Stephen Smale- The Classification of Immersions of Spheres in Euclidean Spaces

8/3/2019 Stephen Smale- The Classification of Immersions of Spheres in Euclidean Spaces

1/19

The Classification of Immersions of Spheres in Euclidean Spaces

Stephen Smale

The Annals of Mathematics, 2nd Ser., Vol. 69, No. 2. (Mar., 1959), pp. 327-344.

Stable URL:

http://links.jstor.org/sici?sici=0003-486X%28195903%292%3A69%3A2%3C327%3ATCOIOS%3E2.0.CO%3B2-N

The Annals of Mathematics is currently published by Annals of Mathematics.

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtainedprior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content inthe JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/journals/annals.html.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.

The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academicjournals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers,and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community takeadvantage of advances in technology. For more information regarding JSTOR, please contact [email protected].

http://www.jstor.orgFri Feb 15 05:09:54 2008
http://links.jstor.org/sici?sici=0003-486X%28195903%292%3A69%3A2%3C327%3ATCOIOS%3E2.0.CO%3B2-Nhttp://www.jstor.org/about/terms.htmlhttp://www.jstor.org/journals/annals.htmlhttp://www.jstor.org/journals/annals.htmlhttp://www.jstor.org/about/terms.htmlhttp://links.jstor.org/sici?sici=0003-486X%28195903%292%3A69%3A2%3C327%3ATCOIOS%3E2.0.CO%3B2-N


2/19

, \vv.%T.s


3/19

328 STEPHEN S M A L Ea,E i-rk(Vn) an d let a based C" i m m e r s i o n f :Sk Enbe given. T h e n thereter is ts a basad imm ers io n g :Sk-+ En such tha t Q(f , g ) = a,. Thzss therei s a 1-1 correspondence betzceen elements of n k (Vn , , )and based regularhomotopy classes of imm ers io ns of Skin En.If n > c + 1 ,a(f , 8 ) can be defined for non-based immersions andTheorem A i s t ru e om i t t ing th e word based wherev er i t occurs . TheoremA is a d irec t ge nera lization of Theore m A of [7] w h e r e k = 2. The caseof Theorem A for k = 1 is included in my thesis [6]. See these papersfo r implications of The orem A w he n Ic = 1, o r 2. Many of the groupsi-rk(V7,) have been computed. See Paec hter [5]. Sin ce rk(l/, ,


4/19

- - ---

329N I M M ER S IO N S O F S P H E R E SThe ques tion of regular ex tens ions is closely re la ted t o th a t of reg ula r

homotopy. In par t icu lar , w e ar e in teres ted in th e fo llowing problem :Suppose f :Sk-I-+ En i s an immersion where S k - I i s th e boundary of th ek-disk D k. W hen can f be exten ded to a n immers ion of D k ? Fro m H.W hitney 's w ork one obtains a n aff irmat ive answ er whene ver n 2 2k.The fo llowing theorems g ive an answ er to th is ques tion unde r th e res tr ic-tion n > k.

THEOREME. If n > k a n immersion f :Sbl -+ Enca n be extended to a nim m ersion of D h if an d only if a(f , e) = 0 where e : Sbl-+EhcEn i s thesta nd ar d zsnit sphere in a k-plane of EnIn a c er ta in sense th e resul ts of th is paper a re local in nat ure . M.

Hirsch us ing these resul ts tog ether wi th obs t ruct ion theo ry has provedtheo rem s on th e re gu lar homotopy classification of ma nifolds inste ad ofspheres . He a lso obtains some sufficient conditions for manifolds to beimm ersible in euclidean space. Fo r example, he proves ev ery closed 3-manifold can be immersed in E' [ Z ] .The above resul ts sugg es t t he fo l lowing ques tions :( 1 ) One problem is to replace En in Theorem A by a n a rb i t ra ry n -manifold M n . I believe on e would g e t a classification of imm ersions of

Skin M n in t e rm s of x h (Fk (Mn ) ) h e re Fk (M n ) s th e bundle of k-fram esover Mn . I don't thin k this will be ver y difficult to prove, following th eproofs in th is paper.

( 2 ) Find explici t rep res ent at iv es of regula rhom otopy classes. W hitn eyhas essent ia l ly done th is for th e case n = 2k. W ha t regu lar homotopyclasses have an imbedding for a repr esen tat ive ?( 3 ) Develop an analogous theor y fo r imbeddings . Presuma bly th iswill be quite hard . Ho we ver, even part ial results in this direction wouldbe interest ing.

