Lectures on the H-Cobordism Theorem

LECTLTHES ON TEE h-COBORDISM THEOREM

JOHN MILNOR

L. SIEBEMvIANN AND J. somw

PRINCE'KIN, NEGJ JERSEY PRINCETON UIVrVERSITY PRESS 1965 Copyright 0x965, by Princeton University Press

Printed in the United States of America Zhese are notes for lectures of John Milnor that were given as a seminar on differential topology in October and November,

1963 at Princeton University.

Let W be a compact smooth manifold having two boundary components V and V1 such that V and V' are both deformation retracts of W. Ben W is said to be a h-cobordism between V and Vt . The h-cobordism theorem atates that if in addition V and (hence) V1 are simply connected and of dimension greater than 4 , then W is diffeomorphic to V X [O, 11 and (consequently) V is diffeomorphic to V' . Ihe proof is due to Stephen Smale [6]. 'Ihis theorem has numerous important applications - including the proof of the generalized ~oincare conjecture in dimensions > 4 - and eeveraJ. of these appear in $9. Our ma,in task, however, is to describe in some detail a proof of the theorem.

Here is a very rough outline of the proof. We begin by

constructing a Morse function for W (02.1), i.e. a smooth function f : w -> [o, 11 with v = C1(0) , v1 = f1(1)

such that f has finitely many critical points, all nondegen-

erate and in the interior of W. 'ihe proof is inspired by the

observation (63.4) that W is diffeomorphic to V X [o, 11 if

(and only if) W admits a brae f'unction as above with no crit-

ical points. %us in 554-8 we show that under the hypothesis

of the theorem it is possible to simplify a given Morse function f until finally all critical points are eliminated. In $4, f is ad jueted so that the level f (p) of a critical point p is an increasing f'unction of its index. In $5, geometrical conditions are given under which a pair of critical points p, q of index h and h + 1 can be eliminated or *cancelleds. In $6, the geometrical conditions of $5 are replaced by more algebraic conditions - given a hypothesis of simple connectivity. In $8, the result of 95 aows us to eliminate all critical points of index 0 or n , and then to replace the critica3.points of index 1 and n - 1 by equal numbers of critical points of index 3 and n - 3 , respectively. In $7 it ie shown that the critical point6 of the same index h can be rearranged among themselves for 2 <- A <- n - 2 (97.6) in such a way that all critical points can then be cancelled in pairs by repeated application of the result of 16. !his completes the proof.

Iko acknowledgements are in order. In $5 our argument is inspired by recent ideas of.Me Morse Ell] [32] which involve alteration of a gradient-like vector field for f , rather than by the original proof of Smale which involves his 'handlebodie8'.

We in fact never explicitly mention handles or handlebodies in these notes. In 66 we have incorporated an Improvement appearing in the thesis of Dennis Barden [33], namely the argument on our pages 72-73 for 'Iheorem 6.4 in the case h = 2 , and the statement of Theorem 6.6 in the case r = 2. 17he h-cobordism theorem can be generalized in sever& direc-

tions. No one has succeeded in removing the restriction that V and V' have dimension > 4. (see page l3.3. ) If we omit the

restriction that V and (hence) V' be simply connected, the

theorem becomes false. (see Milnor [34]. ) But it will remain

true if we at the same time assume that the inclusion of V

(or V' ) into W is a simple homotopy equivalence in the senee

of J. H. C. Whitehead. This generalization, called the s-cobor-

dism theorem, is due to Mazur [35], Barden (331 and Stallings.

For this and further generalizations see especially Wall 1361.

Lastly, we remark that analogous h- and s-cobordism theorems 3 hold for piecewise linear manifolds.

Contents

Introduction P.W The Cobordism Category PJ Morse Fhctions p.7 Elementary Cobordisms p.20

Rearrangement of Cobordiema p.37 A Cancellation meorem p.45 A Stronger Cancellation Theorem p.67 Cancellation of Critical Points in the Middle Dimensions p.83 Elimination of Critical Points of Index 0 and 1 p .100

mp h ,f?nhnwd i am Tken-aw --A @-- A-3 --La --- - - A- - Section 1. The Cobordiem Category I m

First some familiar definitions. Euclidean apace will be

denotedby R~ = x E R, ((5,..., n )Ixi i = l,.., where

R P the real numbers, and Euclidean half-space by

Definition 1.1. If V is eny subeet of R~,a map

f: V R~ is smooth or dif'ferentieble oi claaa C? if f can

be extended to a map g: U Rm, where U 3 V is open in Rn,

euch that the partial derivatives of g of all orders exist end are continuc~~~.

Definition 1.2. A smooth n-manifold is a topological manifold W with a countable basis together with a smoothnese structure 4

- on Me is a collection of' pairs (~,h) satisfying four conditions:

(1) Each (~,h)E cmsiste of? an open set U C W

(called a coordinate neighborhood) together with a homeomorphism h n which maps U onto an open subset of either R~ or R+ ,

(2) The coordinate neighborhoods in 08 caver W. (3) If (ul,hl) and (u2,h2) belong to 4, then

is smooth. (4) The collection r8 ia maximal with respect to property (3) i.e. if any pair (u, h) not In $ is adjoined to The boundary of W, denoted Bd W, is the set of all points in W which do not have neighborhoods haneanorphic to FIn (see

Definition 1.3. (wj Vo, v1) ie a smooth manifold triad if

W is a compact smooth a-manifold and Bd W is the disjoin* union of two open and closed submanifolds Vo and V1 .

If (w; Vo, V1), (w' j Vi, Vi) are two smooth manifold triads and h: V1 ----+Vi 18 a diffeomorphism (i.e. i homeomorphism such that h and h-' are smooth), then we can form a third triad

(W % W l j Vo, V; ) where W % W is the space formed from W and Wt by identifying points of V1 and Vi under h, according to the following theorem.

