te entrenfod hs proves te lea.

om' LASSIFICAON HER . ny paraetration by arengt can be extended to one

/

whc is xia n te sense tt cannot be exteded over anyrger interva as a paraetrzation by rc-length it s ony necessaryto extend as far as possible to te left and ten as far as possible t

te rghtf is not deoorpc to 81, we wll prove that is onto and

ence is a deoorps. or f the open set /) were not l of tere woud be a it point x of /) n / araetring aneigborood f x by rc-lengt and appying the lea e udsee tat cn be extended over rger interval. hs contradcts teassuption that is axal and hence copletes te proof.

o anfolds of ger dienson the casscaton

probe becoes quite fordabe. or -densona anfolds atoroug exposton as been gven by err [. he study f3-densiona anods is ver uc topc of current researc.See Papraopouos [] or copact anfolds of diension 4te csscation probe s actuly unsovabe.* But for hig densona spy connected anfods tere hs ben uch progress inrecent years as exepled by te or of Sae [3 and Wal 3.

* ee Markov [1 9J ; and also a forthcoming paper by Bone, Haken and Poarun ndamena atemaiae


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71/80

BBLGHY

T followig is a miscllaous lis cosising of origina sourcs and of rcommdd xbooks For h radr wh wishs o pursu h sudy of dirntialopology, l m rcommnd Milor [22] Munkrs 25] and onryagin 28]Th survy aricls 23 and 32 should also prov usful For backgroundknowldg i closy lad lds, l m rcommnd Hilon and yli 1 1 ] Hu 16] ang 8] d ham 29 nrod 34 ad rbrg 35

1 ] Allndorfr, C B, Th ulr numbr of a iman maifod, Amer.Jour. Math. 62 (190) 243-248.

2] Aposto, T M, Mahematica Anaysis aing, Mass Addiso-sly,1957.

3 Ausladr, L, ad MacKni, Introduction o Diereiabe ManifodsNw York McGraw-Hill, 963.

4] Brouwr, L J "Ubr Abbildung vo Mannigfatigkitn, MahAnnaen 71 ( 1 2) 971 15 .5] Brown, A B, Functional pndnc, Trans. Amer. Math. Soc. 38

(1935) 379394 ( Thorm 3III)6 Chrn, , A simpl inrinsic proof of h Gauss-Bonn formula for

cosd imania maifolds, Annas of Math. 45 (1944) 747-752.] Diudonn, J Foundaions o Modern Anaysis. Nw York Acadmic

rss, 1960.8] Dubovicki, A Ya , O dirntiabl mappings of a dimnsional cub

ino a -dimnsiona cub Ma. Sborni N 32 (74) (1953) 443464.(In ussian)

9] Fnch, , O toal curaturs of Rimannian manifolds, Jour.London Math Soc 15 (1940) 5-22

10 Goman, C, Cacuus of Severa Variabes Nw York Harpr & ow,1965.

1 1 ] Hilo, , ad yli, Hooogy Theory. Cambridg Uiv rss, 196.


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Biliogapy

2] rsc, . , "A roo o te nonretractty o a ce onto ts ounary,

Proc mer Math oc 4 96 64-65] o, . , "Aungsassen mensonaer anngatgketen, latAae 96 (926 209224

4] , "Vetoreer n mnsonaen anngatgeten, atae 96 (926 225-250

] , "ber e Abbungn von Sren au Spren nrgererDmenson, Fudameta Matematicae 25 (95 427440

6] u, S.-T., omotopy Teory New York: Aaemc Press, 9597] Kerjrt, B. v., Voresuge ber Topoogie Bern : Sprnger, 92

8] Lang, ., Itrodcto to Dretiabe laods Xew ork: Interscene9629] arov, A. A., Insoubty o te probem o omeomorphy, Pro

ceedgs Iter Cogress o ath 98, Cambrge n. Press, 960. 00-06 (n Russan.

20] nor, J "On manos homeomorpc to the 7sper, Aas oath 64 (956 99-405 .

2] , "A survey o coorsm theory, L'Eseigemet math 8 (962)6-2.

22] -, Morse Theory (Annas Stues 5 ) Prneton Unv. Press, 62] , "Derenta tooogy, Lectures o loder Matematcs II , e .