1. The cover ing homotopy theoremA t r ip le ( E l p , B) con sists of topological sp ace s E a n d B with a mapp from E in to B. A t r ip Ie has th e CHP i f i t has the covering homotopyproperty in the sense of Serr e .Le t D h be t he un i t k -d isk in E k (k >= 1)with generalized polar coordi-na tes. T ha t is to say, points of D k will be pairs (t , x) whe re t i s the d is -

t ance f rom t he o r ig in 0 of Ekand x is a, point of t h e boundary D'"of Dk .L e t E,,, = E be th e se t of al l C" imm ersions of D h in E n , n > k. Thes e t E is given t h e C 1 topology, ' i .e . , is metrized by

1 Added in proof: Assume all function spaces have the CZ topology instead of the C1topology.


5/19

330 S T E P H E N S M A L E~ ( f l ) = m a x (P (f(y) , s(?-J))? ~ ( d f , ( Y ) , d g v ( Y ) ) I ~ c ~ ~ , i l \ Y \ I!?C =wh ere f,g e E , 5 i s the met r ic on E",E" being considered as i ts owntan ge nt vec tor space, and Dk is th e ta ng en t space of D La t y e D L .L e t B, = B be t h e s et of al l pairs (g, g ') w her e g is a Cmimm ersion ofD~ in Emand gf is a Cmcross-section in the bundle of transversal vectors

of g(Ljh). Th us gf is a C - ma p of ghin to En- 0 such t h a t g l (x ) does no tlie in @(,, the tan ge n t p lane o f .q(hh) a t g (x ). The se t B i s g iven thefollowing me tric. Fo r (g, gf ), h, h ') e B le twh ere p i s a s above and p, is defined a s p above excep t tha t y is onlyallowed to r an ge o ver 5,.A map n E -+ B is defined a s follows. Fo r h e E le t n(h) = (g, g')w he re g is th e restrict ion of h to 5h:nd g l (x ) = h , ( l , x ) ( the subscr ip t tme ans differentiation w ith re spect to t) . Th e goal of this section is toprove tha t (Elx, B ) is a fiber space in th e sense of Se rre .

THEOREM1.1. The t r ip le (Elr,B ) hccs the CHP.PROO F. Some of th e constructions in th is proof ar e s t ra ightfor wa rdgenera lizations of thos e of [7]. This proof is ess enti ally ind epe nd ent,however, a nd so me wh at more detai l i s g iven he re th an in [7].In the art icle La classijkation des irmmersio?zs, Seminaire Bourbaki,December 1957, R . Thom ha s a n inte res ting exposit ion of th e proof of1.1.

A ro ug h acco unt of ou r proof is a s follows.We a r e giv en a h om otopy h,O : P -+ B , hence h:(p) fo r each p an d v i s animmersion of a sp he re h,(p) w ith a tra ns ve rsa l ve ctor field h:(p). F ur -th er m or e, h:(p) fo r each p is covere d, i.e., h,(p) is th e boun dary of a nimmersed disk h(p) e E an d h:(p) is a tra ns ve rsa l field induced b y th eimmersion of th e disk. Th e problem is to follow th e homotopy h:(p) bya n imm ersion of a disk Z,(p).

I n ou r constru ction of h,(p), Equ ation (17), th e facto rs a (t ) , jS(t), a ndM ( v ) ar e in t roduced mainly so th at va rious boundary condit ions a re m etand th e regu lar i ty of is preserved.The f i r s t and las t t e rm s of (17) roughly speaking a re used t o take careof th e t ra nsve rsal fie ld pa rt of the homotopy. In part icular , th e t ra ns-form ation Q,(p) is th e principle elem ent he re. This pa rt of th e coveringhomotopy could be tak en ca re of directly by a n isotopy of A". T h e l a t t e r ,in fact , i s w ha t Thom does .Th e second t e rm of (17), a(t)[h,(p)(x) - h,(p)(x)], is just what makes


6/19

331N IMMERSIONS OF SPHERESthe immersion of the disk h,(p) project onto the immersion of the sphereh,(p). However, in general the introduction of this term will cause t hemap of Dkto have critical points or points where the Jacobian has rank< Ir. To counteract this, the term ,i(t)M(v)z~(v,, x) is added. The effectof this term is, roughly, to blow up the immersion whenever i t mighthave become critical.The above construction is used in [6], but in simpler form. The readermight see the idea of this proof by looking, therefore, a t that paper also.Let h\ : P -+B be a given homotopy where P is a cube (recall that i t issufficient to prove the CHP for cubes) and let h : P -t E cover h!. ' Wewill construct a covering homotopy z,: P -t E.

We write h!(p) = (h,(p), h:(p)) (recall t he definition of B).Let ~ , ( v ,p, x) be the distance from h:(p)(x) to the tangent plane ofhu(p)(Dh)a t x and let E, = min (E,(v,p, X)[ v , p, x). Let

E~ = min :[yvh,(p)(x)[[ 1 V I = 11, v, 11, V e DE,and take E = (1110) min {E,,E,,1:. The symbol y,h,(p)(x) means thederivative of h,(p) with respect to V at x.

We define a linear transformation of E n ,Q,(p)(x) for p e P , x e D'"ndsufficiently small v (we clarify this later) as follows. Le t V,(p)(x) be the2-plane of Enspanned by the vectors h:(p)(x) and h:(p)(x), if i t exists, andlet a,(p)(x) be the angle from the first to the second of these vectors. LetQ:(p) :Dh+Rn (the rotation group) be the rotation of En which takesV,(p)(x) through the angle a,(p)(x) and leaves the orthogonal complementfixed (if V,(p)(x) does not exist then Q,*(p)(x) s to be the ident ity rotatione). Finally, define Q,(p) :hh-t GL(n,R) to be the rotation Q,*(p) multi-plied by the scalar 1 h:(p)(x) [ / [h:(p)(x) 1. We will consider Q,(p)(x) as act-ing on Emon the right. I t is immediate that Q,(p) is C mwith respect tox and that( 1 ) hXp)(x)Q,(p)(x)= h:(p)(x) .