Theorem 1.4. There- exists -a smoothness structure -for W % W1 compatible with--- the given structures (i.e. ---so that each inclusion map- W 4 W U, Wf, Wl -4W % W' -is -a diffeomorphism --~ntoits image.) 8 is unique up to a differnorphiam leaving Vo, h(vl) = Vi , and V; fixed. -.-- -

The proof will be given in $ 3 .

Definition 1.5. Given two closed smooth n-manifolds M0 knd % (i.e. Mg , 5 compact, Bd Mg - Bd M.,. = I),a cobordiam from Mo to ie a 5-tuple, (wi Vg, V1j ho, hl), where (wj Vo, V1) is a

l am& menifold triad and hi: Vi 4 M is a diffemorphism, 1 = 0, 1.

TWOcobordims (wj Vo, Vlj ho, hl) and (w' VA, Vij h& hi) from

to M1 are equivalent if there exist1 a diffeomorphism g: W -4W1 *o '3 carrying Vo to Vt, and V1 to I 'i such that for i = 0,l the following triangle ccamrmtes:

Then we have a category (see Eilenberg and Steenrod,

[2,p.1081) whose obdects are closed manifolds and whose morphism

ere equivalence classes c of cobordisms. This means that cobordiems

satisfy the following two conditions. They follow easily *can 1.4 and 3.5, respectively.

(1) Given c obordism equivalence classes c from M to 0 5 and ct from y to , there is a well-defined claas ccg from M to %. This composition operation is associative. 0 (2) For every closed manifold M there is the identity

cobordism class P the equivalence class of L~

That is, if c is a cobordism class frm to l$, then

Notice that it ie possible that cct = rM , but c is not bM . For example c is shaded. cc is unshaded.

Here c has a right inverse c', but no lert inverse. Note that the manifolds in a cobordism are not assumed connected.

Consider cobordism classes from M to itself, M fixed.

These form a monoid HM , 1.e. a set with an aesociative cclnposition

with an identity. The invertible cobordisms in HM f om a group

We can construct soneelements of GM by taking M = M1 G~ . below.

Given a diffemorphism h: M d M1, define ch as the

class of (M x Ij M x 0, M x ly j, hl) where j(x,0) 3 x and

hl(x,l) = h(x), x e M .

Theorem 1.6. chch, = c hth -for -any two- diffeomorphisme h: M &MI -and h': M' & M" .

Proof:- Let %I-. M x I % Mt x I and let Jh: M % I d W, Mt jh, : xI+W be the inclueian mapa in the definition of

Chcht Define g: M x I \ W as follme:

Then g is well-defined and ie the required equivalence. Def initim 1.7. Two differnorphiems h hl: M ---+MI are (smoothly) isotopic if there exists a map f: M x I d Mt such that

(1) f is smooth,

(2) each ft, defined by ft(x) 2 f(x,t) , is a diffeomorphism,

Two dlffeomorphisms %, hl: M -+ MI are pseudo-isotopic' if there is a diffecanorphism g: M x I *-a Mt x I such that

Lemma 1.8. Isotopy -and pseudo-isot opy -are equivalence

Proof:- Symmetry and reflexivity are clear. To ahow kransi- tivity, let ho, hl, h2: M --+ M' be differnorphiems and assume we are given isotopies f, g: M x I MI between h and hl 0 and between hl and h2 respectively. Let m: I -3 I be a smooth monotonic function such that m(t) = 0 for o <- t <- 113, and m(t) = 1 for 213 <- t <- 1. The required isotopy k: M x I 4 MI between h and hl is now defined by 0 for 112 <- t <- 1. The proof of transitivity for pseudo-isotopies is more difficult and follars from Lermna 6.1 of Munkres [5,p.593. * In Munkrest terminology h is "I-cobordantn to hl 0 . (see [5,p.62]. ) In Eirsch~sterminology h is "concordantn to hl. D It is clear that if ho and hl are isotopic then they are

diffemorphism, as follows frm the inverse f'unctian theorem, and is hence a pseudo-isotopy between h0 and hl . h he converse 11 for M = S , n >- 8 is proved by J. ~erf[39J, ) 1t follars fran this remark and frm 1.9 below that if ho and 4 are isotopic, then c = c . ho hl

Theorem l.9, c = c ho -is pseudo-isotopic to- hl . ho hl - -Proof: Let g: M x I ---+Mt x I be a pseudo-ieotopy and hl. Define between h0 hi1 x I: Mt x I -M x I by

(hi1 X l)(x,t) = (h;l(x),f) . Then (hi1 x 1) 0 g is an equivalence between c and c . hl ho The converse is similar. Section 2. Morse Functions

We would like to be able to factor a given cobordism into a canposition of simpler cobordiema, c or example the triad in Mere 2 can be factored a~ in Mgure 3,) We make this notion precise in what follows.

FIGURE 3 Definition 2.1, Let W be a smooth manifold and f: w 4 R a smooth function, A point p t: W 18 a critical point.- of f if, in some coordinate system, = 0 . Such a point Is a non-degenerate ceitica1 point if det ( a2f 1 ) 0 or example, if in Figure 2 -13P f is the height function (projection into the z-axis), then f has

four critical points pl, p2, p3, pq, all non-degenerate,

Lemma 2.2 (~orse). -If p --is a non-degenerate critical point of f, then in same coordlnate system about p, 2 2 f(xl, ..,,x n ) = constant " X1 - 0.. - x + <+I + ... + x2n for

some A between 0 and n ,

h is defined to be the -index of the critical point p,

Definition 2.3, A Morse function on a smooth manifold triad

(wj Vo, V1) is a smooth function f: W ----5 [a,b] ouch that

(2) All the critical points of f are interior (lie in W - Bd W) and are nm-degenerate,

As a coneequence of the Morse Lemma, the critical points of

a Morse function are isolated, Since W is compact, there are only finitely many of them. Definition 2.4. The Morse nuniber p of (wi Yo, v1) i8 Cc the minimum wer sll Morse function8 f of the number of critical points of

'Phis definition is meaningf'ul in view of the following existence theorem.