T. L. Saaty, Xew ork : Wey, 964 p. 65-24 orse, A. P., "Te beavor o a uncton on ts crtca set, Aas of

Mat 40 (99) 6270.25] unres, J R., Eemetar Dretia Topoogy (Annas Stues 54)

Prnceton Un. Press, 9626] Papakyraopouos, C. , "The theory o treemensona manos

snce 950" Proceedigs Iter Cogress o ath 98, CambrgeUnv. Press,

960,pp. 4-44027] Pontryagn, L. S., "A casscaton o contnuous transormatons o a

compex nto a spere, Doady Aad Nau S SSR (Comptes Redues)9 (98) 47-49.

28] "Smooth manos an ther appcatons n homotoy theory,Amer Mat Soc Trasatios Ser. 2 II (959) 4 (Transate romTrudy Ist Steo 45 (955) )

29] Rham, G. , aris diretiabes Pars : ermann, 955.0] Sar, A., "The measure o the crtca ponts o erentabe maps,

Bu Amer Math Soc 48 942) 88890] Smae, S., "Generae Poncar's conjecture n mensons greater thanour, Aas o Mat 74 (96 9406.

2] , "A survey o some recent eveoments n erenta topoogy,Bu Amer Math Soc 69 ( 963) , 131-145.

] Sva, ., Cacuus o Maiods New ork Benjamn, 965

4] Steenro, X., The Topoogy o Fibre Budes Prnceton Un. P


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boaph

[35] Sternberg S Lectures on Derentia Geometry. New ork Prente

a 196436] Tho R "Queques ropts goaes es varts rentabes

ommenarii Mat eet 2/ 1954, 17863] Wa C T C " Casscaton o (n connecte 2anos

nnas o at. 75 (962) 163189[3/] Whtney A unton no t constant on a connecte se t o rtc

onts Due ath. Jr. 1935 514517


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PAGE-BARBOR LECRE SERIES

Th ag-Barbor Lctr Fondation was fondd in 1907 by a gift fromMrs Toma lon ag n arbor) and h Honorabl Thoma Nsonag for h rpos of brnging o th nivsty of irgnia ach ssson asris of cr by an mnn prson n om ld of scholarly ndavor Thmatal o wr prsntd by rofssor John W Milnor in Dcmb a forn r ponord b h Fondaton


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DE

antipodal map 30 52bondary 12Brower J 35 52

degree 20 28ed point thorem 1

Brown A B 10 1chan r or derivatves 3 7cobordism 42 43 51codimension 42cohomotopy 50coordinae system 1corrsponding vecor elds 32criica point 8 1 1critical vale 8 1

degree of a map 28mod two 2

derivative of a map 27dieomorphism 1dierenia topoogy 59dimension of a manifold 1 5 7disk 3 1ler nmbr 35 3 6 4 framd cobordism 43

framed sbmaniold 42 44 46Fbini horem 17fndamental theorem o algebra 8Gass-Bonnt thorem 38Gass mapping 35 38hafspace Hirsch M 4

homotopy 20 21 5Hopf , 31 35 36 40 51 54inde (of a zero of a vector ed) 32

34inde sm 354

inverse fnction theorem 4 8inward vector 2isotopy 1 22 34linking 53Morse A 1Mors M 40nondegenerat zero of a vctor eld)

37 40normal bnde 53

norma ecors 42orientaion 26

of a bondary 27of () 8

oward ector 26paramrizaion 1

by arc-lengh 55oincar H 35ontryagin 16 42

ontryagin manfold 4prodct neghborhood heorem 46regar point 7reglar vale 8 1 1 13 14 2 7 40 43ard A 1 16smooh anifods

with bondar


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mooh manfods cotth cassication problem, 57o imension ro 2o imension on 14 55orinted 26

smooth maps smooth mappings 1sphr 2

stereograpic proection 9 4tangent bunde 53tangent sace 25

at a boundar point 12 26tangent vector vector eds 30 3241

Ine

eerstrass approximation teorem 4

IDEX OF SYBOLS

deg / y 27 1dx 27 Hm 1Dm 14 Rk 1 1 2 Snl t*\, 43 Tx 25

#y 8 ! 14


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TPLGY FRM TE DIERENTIABLE VIEWPOIT

was compose an print by Wiliam Byr Prss Inc., Ricmn, Virginiaand boun b Rusl Rtr Co, Inc, N York, N Y

Th ypes ar Morn Number 8 and Cntury chobookand t paer is Oxord's Bookbuiders Patsign s by Edwar Foss


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milnor - topology from the differ en ti able viewpoint

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