See [7] for the following.LEMMA.2. Let n > Ir 1,G,,, t h ~ rassman manifold of oriented k-planes in En and Sn- 'the unit vectors of En.Let a map w : Q -+ G,,, bagiven which is homotopic to a constant zchere Q is some polyhedron. Then

there is a map ZL : Q -t Sn-Isuch that for all q e Q, z~(q)s ~zormal o theplane zc(q),Note t hat if u! is C - we may assume that ZL is also.Now apply 1.2 taking for Q, I x P x hhand for w(v, p, x) the plane

spanned by h:(p)(x) and the tangent plane of h,(p)(x). Because (h,(p),hi(p))


7/19

332 STEPHEN SMALE is in the image of 7r :E -,B, one can show w is homotopic to a constant.Thus one obtains a map u :I x P x D *+SSn.Choose6>0so that for 1 v - v ~ ( ~ 6 a n d a l l p ~ ~ , x e ~ a n de a , l v l = l ,the following conditions a re satisfied.( 2 ) The angle between h:(p)(x) and h:,(p) is less than 180" (this insuresthat Q,(p)(x) is well-defined).

(If quantity on right of inequality of (5) is undefined, omit (5)).I t is clear that such a choice for 6 may be made and t hat our choice of6 depends only on h,(p) and not z(p). I t is easy to check th at (3) implies

We choose now to, 112 < to< 1, so that for all t e [to,11, v 5 6, p e P,V e D:and I V I = 1, the following conditions hold.

(See Lemma 5.1 of [6]),

(If quantity on right of inequality of (9) is undefined, omit (9)). It isclear such a choice for tomay be made. Set t, = to+ (1/3)(1- to).Real Cmfunctions on I, a( t) and P(t) ar e defined satisfying t he follow-ing conditions :

(10) a(t) = 0 ost s t , .(11) a(1) = 1 al( l) = 0 .(12) I a(t) 1 5 1 I aJ(t)I < 2/(1- to) .(13) P(t>= 0 o s t 5 6 , .(14) P(1) = p'(1) = 0(15) I P'(t) I > 10 l a'(t) l t , $ t S l .(16) 1 P(t) 1 s 20 .


8/19

33 3N IMMERSIONS OF SPHERESPrimes in this case denote th e derivatives. As in [qwe leave to thereader the task of constructing such functions.LetThen for v (= 6 th e desired covering homotopy G ( p ) is defined as fol-lows.

z v ( ~ )( t , = P(P)t J - % ( P I ('9 Ce + a(t)& U ( P ) (x )-(17) + a ( t )ChU(p)(x) h,(p)(x) l+ P ( t ) M ( v ) u ( v ,P , x ) + % ( P I ( 1 , s ) .We write down the following derivatives for reference.

h,,(p)( t ,X I = Z , (P )( t ,4 Ce + a ( t ) Q ,(P )(4- 41(18) + [@P) ( t ,2 )- Go ( 1 , x ) l a l ( t )QU(P) 4- e )+ a ' ( t ) [ h : h , ( ~ ) ( ~ )o ( ~ ) ( ~ ) lP1( t ) M( v ) -u (v,P , X )

For V e DE~ v h , ( ~ ) ( t ,= X ) v & ( P ) ( ~ ,e + a ( t ) ( Q v ( ~ ) ( ~ )11) [ ~ v L ( t , - ~ 1 1 -+ G(P)t ,2 ) - ( X I( P ) ( ~ ,X ) I ~ ( ~ ) V ~ Q , ( P )(19) + a( t )Cvvh, (p) (x )- vvh, (P) (x ) l+ P( t ) M( v ) v ,u ( v , P , x )

+ v v @ p ) ( l ,$1 . We will prove that & ( p ) has the following properties.

-h,'(P) (1 ,4 = h: (P)(4 . & ( p ) is regular .

First we show how 20-24 imply Theorem 1.1. Properties ( 20 )and ( 2 4 )yield that the homotopy zg:P -+ E is well-defined, ( 21 ) says that h, is ahomotopy of % and (22), ( 2 3 ) imply that E,, covers h". Thus it only re-mains to prove (20-24).Property ( 20 ) follows from the fact that all the functions used to de-fine & ( p ) are C" in x (17) .One can check ( 2 1 ) immediately frdm (17) .Onecan obtain ( 22 ) from ( 17 ) noting ( l l ) ,( 14 ) and that h o ( p ) ( x )=7 6 i ~ ) ( l J $1.Property (23) follows from ( 1 8 ) using ( 1 1 ) ) ( I ) , ( 14 ) and the facth t ( ~ ) ( l J) = h:(P)(4.