Theorem 2.5. hrery- smooth manifold triad- (wi Vg , V1) possesses --a Morse function,

me proof will occupy the next 8 ~ages.

Lemma 2.6. mere exists a smooth function f: W --+ [0,1] with f91(0) = V fW1(l) = V1 BUC~that f has no critical - 0' , -- -- point in a neighborhood of the boundam of W.

Proof: Let U1, ..., % be a cwer of W by coordinate neighborhoods. We may assume that no Ui meets both V and 0 V1, and that if meets Bd W the coordinate map Ui n hi' Ui d R+ carries Ui onto the intersection of the open unit ball with

On each set Ui define a map

meeta V [respectively vl] let fi = Ihi Ui d where L is the map

xn] x n (respectively 1 - .

If Ui does not meet Bd W , put fi a 112 identically . Choose a partition of unity (qi) subordinate to the cwer { Ui)

(see Munkres [5,p.181) and define a map f: W d [0,1] by

where fi(p) is understood to have the value 0 outside U men i f is clearly a well defined smooth map to [0,1] with fn1(0) = V - 0' (1)= V . Flnally we verify that df' 0 on Bd W. Suppose q E V [respectively q E V 1. Then, for saue i, cpi(q) 0, 0 1 1 n s € Ui. let hi(p) = (X (P).x (p)) Then

Now f (z) has the 0~mevalue, 0, [respectively 11 for all j J k acpj a k and C - = 7 ( Z '91) = 0. So, at q, the first sumand jal axn ax j=1 ie zero. The derivative af i (q) equals 1 [respectively -11 ax and it is easily seen that the derivatives 3 (q) all have the ax afi same sign es 7 (q) , j = 1, , mu8 af (q) + 0. 13 ax axn follows that df' f. 0 on Bd W, and hence df 0 in a neighborhood of Bd W.

The remainder of the proof is more difficult. We will alter f by stages in the interior of W eliminating the degenerate critical points. To do this we need three lemmas which apply to Euclidean space, Lennna A (M. Morse). -If f --is a c2 mapping ---of an open I I. subset U c Rn to the real line then for almost all linear c. 7- - -- mappings L: R~ & R, the- mction f + L has-- only nondegenerate critica1 points.

By "almost all" we mean except for a set which has measure zero in H-(R~,R) 2 Rn .

-Proof: Consider the manifold U x H~(R~,R). It has d(f(x) + ~(x))= 0 means that L = -df(x) it is clear that the correspondence x 3 (x, -df(x)) is a diffeomorphism of U onto M.

Each (x,~)e M corresponds to a critical point of f + L, and a2f this critical point is degenerate precisely when the matrix ( -1 is singular. Now we have a projection a: M & H~(R~,B) sending (x,~)to L. Since L = -df(x) , the projection is nothing but x & -df(x) . Thus T is critical at (x,~)e M 2 precisely when the matrix ~IJ= f/ax ax ) is singular. It -(a i3 follows that f + L has a degenerate critical point (for same x) if and only if L is the image of a critical point of

1 ~f- t: Ee R~ --is any c msp,----the image of the --set of critical points -of r has- measure zero in Rn. This gives the desired conclueicm. Lemma B. -Let K be-- a cornpact eubeet ----of an open set U in R*. If f: U d R is c2 and has only nondegenerate - .111 .L --- critical points -in K, then---- there is a nmiber 8 > 0 --such that -if g: U 'R -is c2 ---and at all point~of K aatiefiee

1, 3 = 1,. .. ,n , -then g likewise has-- only nondegenerate critical

6 -Proof: Let + ...

a2f Then Idf( + I det(?, ,_)I is strictly positive on K. Let p > 0

be its minimum on K. Choose 8 > 0 so small that (1) implies that

and (2) -lies that

at all points In K. The result followe.

Lema C. Suppose h: U d Uc 18 a differnorphiem of --one open subset -of -onto another -and carries the- ccanpsce -set K C U -anto Kt C U*. Given-- a number 6 > 0, there--- is a number 6 > 0 ----such that if a smooth map- f : U' d R satisfies -at all points -of Kt C U' , then- f 0 h satisfies

--at all points

b2foh Proof: Each of f 0 h, afoh ax,' q8q ie a polynomial

f'unction of the partial derivatives of f and of h from order

0 to order 2; and this polynomial vanishes when the derivatives

of f vanish. But the derivatives of h are bounded on the compact set K. The result follars.

2 The C topology on the set F(M,R) of smooth real-valued

functions on a compact manifold, M, (with boundary) msy be defined

as follows. Let (U'] be a finite coordinate covering with n coordinate maps ha: U' ---d R and let (C ) be a ccmpact , Q refinement of (u,) (cf. Munkres [5, p. 71). For every positive

constant 8 > 0, define a subset ~(6),of F(M,R) consisting of

all maps g : M ---+R such that, for all 9

-1 at all points in h (C ), where arid i, j = 1, n 0 a gO: = ghG .*., If we take the eets ~(6)as a base of neighborhoode of the zero function in the additive group F(M,R), the resulting topology is called the c2 topology. The sets of the form f + ~(6)= ~(f,6) means that, for all a ,

at all points of ha(c,) . It shuuld be verified that the topology T we have constructed does not depend on the particular choice of coordinate covering and cmpact refinement. Let TI be another topology defined by the above procedure, and let primes denote things associated with this topology, It is sufficient to show that, given any set ~(6)In T, we can find a set B' (8') in Tt contained in ~(6). But this is an easy consequence of Lemma C.