9/19

334 STEPHEN SMALETo prove th e reg ular i ty of & ( p ) i t is sufficient to s how y , h , ( g ) ( t , x ) f 0where V E Dh,,,. The n V can be wr i t te n V = V ,+ V , where I., i s t he

projection of V in to ~ j :and V , is th e projection of V in to the vector spaceorthogonal to D! in D:,,,,. The n( 2 5 ) t v h , ( p ) ( t ,x)= p th , t (p ) ( t , x ) + p , ~ w & ( ~ )t ,2 )where W is V , normalized and p , , p , a re appropr ia te sca la r s . E i the rp , f 0 or p, f 0 .

L E M M A1.3. T he re i s a vector b' of En , I b' I < E (see begin ning of proofof 1.1 for defin i t ion of E ) such tha t

PROOF. From ( 1 2 ) , ( 3 ' ) and ( 8 ) i t fo llows th a t

By ( 1 2 )and ( 9 )I [%P) ( t 1 x )- h ( p ) 1 ,x ) l a ( t ) v , , (P I ( x ) I < &I10.

By ( 1 2 )and ( 4 )I 4 6 ) v w h , ( p ) x )- vwho(p) x ) l < El10

an d finally by ( 1 6 ) and (5)

Then by ( 1 9 ) and th ese fo ur inequal it ies we ha ve 1.3.L E M M A.4. Th ere exist vectors in E m ,, Z, scalars A , A' w her e I b I < E ,

A > 10 a' a nd I ii I = 1 , a n d u = u (v,p , x ) such tha t

PROOF. W e can easily o btain from ( 1 8 ) ,

where ZL is a un it vector .Then f rom ( 1 2 ) , ( 3 ' ) and (7)i t fo llows th a tI [ h t ( p ) ( l 1 h , ( ~ ) ( t , - I) - x)I [e + a ( t ) ( Q , ( ~ ) ( x ) e)I < 2 ~ 1 1 0.By (1 2 ) and ( 3 ' )


10/19

ON IMMERSIONS O F SPHERESI h t ( p ) ( l , ) a ( t ) [ Q v ( p ) ( x ) el I < 4 0

and by ( 1 2 ) , ( 6 )and (3') we haveI a ' ( t )c ~ ( P ) h ( p ) 1 ,s>ICQ,(p) x )- el I < 4 ~ 1 1 0.t , x ) -By ( 1 5 ) and no t ing t h a t h : ( p ) ( x )= h , ( p ) ( l , ) and Q , ( p ) ( x )= e t he aboveinequalities yield 1.4.

L E M M A1.5. Let a , b, u, C, a' and b' be vectors i n E n , A , A' scalars w it hthe following properties

I b I , I b' I < ( l l lO)P (a , a ') all real sI b' I < (1110) 1 a' I , I ZL 1 = 1 , I u I = I, A > l 0 A r

and u norma l to both a and a ' . T h en a + b + AZL+ A'G and a' + b' arel inearly independent .PROOF. If th e lemma is fa lse then th ere i s a scalar v s uc h t h a t v ( a + b + AZL+ 1 ' E ) = a' + b'

or( 2 6 ) A v ( u + ( l r / A ) E )= a' + br + v ( a + b) .

Since I ( A 1 / l ) G < (1110) 1 u 1 , Z L + ( A r / A ) Uhas ang le less than 25" f romZ L and t hu s s ince u is normal t o a an d a' th e t e rm on th e l e f t of Equa t ion26 i s a t an angle of gre a te r tha n 65" f rom t he a - a' plane ( th e case1 = 0 offers no difficulty). On the other hand a + b has a n ang l e le sst han 25" f rom a , and a' + b' has an ang l e l e s s t han 25" f rom a'. Thisimpl ies th a t th e term on th e r ight of Equat ion 26 h a s a n a n g le fr o m t h ea - a' plane less than 50" . Thus Equat ion 26 is false, and hence 1.5 isproved.From the l as t thre e l emmas we a re now able to prove the regula r i ty ofh , (p ) . By ( 2 5 ) i t is sufficient to show t h a t h , ,( p ) ( t , x ) and y , h , ( p ) ( t , x ) a r el inearly indepen dent . This fa ct follows fro m 1.5 making th e subs t i tu tionsf rom 1.3 and 1.4, a = h , ( p ) 1 ,x ) , and a' = y,, h , ( p ) 1 ,x). One also uses t h edefinitions of e and u to check the hypotheses of 1.5. Thus we haveproved ( 2 4 ) .

The above construct ion may be repeated if 6 < 1 using h 8 ( p )in place ofh. This yields a covering homotopy for v 2 26. I terat io n yields a cover-ing homotopy h, for a l l v I. This proves 1.1.

2. The weak homotopy equivalence theoremL et f , : D L -+E n be a Cmimmersion, n > k . Denote by I'=I',,,(f,) th espace wi th th e C' topology of all C - immersions f of D k in E n such t ha tf agrees wi th f , on bkand d f with dS , on th e restr ic tion of th e t an ge nt


11/19

336 STEPHEN SMALEbundle of D k to D Let x = (x , , ..., x,) be rectangular coordinates onE k 2 D k and if J : D k -+ E n s an immersion denote by f , & ( x ) the deriva-tive of f ( x ) along the curve x,. Let f, : Dk -+ V. , (the Stiefel manifold)be the mapf,(x) = (f z , ( x ) , ...,f , , ( z) ) and r' = r : , , , ( x ) the space with thecompact open topology of all maps of Dk into V.,, which agree withf, on@. Let be the map Q ( p ) ( x )= (f x J x ) , . ,f z , ( x ) ) . The proof of thefollowing theorem is the goal of this section.

THEOREM.1. I f f , : Dk-+En i s the standard imm ersion of Skin a k + 1plane of E n then : l',,,( f ,) -+ l",,,(f,)i s a weak hom otopy equivalence.4 map f : X -+ Y is a weak homotopy equivalence if its restriction to

each arcwise connected component of X induces an isomorphism of homo-topy groups and it induces a 1-1 correspondence between arcwise con-nected components of X and Y.One could probably prove 2.1 without much trouble even i f f , : Dk-+Enis an arb itrary immersion.Let x , be the South pole of b+'.hen impose a coordinate system

X = ( X I ,..., X , ) , X E Dk+l- x - , on h k + l - x , such that r ( X )= (zf~:)1is the equator Q of bk+" .rom now until the end of this section weidentify x and 2, xi and X i ; identify Dk with the upper hemisphere ofof 6 '' and bkwith Q. Let

A = {( t ,x ) e Dk+' t = 112, r ( x ) >= 1 or x = z-)and let g, : Dk++'+ E n be the standard inclusion into a k + 1 plane of E n .

A subspace B = B,+,,. y , ) of B,,,, .(g , ) is defined as follows. An ele-ment ( y ,g ') of B belongs to B if g restricted to L j k+ ' l n A is y,, and g'restricted to bk+lA is g,, (the derivative of g , with respect to t ) .

Let A' = ( ( t , z ) e Dk+"x f x,) and define 5, :A' --+ V,,,+lby G(Y) =(gozl(y),...,g,x,(y>,g,+(y)), e A'.Let B r = B:++',,(g,)e the space with the compact open topology of allmaps of D, fl A' into V,,,,, which agree with g, on D"" n A' n A.

A map U' :B -+ B' is defined by. u ( g , g r ) ( x )= ( gx , ( x ) ,.. ,gz,(x), gr (x) ) , g , E B, x E A' n @+I.Let f , : Dk -+E n be the restriction of g,.

THEOREM.2. I f @ : I f , ) -+ r:,,(f,) is a z~teakhomotopy equivalence,then U' : B,+l,n(g,)-+ BL+,,(go) i s also a z~teakhomotopy equivalence ( n> k ) .PROOFOF 2.2. Let r ' q e the subspace of l7 = r,,,(f ,) of those immer-sions which have all derivatives (of all orders) agreeing with the


12/19

O N I M M E R S I O N S O F S P H E R E S 337derivatives of f o on D k . The restriction of @ to 1'" will still be denotedby @.

L E M M A2.3. T h e i n cl u si o n i : I -+ I' i s a we ak h omotopy eqzi ivalence.The proof of th is lemm a is not difficult an d will be le ft to th e re ad er .The idea of t he proof is th a t any com pact subset S of I' can be deformedso tha t e lements of S agree wi th go on a neighborhood of th e b oun dary ofD k .

Define maps p :B + I?" and p' :B' + I" a s follows. Fo r ( g , g ') e Ble t p ( g , g ') be g res tr ic ted to D k = upper hemisphere of @ + I . F o rf B ' , f : @" n A ' + V n , , + l , f ( y ) = ( f l ( y ) , ... , f , ( y ) , f k + l ( ~ ) )e t p ' ( . f ) ( x ) =(f l ( x ) , . .,f , ( x ) ) fo r x E D k . I t i s eas ily checked th a t the p is well-defined, continuous and th a t th e following diagram comm utes .

We c la im t h a t( 1 ) (2,, I?" ) has the C H P ,( 2 ) (B',p', I? ' ) has the C H P , and( 3 ) TPrestr icte d to a fiber is a homeomorphism betw een correspondingfibers.

PROOFO F (1). L e t h, :P+ I'" be a given homotopy a nd :P-+BcoverhUwhere P is a polyhedron. We will co nst ruc t a cove ring homotopyQ:P-+B.

L e t q , : V,,,,, + V,,, be defined by dropping th e las t vector of a fr am eand q , : V,,, -+ G,,, be the ma p which sends a k-f ram e into th e or ientedk-plane i t spans. Then q = q,q, ha s the C H P .Define h f : P x D k + G,,, s o t h a t h:(b, x ) i s the tan gen t p lane of h,(b)a t x t ransla ted to the or igin. Le t h" : P x Dk-+ V,,,,, be the m a p

h * (b , x ) = (h,,(b)(x), ..., h, ,(b)(x>, h ' ( b) (x) ) , x c D kwhere %(b) ( x )= (h (b )( x ) ,h '(b)(x) ) . Then s ince hO covers h, i t fo llows th a tq?i* h f . By the C H P of (V,,,,,, q , G,, ,) we obtain a covering homotopy% P x D k -+ V,,,,, f rom h* a nd hz.