We first caneider a closed manifold M, i.e. a triad (M , ) , since this case in somewhat easier.

Theorem 2.7. -If M Ais I-a canpact manif old without boundary, the-- Morse fUnCt%ma----- form an open dense subeet -of F(M,R) --in the c2 topology.

Roof: Let (lTl,hl), .. . , (uk,hk) be a finite covering of M by coordinate neighborhoods, We can easily find compact sets Ci C Ui such that C1, Cp .. . , Ck cwer M. We will say that f is "good" on a set 8 C M if f hae no degenerate critical point8 on S. 15.

I) The set of Morse flulctions is open. For if

f: M R is a Morse function, Lemma B says that, in a

neighborhood Ni of f in F(M, R), every function will be good in Cia Thus, in the neighborhood N = Nl fl ... n l& of f, every function will be good in C1 U .. . U Ck = M.

11) The set of Morse functions is dense. Let N be a

given neighborhood f E F(M, R). We improve f by stages. Let

h be a smooth function M 4 0,such that h = 1 in a neighborhood of C1 and A = 0 in a neighborhood of M - U1 For elmost all choicee of linear map L: # -R the function - - fl(p) = f (P) + A(P) ~(h&p)) will be good on C1 C U1 (Lemma A). We assert that if the coefficients of the linear map L are

sufficiently small, then fi will lie in the given neighborhood

Firat note that fl differs f'rm f only on a compact set

K = Suppod h C U1. Setting ~(x)= L(X~,.. . ,x n ) = z lixi , note

for all x E hl(~). By chooaing the li sufficiently small we

can clearly guarantee that this difference, together with its first

and second derivatives, is lees than any preassigned e throughout

the set hl(IC). low if e is suificiently small, then it follows

from Lemma C that fl belongs to the neighborhood N.

We have obtained a function fl in N which is good on of C1' Amlylng Lennns B again, we can choose a neighborhood N1 f,, N, C N . ao that any Function in N- ie. still aa& rm I! - - - -- At the next stage, we simply repeat the whole procees with

and N1, to obtain a function f2 in Nl good in C2, and a fl neighborhood, N2 of f2, N2 C N1, such that any function in N2 is still good on C2. The function f2 is automatically good on

since it lies in N1. Finally we obtain a function C1 fl( E Nk C Nkml C ... C N1 C N which is g00d On C1 U .. . U Ck = M . We are nuw in a position to prove

Theorem 2.5. On--- any triad (w, Vo, V1), there- exists -a Morse Function.

-Proof: Lema 2.6 prwides a function f: W & [0,1] such that (i) fol(0) = Vo , fwl(l) = V1

(ii) f has no critical points in a neighborhood of

We want to eliminate the degenerate critical points in W - Bd W, always preserving the properties (i) and (ii) of f.

Let U be an open neighborhood of Bd W on which f has no critical points. Because W is normal we can find an open neighborhood V of Bd W ' euch that C U . Let [Ui) be a finite cover of W by coordinate neighborhoods such that each

.I set Ui Lies in U or in W - V . Take a cmpact refinement (Ci) of (Ui] and let Co be the union of all those Ci that lie in

U. Just as for the closed manifold of the last theorem we can use

Lemma B to show that in a Oufficiently small neighborhood N of Also f, no Amction can have a degenerate critical point in Co. f is bounded away from 0 and 1 on the compact set W - V. Hence, in a neighborhood NI of f every function, g, satisfies the condition 0 < g < 1 on W - V. Let No = N n N' , We may suppose that the coordinate neighborhoods in W - V are ul' ..., $ . From this point we proceed exactly as in the previous theorem. With the help of Lemma A we fnnd afunction fl in No which is good (i.e. has only nondegenerate critical points) on C1, and a neighborhood N1 of fl , N1 C No in which every function is good in C1. Repeating this process k times we produce a

which is good on Co U C1 U U Ck = M Since f NO N' and ... . k C C fXlv = fl~, fk satisfies both conditions (i)and (ii). Hence is a Morse function on fk (w, Vo, v~).

Remark: It is not difficult to show that, in the id topology, the Morse functions form an open dense subset of all smooth maps

For some purposes it is convenient to have a Morse

function in which no two critical points lie at the same level.

Lemma 2.8. Let- f: W d [0,1) be--- a Morse function -for

f can-- be approximated by--- a Morse function g with--- the same critical point a --such that -for Roof: Suppose that f(pl) = f(p2) . Construct a smooth function A: W d [o, 11 such that X = 1 in a neighborhood U of pl and X = 0 outside a larger neighborhood N, where

nCW - Bd W and contain8 no pi for i f 1. Choose > o so small that fo = f + elA has values in [0,1] and fo(pl) fo(pi) , i f 1. Introduce a Riemannian metric for W (see Munkres [5, p.24]), and find c and cl so that 0 <- c <- lgrad fl throughout the compact set K = closure (0 < h < 1) and lgrad hl <- ct on K. Let 0 C E C min(el, c/ct) . Tben fl = f + Eh ie again a Morse function, fl(pl) # f(pi) for i f 1, and fl has the same

critical point 8 as f. For on K,

Andoff K, lgradhl -0, so lgradfll = lgradfl . Continuing inductively, we obtsin a Morse function g which separates all the critical points. This cmpletes the proof.

I Using Morse fhnctione we can now express any "canplicated"

cobordism ae a camposition of f'simplern cobordiams.

I Definition, Given a smooth function f: W d R, a

critical value of f is the image of a critical point.

Lemma 2.9. -Let f: (w; Vo, V1) d ( [0,1], 0, 1) be- -a Morse function, -and suppose that- 0 < c < 1 where- c ---is not a critical value-- of f. Then-- both f-l[o,c] -and f-l[c,l] -are smooth manif old6 with- boundary. Hence the cobordism (w; Vo, Vlj identity, identity) from vo to V1 can be expressed ae the composition of two cobordisms: one from Vo to fD1(c) and one frm fw1(c) to V1. Together with 2.8 this proves:

Corollary 2.10. -Any cobordism can-- be expressed --as a composition -of cobordism with-- Morse number 1,

Proof of 2.98 This follows immediately fram the implicit function theorem, for if w E f-l(c) , then, in some coordinate system 5, x2, ..., xn about w, f looks locally like the Section 3. Elementary Cobordisms 7d

Definition 3.1. Let f be a Morse function for the triad (!V V) . A vector field on P is a gradient-like vector- --field for f if

1) e(f) > 0 throughout the complement of the set of critical points of f , and

2) given any critical point p of f there are coordinates

3 ( = (xl, .> 5, :: ) in a neighborhood U of p so .. .. . n that f = f (p) - 13 + la2 and 5 has coordinates - .. ., -\, '~1+~,.. . , xn) throughout U .

Lema 3.2. For-- every Morse function f ---on a triad (W"; V, V1) there- exists -a gradient-like vector field .

Proof. For simplicity we assume f has only one critical point p , the proof in general being similar. . By the Morse Lema 2.2 we may choose coordinates (2, = (x, x xh+l~ -1 in a neighbor- .. . xn ) hoocl U of p GO that f = f(p) !?/ + 13 throughout u,. Let u 0 - be a neighborhood of p such that U C Uo.

Each point p* e W - Uo is not a critical point of f . It follo~~sfrom the Implicit F'unction Theorem that there exist coordinates xi, .. . , xtn in a neighborhood Ut of pt such that f = constant + xi in U* . Using this and the fact that t! - uo is compact, find neighborhoods U1, .. ., Uk such that

2) u n U, = fi, i = 1,..., k , and

i i i has coordinates x and f = constant + x on 3) ui , .. . , n 1 Ui, i = l,...,k .

On Uo there is the vector field whose coordinates are (-xl, .. . , -5,x+~, .. . , xn) , and on Ui there is the vector field with coordinates 0 .. . 0 , i = 1.. k . Piece together these vector fields using a partition of unity subordinate to the cover

uo, 5, a**, uk, obtaining a vector field on t! . It is easy to check that 5 is the required gradient-like vector field for f .

Remark. From now on we shall identify the triad (w; Vo, V1)

with the cobordism (w; Vo, V1; io, il) where iO:Vo ---A Vo and

il: V1 +V1 are the identity maps.

Definition 3.3. A triad (v; Vo, V1) is said to be a product

cobordism if it is diffeamorphic to the triad (V0 x [0,1]; V0 x 0, V 0 x 1) .

Theorem 3.4. ---If the Morse number p ---of the triad (w; Vo, v~) --is zero, then- (bl; Vo, V1) -is -a product cobordism.

-Proof: Let f: W & [0,1] be a Morse function with no critical points. By Lemma 3.2 there exists a gradient-like vector field 5 for f.

Then ~(f): W d R is strictly positive. Multiplying E at each

point by the positive real number l/~(f), we may assume ~(f)= 1 identically on W .

- If p is aay point in Bd W, then f expressed in some coordinate system 5, ..., xn , xn 2 0 , about p extends to a smooth function g defined on an open subset U of Itn. Correspondingly, t expressed in this coordinate system also extends to U . Ihe Fundamental edstence and uniqueness theorem for ordinary differential equations (see e.8. Lang 13, p.551) thus applies locally to W . Let cp : [a, b] -> W be aay integral curve for the vector field . %en

is identically equal to 1 ;hence

f (

Making the change of parameter, 9(s) = p(s - constant), we obtain an integral curve which satisfies

Each integral curve can be extended uniquely over a maximal interval, which, -since W -is compact, must be [0, 11. Zhus, for each y E W there exists a unique maximal intewal curve

which pasees through y and satisfies f ($ (6)) = e Wthermore , Y . (s ) is smooth as a function of both variables (cf. 65, pages 53 - 54)- Y The required diffeomorphism

h: Vo X [0,1] + is now given by the formula

h(y0, 4 = P (4, yo with

Corollary 3.5. (collar Neighborhood Theorem) -Let W be-- a compact smooth manifold with-- boundary. There exists -a neighborhood of Bd W (called a collar neighborhood) diffeamorphic to - - -, Bd W x [0,1) .

-Proof. By lemma 2.6, there exists a amooth fuoction f: W 4 R+ such that fo1(0) = Bd W and df { 0 on a neighborhood U of Bd W . Then f is a Morse function on fW1[0, ~/2], where r > 0 is a lower bound for f on the compact set W - U . Thus Theorem 3.4 guarantees

A connected, closed submanifold C wn - Bd fl is said to be two-sided if some neighborhood of # on $ is cut into two caoponenta when 8-' is deleted.