Define :P -,B a s follows. L e tR ( b ) ( z )= ( h , (b ) ( x) , Z t b ) ( x ) )where


13/19

STEPHEN SMALE

Now it can be checked that hi(b)(x) is well-defined and is the desiredcovering homotopy. This proves (1). One proves (2) the same way.Lastly, (3) can be seen as follows. If g e l'*,g :D k-,En is regular,

and then p-](g) is the space of all g' :D Ensuch tha t for each x e Dk,g1(x) is transversal to the tangent plane of g at x and gr obeys a boundarycondition. The fiber over @(g) e I" is the same while the restriction ofTV is a homeomorphism between these fibers. To finish the proof of 2.2we note th at '-V maps the exact homotopy sequence of (B, p, r")into theexact homotopy sequence of (B', p', I"). By the five lemma the theoremfollows using (3) above, the given condition on (D and 2.3. This proves2.2.Let g, :D k-,En be the standard inclusion of D q n t o a k-plane of En ,and f , be as in 2.2.

LEMMA.4. If @ : l ,(g,)-,r:,,(~,) is a z~teak omotopy equivalence thenso is @ : r, n(f")+ l n(x) .The proof of this lemma offers no trouble and we leave the proof of i tto the reader. One can use for example a diffeomorphism of En.

THEOREM.5. If T : -,B;,,(g,) is a weak homotopy equivalencethen so is @ : l',,,(g,) --+ rL,,(y,).Before we prove 2.5 we note that 2.1 follows from 2.2, 2.4 and 2.5 byinduction on k keeping n fixed. The first step, that T : B1,n(g,)+~:,n(g,)is a weak homotopy equivalence, is trivially checked. In fact , roughlyspeaking, my thesis [6] contains the second step and [7] is the third stepin this induction.

PROOFOF 2.5. An outline of the proof is contained in the followingdiagram. The spaces and maps will be defined as the proof proceeds.

I x r # A 1.8 R ]e#Z '$ A!zt(B')+- nt (E', 8'')* T ~ - ~ ( F ' )

Let 2, be a (k-1)-frame of SL-'=D+hose base point is a x,, the southpole of Sk-'. The g,,(x,) is a (k - 1)-frame say y, of En (with base point)and g,,(x,) is a vector say jj, transversal to the plane of yo. Let B, be the


14/19

339N I M M E RS IO N S O F S P H E R E Ssubspace of B of ele m en ts (f , f ) wh ere f ,(xo) = yo and f '(x,) = Yo. L e tE, = T;-'(B,) c E. Then by 1.1( 4 ) (E,, n,B,) has th e CHP. (We sometimes denote th e restr ic t ion of a map by th e same symbol as the original map ).

We let E be th e sub space of E,,of immersions f : D k -+ Emwhich agreewith g, on A . Note tha t 2 c B,,and l e t 5 : E -t be the restr iction ofT;. L e t F = F-'E(g,). Th en w e will pr ov e( 5 ) For all i, E, : E,(E , F ) -+ ;r,(B,E(g,)) is a n isomorph ism onto.

For the proof of (5) consider - .iE- E,,where j and j' a re inclusions. Th en it is sufficient fo r (5) to show fo r a l li 2 0( 6 ) j, : ;r,(E, F ) -+ ;.;,(E,,F) is 1-1 onto,( 7 ) T;$ : z i (EU, ) -+ ni(Bu) s 1-1 onto, and( 8 ) j: :T;,(B)+ r I (BO)s 1-1 onto.The t ru th of (7) follows fro m (4).

By the exac t homotopy sequence of th e pa i rs ( E , F ) a n d ( E,, F ) f o r (6 )it is sufficient to sho w T;,(E,)= T;,(E)= 0 f o r a11 i 2 0 .

L e t g :Sj --+ E be given. I t is sufficient to show th a t g is homotopic toa point . Fo r eve ry e > 0 th ere is a dif ferent iable s trong d eformation re-trac t ion H, of D k into N, where N, is diffeomorphic to D k ,N , 3 A a nd fo reve ry y e N,, d(y, A) < e. Then fo r each such E, th ere is a homotopy g,of g defined by g,(p )(y )= g(p)(H,(y)) ,p e S and y e D k . On th e o therhand, for each E > 0 we hav e th e homotopy h , : SJ-+E between g, andthe f ixed map f,,defined by h,(p)(y ) = ( 1 - t)g,(p)(y) + tf,(y). W e leav ei t to the reader to show tha t h ,(p) will be regular (hence h, will be well-defined) if i has been chosen small enough. Th us g is homotopic to apoint . This proves z j ( E ) = 0 .In a similar fashion one proves ;r,(E,) = 0.PROOFF (8). We wish to show i : -+ B, is a weak homotopyequivalence. L e t f :P -+ B, where P is a polyhedron. I t is sufficient fo r

th e proof to show th ere is a homotopy FAP --+ B , such tha t ( a ) F, = f ,(b) if f(p ) e B hen FA(p) = f (p) , and (c) F3(p ) e B.