Corollary 3.6. (The- Bicollaring- Theorem)

Suppose that every cmponent of a srnocrth submanifold M of W is compact 7------and two-sided. Then there exists a nbicollar" neighborhood of M in W --..-.-. -I___ - - - diffemorphic to- M x (-1,l) ----in such a wey that M corresponds to- M x 0 . -Proof. Since the components of M may be covered by diejoint open sets in W , it suffices to consider the case where M has a single c canponent . Let U be an open neighborhood of' M in W - Bd W such that is compact and lies in a neighborhood of M which is cut into two cumponents when M is deleted. men U clearly splits up as a union of tvo submanifolds U1, U2 such that U1 fl U2 = M is the boundary of each. As in the proof of 2.6 one can use a coordinate cover and a partition of unity to construct a smooth map

such that dq 0 on M , and cp < 0 on U - Ul , cp = 0 on M , > 0 on a - U2 . We can choose an open neighborhood V of El , with C U , on which cp has no critical points. Let 2d' > 0 be the lub of cp on the compact set - V . Let 26' < 0 be the glb of cp on the compact set o2 - V . Then dl[et, elt ] is a compact n-dimensional sub-manifold of V with boundary pt ) u p ) , and cp is a ~orsefunction on -1 cp [el, G" 1 . Applying Theorem 3.1 ve find that dl(et, etl) is a "bicollar" neighborhood of M in V and so 8180 in W .

Remark. The collaring and bicollaring theorems remain valid without the compactness conditions. (~unkres[5, p. 511).

We now restate and prwe a result of Section 1. Theorem 1.4. Let (wj Vo, vl) and (wtj Vj, v;) be two smooth - - _.- manif old triads -and h: V1 -4Vi -a dif'feamorphism. Then-- there exists -a smoothness structure -for W % Wt compatible with--- the given strut- tures on W and Wt. ie unique up to a diffeamorphism leaving ------Vo, h(vl) = Vi , -and V; -fixed. proof.- Existence : By Corollary 3.5, there exist collar neighborhoods Ul, Ui of Vl, Vi in W, Wt and diffemorphisms

,jt: k?' -W % Wt be the inclusion maps in the definition

To define a smoothness structure on 8 manifold it suffices to define compatible smouthness structures on open sets cwering the manifold,

and the smoothness structures defined on these sets by j, j and g respectively, are cmpatible. This completes the proof of exietence.

Uniqueness: We ahm that any smoothness structure 2 on

W % Wt compatible with the given structures on ?I and Wt is isomorphic to a smoothness structure constructed by pasting together collar neighbor-

hoods of Vl and Vi as above. The uniqueness up to diffeanorphism

leaving Vo, h(vl) = Vi , and Vh fixed. then follars essentially f'rm Theorem 6.3 of Munkres [5, p. 621. By Corollary 3.6 there exists a blcollar neighborhood U of (v)= j (1) in W % W1 and a diffeomorphism g: V1 x (-1,l) U with respect to the 8moothnese structure 8,so that &,o) =J (x) , for x E Vl. Then fl(u tl J(w)) and j n (w)) are collar neighborhoods of V1 and Vi in w and W'. This canpletes the proof of uniqueness.

Suppose now we are given triads (WJ Vo, v~), (wt j Vi, Vi) with Mrse functiuns f, f1 to [0,1], (1,2], respectively. Construct gradient-like vector fields 6 and on W and Wt, respectively, normalized so that t(f) = 1 , kt (11 ) = 1 except in e small neighborhood of each critical point,

Lemma 3.7. --Given a dlffemorphism h: V1 Vt1 -there is a unique smoothness structure -on W L$ W' , compatible with--- the given structures on W , W', so that f and ft piece together to give a - -- - -L-- smooth f'unction -on W W Wt -and 5 -and E' -piece together to give -e emooth vector -field.

Proof. The proof ie the aeme as that of Themem 1.4 above, I-L- except that the emoothness structure on the bicollar neighborhood must be chosen by piecing together integral cmsof and gt in collar

neighborhoods of V1 and Vi , This condition also prwee uniqueness.

(~oticethat uniqueness here is much stronger than that in !Theorem 1.4. )

This construction gives an immediate proof of the following result.

Corollary 3.8. p(w % W' j Vo, v;) <_ p(~;VO, V1) + P(W* Vi, V$) where p is the Morse n-er of the triad. -- _.-- Next we will study cobordisme with Morse number 1 .

Let (wj V, V' ) be a triad with Morse function f: W -> R and gradient-like vector field for f . Suppose p e W is a critical point, and Vo = f-'(c0) and Vl = fal(cl) are levels such that co < f(p) < cl and that c = f(p) is the only critical value in the interval [c~,cl] . OD' the Let r denote open ball. of radiue r with center 0 in R', R', aad set OD: = oDP.

Since E is a gradient-like vector field fcn. f , there exists a neighborhood U of p in W , and a ccordinate diffemorphism

4 3 4 2 g: OD" -U ao that fg(x, y)=e 1x1 + and so that 5 2e - lq2 has coordinates ( -xl, .. . , -5, .. ., x,) throughout d; for

vo and f-'(c) and Ve lies between fwl(c) and V1 . The ~ituation is represented schematically in Figure 3.1, Iat S denote the boundary of the closed unit di~c5 in R'.

I Definition 3.9. The characteristic embedding n-h %- &-' x OD d Vo is obtained ae follows. First define an A-1 embeddiog cp: S x OD~-' d V by du, 0v) g(cu cosh 8, cv sinh 8) -€ - for u E sA-', v E 9n-A-', and 0 -< 6 < 1 . Starting at the point cp(u, 8v) in V the integralcurve of 5 is anon-singular curve which leads -6 I from cp(u, 8v) back to some well-defined point cp (u, 0v) in Vo. Define L I the left-hand ephere SL of p in Vo to be the image %(sA-l x 0).