15/19

340 STEPHEN S M A L EThe homotopy FAs defined in st ag es with some of th e de tails omitted.Firs t , i t can be shown th a t the re is a neighborhood N of x, an d a hom o-

topy FAP -,B,, 0 1A 11 sa t i s fy ing (a ) and (b ) and such tha t F , (p )ag ree s with g, on AT.There i s a number e > 0 such th a t if g c B, and satisfies p,(g, g,) < iwh ere

p*(g , gU)=max{P(g(y) , u (~ )) , ( v , ~ ( Y ) ,rgo(1 / )) Y c ikA , vc i : I V I = l )th e n G,(y) =Ag(y)+(l - 2)g,(y) is re gu lar w ith 0I,5 1 , an d y e D,n A.Fur thermore , i f E is small enough one can use th e ma p G, to define ahomotopy H, e B, w h ere 0 5 A 5 1 such th a t H ,=g , H,=g , and if g= g , ,H,=H,. To define H, from G, one uses a ribbon around t h e equa tor of D'".Ev ery thi ng in thi s paragra ph is valid on a compact s et of such g allsa tis fy in g p,(g, go)< E .

Taking E a s in th e las t pa rag rap h one can define F, : P -.B, 15 i,12such t h a t F l(p ) is th e previously defined Fl (p) , ,o,(F,(p), go)< e, and ifF l (p ) E B,F,(p) = Fl(p ) . Here F i s take n as a s t re tchin g of EnmovingF l( p ) ex ce pt in a neighborhood of x,. Also F A , 15A 5 2 involves a simplere-param eterization of Dl.Final ly , FA o r 2 1 5 3 is defined d irectly byth e H, of the las t para grap h. This proves (8).Le t A and A' be as before and us ing the f ixed ma p go, define a noth erfixed map go A' -, V,,, by

?u(Y)= (goZl(Y),. . ~ O ~ , - , ( Y > , y E A'.t(Y)) ,Let E:,,(ij,) =E' be th e space w ith th e compact open topology of allmaps of A' into V,,, which agree wi th gUon A n A'. L e t BL,,(g,) = B' be

th e space w ith th e compact open topology of all m aps of Dk n A' intoV,,, which ag re e w ith g,on Lj" n A' nA. Define ;r' :E'+ B' by restrict-ing a map to Dkn A'. L e t Fk,,(g,) = F' = ;.;'-'(rt(?,)) .( 9 ) The tr ip le (E ' , r ' , B ' ) has the CHP.To prove (9), let h, :P -+B' be a g iven homotopy w her e P is a polyhed-ron and le t x:P -t E' cover h,. A covering homotopy h, :P -.E' isdefined by

and if r + 1


16/19

O N I MM E RS IO N S O F S P H E R E S

whe r e

and

I t can be checked witho ut much troub le t h a t t his is a good covering homo-topy. We show now,(10) 7T$(E1)= 0 , i 2 0PROOF.L e t H, be a s t ro ng deformat ion re t rac t ion of Dk in to A , i.e., a

homotopy H, : Dk+ Dk suc h t ha t H , is t h e iden t i ty , Hl(x) c A fo r x E Dkand if x E A, H ( x ) = x f o r a l l t . Then define a homotopy H, : E' -2 E'by H,( f )(x)= f (H,(x)). I t i s easily checked th a t Zci s a s t rong de forma-t ion retract ion of E' i n to th e po in t X H ~of E'.A m a p cp :E + E' is defined a s follows.

~ ( g ) g , ( y ) ) , g c E, y e A'.y)= (gzl(y),..., gZk-,(y),Then t h e following diagram com mutes .

L e t cp :F + F' be th e restr ict ion of cp .I t is e a sy t o c hec k t h a t cp i s con t inuous and th a t d i agrams A and B a tthe beginning of the proof of 2.5 commute . Then we have(11) F , :ni- l ( F )-t T ; ~ - ~ ( F ' )s 1- 1 onto.

For (11) f i rs t no te th a t cp, : zi ( E ,F ) -t zd(EP, ' ) is an isomorphismonto since F.;, Then A a nd A' a r e i somorphismss, by (5), and ;r' i s by (9).onto because ;r,(E)= 0 (proof of (6)) and n,(Er)=O by (10). This proves(11).(12) There a re maps a, :F -2 I? and a, :F'+ 1'' which a r e weak homo-topy equiva lences an d such th a t th e following diagram comm utes.


17/19


18/19

O N I M M E R SI O NS O F S P H E R E S 343onto the plane i t spans. Then we have the fol lowing commutat ive dia-g r a m .

Here th e ver tical maps a re Hurewicz maps ;f and g a re imm ersions of Skin E i k ;S,g :Sk+ G-, ,ar e induced by f , g by t rans la t ing ta ng ent p lanesto th e origin of EZk.Then-6 n d df , , a re induced by f,

Suppose now I, = Ig. We wish to prove th a t f and g a re regula r lyhomotopic . For this we use th e resul t f rom [3] w h i c h s a y s t h a t f o r a nimmersion f : M k + E L k w i th M k compact oriented an d I, defined) m,(f )=21f where w,(f ) i s t he k th Stiefel-Whitney class w ith int eg er coefficientsof th e normal sphere-bundle over M k . Thu s in our case W,( f ) = ~ , ( g ) .But by [3] t h i s im plies t h a t f , = 2,.