Notice that SL is duet the intersection of Vo with all integral cuntes of 5 leading to the critical point p . The --left hand disc DL is a smool SL~

of these integral curves beginning in gL and ending at p . n-A-1 Similarly the characteristic embedding %: O$ x S v1 n-X-1 - is obtained by embedding O$ X S 4 VE by

(w,v) ~(EUsinh 8, cv coah 8) and then translating the image to

V1. The right-hand sphere SR of p in V1 is defined to be Qk (0 x snwh-l). It is the boundary of the right-hand -disk DR, defined as the union of segments of integral curves of g beginning at p and

ending in SR.

Definition 3.10. An elementary cobordism ia a triad (w; V, V1 ) poseessing a Morse f'unction f with exactly one critical point p .

Remark. It follows from 3.15 below that an elementary cobordism

(w; V, Vt) is not a product cobordism, and hence by 3.4 that the Morse

number p(~jV, Vt ) equals one. Also 3.15 implies that the -index of the

elementary cobordism (w; V, Vt ), defined to be the index of p with res- Figure 3.2 illustrates an elementary cobordism of dimension

, = 2 and index h = 1 .

Figure -3.2

Definition 3.11. Given a manifold V of dimension n-1 and an embedding 9: x V let x(v,~) denote the quotient n-A-1 1 manifold obtained from the disjoint sum (V - (p(&-' x 0))+ (0$ x S A-1 n-A-1 by identifying cp(u, 8v) with (Gu, v) for each u ES , v €9 9 0 < 8 < 1 . If Vt denotes any manifold diffeomorphic to X(V, q) then we will aay that Vf can be obtained from V by surgery of type (A, n-A)

Thus a surgery on an (n-1)-manifold has the effect of removing

an embedded sphere of dimension A-1 and replacing it by an embedded sphere of dimension n-1 . The next two results show that this correa- ponds to passing a critical point of index A of a Mcrse futlction on an

n-manif old. Theorem 3.12. -If Vt = x(v,(~) can-- be obtained -frm V by- A-sw ery of type (A, -A), then-- there exists -an elementary cobordism (w; V, V1 ) ---and a Morse function f: W ---4R with- exactly -one critical point-' -of -index h . -Proof. Let \ denate the set of points (s in R"-' R"-' 4 2 + 2 Rh x = R" which satisfy the inequalities -1 <- - Ix 1 + 1 y 1 5 1 , and IX4 I Iy4 I < (sinh l)(coeh 1) . Thw \ is a differentiable manifold 42 -+ 2 with two boundaries. The "left" boundary, -1 x I + 1 y ( = - 1 , is diffcomorphic to Sh-1 x 0~~"'under the correspondence

(u, m)C--, (u cosh 8, v sinh 8) , 0 <- 8 C 1 . The "right" boundary, n-h-1 under the correspon- -l?12 + lf12 = 1 , is diffeamorphic to O$ x S

dence (eu, v) C--j. (u sinh 0, v oosh 8) . Consider the orthogonal trajectories of the surfaces - 13 + 13 = constant . The trajectory which passes through the point (s can be parametrized in the form t 4 (tG t-la . If -+ x or is zero this trajectory is a straight line aegment tending to

the origin.. For 2 and f different from zero it is 8 hyperbola vhich

leads from some well-defined point (u cosh 8, v einh 8) on the left boundary of \ to the corresponding point (u sinh 0, v coah 8) on

the right boundary.

Construct an n-manifold W = w(~,(p) as follows. Start with the A-1 n-A-l disjoint sum (V - cp(9hm1x0)) XD~+\. For each up S , v cS J

0 < 0 C 1 , and c E identify the point (cp(u, w), c) in the first summand with the unique point (3 fi e \ such that

(2) (3 3 lies on the orthogonal trajectory which passea throw the It is not difficult to see that this correspondence defines a

It follows frm this that w (~,cp) is 8 well-defined smooth manifold.

This manif old w (v, cp) has two boundaries, corresponding to the can be identified with V , letting z e V correspond to:

A-1 ((2, -1) E (V - cp(sA-I x 0)) x 9 for z ,!! .(s X 0).

I(U cosh 0, v sinh 8) E \ for z = q(u, BV) .

The right boundary can be identified with x(v,(~): letting h-1 z e V - CP(S x 0) correspond to (2, +1) 3 and letting (&, v) E O$ X Sn-h-l ~0rre~p0IIdto (u sinh 8, v cosh 0) . A function f: w(~,cp) d R is defined by:

A-1 for (z,c) e (V - rp(~ x0))xD'

It is easy to check that f is a well-defined Morse function with one critical point, of index A . This completes the proof of 3.12.

Theorem 3.13. Let (v; V, V') be an elementary cobordism with - II- - characteristic embedding (PI: &' x OD~-' d V . Then- (w; V, Vc) -is diffeomorphic to--- the triad (w(v, t);V, X(V, t)).

-proof. Using the notation of 3.9 with V = Vo and Vt = V1, ve 2 know from 3.4 that (fo1( [c0, C-a2]); V, v-~)*d (fgl( [C+E , ~~1);v~,V'

are product cobordisms. Thus (wj V, V8) is diffeomorphic to (wp; V-,, v 2 2 where h' = fol([c-E C+E I) Since (w(v,%); V, %(v,(pL)) is clearly € , . diffemorphic to (U(V ,cp); VDE, X(V ,q)) it suffices to show -E -e ,

(we; V-€' VE) 18 diffeomm?hic to (w(v ye); v-,, x(v ,q)) -E -E . Define a diffemorphism k: w(~, cp) -4WE as follows. For -e each (z, t ) r (V cp(sh-l x 0)) x DI let k(z, t ) be the unique point -€ - of GI such that k(z,t) lies on the integral curve which passes through e 2 the point z and such that f(k(z,t)) = E t + c . For each (33E \

set k(Za ;. g(~zea . It follows from the definitions of cp and of w(v_~,(P) , and the fact that g sends orthogonal trajectories in 'x to integral curves in that we obtain a well-defined diffemorphism WE) from U(V , 9) to W This cmpletee the proof of 3.13. -E E .