I t is easily checked th a t p&( f , g) = f,(S,) -g,(S,) w her e S, is a gen er-at or of x,(Sk). Then by th e previous diag ram h,p,R( f , g) =Z(h,(S,)) --g,(h,(S,))=O since& =?,. Th en by th e dia gra m p,h,Cl( f , g)=O. By [3] p,is 1-1 and since h, is 1-1 t h i s i m plies a ( , g) = 0. Thus f and g a r eregu larly homotopic by The orem A. This proves Theorem C.Theorem D is proved as follows. L et p : V,,,,, -+Gk+ , , ,= Sksend a k-f r am e into the k-plane i t span s. Since p is a f iber map, we h ave th efollowing exact sequence.

P# AX , ( V , + ~ , ~ )4 xk(Sb) n, - , (Fiber = R,) .3The map h s zero since Skis parallelizable. Hence p, is onto. Th ere for eby Theorem A, th ere exis t immersions f , g Sk-,Ek+Iu ch t h a t p , a ( f , g )is a gene rat or of x ,(Sk). Since p,Cl(f, g) = f,(S,) - &(AS,) e i th e r f or g haseven normal d egree , and the re exis ts a n immersion of Sk in Ek+lwi thnormal degree zero [4]. This proves T heorem D.Lastly we prove Theorem E. If Cl( f , e) = 0 then f is regu larly homo-topic to e . Fu r ther m ore , this regular homotopy can be covered by atransv ersal vector f ield f ' by th e arg um en t of Theorem 2.2 (1). By 1.1,( f , f r ) i s in the image of x ; hence th e desired extension exis ts .Conversely, in order th a t f have an ex tens ion i t mus t l ie in th e imageof x (w ith some f r ) . Bu t i t follows fr om th e proof of 2.5, Sta tem en t 6t h a t E is arcwise co nne cted. This implies Cl( f , e) = 0.


19/19

344 STEPHEN SMALEAddenda

H ere w e note t h a t th e solution of an oth er problem posed by Milnorfollows from our work. On page 284 of [4] he asks :Let n be a dimension for which S" is not parallelizable. Can someparallelizable n-manifold be imm ersed in E " + l w ith odd d egre e ? Cansome (necessarily parallelizable) n-manifold be immersed in En+ 'bothwith odd and with ev en degree ?The answ er to the f i rs t and hence to the second question is seen to beno, a s follows.As in t h e proof of The orem D, Section 3, consider the homomorphismp# : x n ( V m + , , J-,xn(S w). I t is suff icient to consider the case of n odd w ithS" not parallelizable. By Theorem A, th e image of p, consists of evenele m en ts of x,(Sm) since on ly odd de gre es of imm ersions of S" in E m + 'a re possible [4]. Now if M " is parallelizable and f:M " -,En+ 's a n im-mersion, then f induces a map F : M"+V,+, , , wi th pF = f. Then fm u s thave e ven normal deg ree , proving our asser tion.

1. S. S. CHERN,L a ge'ome 'tric des sous -varie' te 's d 'u n espace e uc l ide an a p lu sk r s d im en -sions, L'Enseignement Mathematique, 40 (1955), 26-46.2. M. W. HIRSCH,O n i m m e r s i o n s a n d r e g u l a r h om o to pi es o f d i f e r e n t i a b l e m a n i f o l d s ,

Abstract, Amer. Math. Soc., 539-19, Notices Amer. Math. Soc., 5, No. 1 (1958),62.

3. R . LASHOFA N D S. SMALE,O n t h e i m m e r s i o n o f m a n i f o l d s in e u c l i d e a n s p a c e , Ann. ofMath., 68 (1958), 562-583.

4. J. MILNOR,O TL h e i m m e r s i o n of n - m a n i f o l d s i n ( n + 1) - space , Comment. Math. Helv.,30 (1956), 275-284.

5. G. F. PAECHTER,he groups K ~ ( V ~ , ~ ) ( I ) ,uart. J . Math., 7 (1956), 249-268.6. S. SMALE, e g ul ar c u rv es on R k m a n n i a n m a n i f ol d s , Trans. Amer. Math. Soc., 81

(1958), 492-512.7. A c las s i f ica t ion o f imm ers ion s o f the two- sphere , Trans. Amer. Math. Soc., to

appear.8. --, On classifying immersions of Sf, in euclidean space, Summer Conference on

Algebraic Topology, Chicago, 1957, mimeographed notes.9. H. WHITNEY, The se l f - in ter sec t ions of a smooth n-m ani fo ld i n 2n- space , Ann. of

Math., 45 (1944), 220-246.

Date post:	06-Apr-2018
Category:	Documents
Upload:	gremndl
View:	222 times
Download:	0 times

Stephen Smale- The Classification of Immersions of Spheres in Euclidean Spaces

Documents