Theorem 3.14. -Let (w; V, V' ) be-- an elementary cobordism possessing --a Morse function with-- one critic& -point, --of index A . -Let DL be-- the left-hand -disk associated to--- a fixed gradient-like vector -field. Then- V U DL -is -a deformation retract -of W.

Corollary 3.15. H,(w,v) -is ismorphic to-- the integers Z -in dimension A --and is zero otherwise. -A generator -for v -is represente 1 by- DL.

Proof of Corollary. I --

r\, We have H,(w,v) = H,(V U DL, V)

H*(DLY sL)

(2 in dimension A ,% = (0 otherwise

where the second iscnnorphism is excision. Proof-- of Theorem -3.14. By 3.13 we may assume that for the charac teristic embedding . sh-' x - V we have

modulo identifications, where now % is the disk

Let

be the cylindrical neighborhood of DL . We define deformation retractions rt from W to V U C and *;. from V U C to V U DL . (~sret E [o, 11.) Composing these gives the desired retraction,

lst Retraction: Outside \ follow trajectories back to V . In LA follow them as far as C or V . Precisely: For each (v, c ) E (V - %(sh-' X 0~~~')) x D' define

For each (s7) E LA define

where p = p(G t) is the maximum of 1/(101?( ) and the positive real solution for p of the equation Since for 13 2 m the equatiotl has a unique ~olution> 0 which varies continuously, it follars easily that rt is a well-defined retraotion from W to V UC.

Fiaure 1.3. The retraction from W to V U C .

2nd Retraction. ~utsideof C define r; to be the identity.

(case 1).

In C move along straight lines vertically to V U DL, moving more slowly near V n C . Precisely: For each (3$) E C define

for r$Z 3 = for 1<-

-+ 3 2 112 where a = a(x, y, t) = (lot) + t(( 13 - 1)/ 13 2, . One verifies that =i remains continuous as (31* 4 1, (f12 4 0 . Note that the two def- Flgure 3.4. The retraction from V U c to V u DL*

Remark. We now indicate briefly how most of the above results can be generalized to the case of more than one critical point.

Suppose (w; V, V' ) is a triad and f: W & R a Moree functior with critical points pl, .. . , %, all on the same level, of indices hl, ..., $ . Choosing a gradient-like vector field for f , ?re obtain diejoint characteristic embeddings pi: sAi-' x OD'-~~ -L V , i = 1.. k . Construct a smooth manifold o(vb, (4) . as follows. k hi-1 1

Start with the disjoint sum (V - U qi(s x 0)) x D + 5 + ... id 1 + %* n-hi-1 For each u e v g S , 0 < e < 1 , and c g DI identify the point pi(^, BV), c ) in the first summand with the unique point (%ae such that

(1) -1q2 + lf12 5 c , and

(2) ($,a liee on the orthogonal trajectory which passes through As in Theorem 3.13 one proves that W is dif femorphic w(vj rpl, .. ., s). It follows frm this, as in 3.14, that V IJ Dl U .. . U Dk is a deformation retract of W , where Di denotes the left hand disk of pi, i = 1.k . Mnally, if Xl= X2= ... = Xk- then H+(w,v) la iscmorphic to Z @ .. . @ Z (k Sunnnands) in dimension X and is zero otherwise. Generators for H,,(w,v) are represented by .. ., Dk. These generators of %(w, V) are actually completely determined by the given Morse function without reference to the given gradient- like vector field - see [4, p. 201. Section 4. Rearrangement of Cobordisms

Frm now on we ahall use c to denote a cobordism, rather than an equivalence class of cobordiems as in Section 1. If a composition cct of two elementary cobordisms is equivalent to a composition dd' of two elementary c obordiw~such that

and index(c ) = index(d) then we say that the canposition ccl can be rearranged. When is this possible ?

Recall that on the triad (w; V V1) for cc* there exists a 0' Morse function f: W -+ [0,1] with two critical pointa p and p1 , 1 index(p) = index(c ) , index(pt ) = index(c I) , such that f (p) < < f (pl ) . Given a gradient-like vector field for f , the trajectories from p -1 1 meet V = f () in an Imbedded sphere 'R called the right-hand sphere of p , and the trajectories going to pt meet V in an imbedded sphere Sfi , called the left-hand sphere for pl . We state a theorem which guarantees that cct can be rearranged if SR rl S; = 4 .

Theorem 4.1. Preliminary Rearrangement Theorem. Let (w; Vo, vl) be a triad with Moree function f having two critical points p, p' . Suppose that for some choice of gradient-like vector field 5 , the cmpact set of points on trajectories going to or frm p is disdoint K~ frcm the compact set K of points on traJectories going to or from pl. P' If f(~)= [0,1] and a,al E (0,l) , then there exists a new Morse (a) 5 is a gradient-like vector field for g , (b) the critical points of g are still p, p1 , and g(p) = a ,

g(p') = a' ,

(c) g egrees with f near V U V1 and equals f plus a constant 0 in erne neighborhood of p and in same neighborhood of p' . (see Figure 4.1)

Proof: Clearly trajectories through points outside K = K U K , - = A all go fram Vo to V1. me function x: W - K +Vo that assigns to each point q in W - K the unique intersection of its trajectory with Vo is smooth (cf. 3.4) and when q lies neex K , then n(q) lies near K in Vo. It follows that if b: Vb-+ [0,1] is a smooth function zero near the left-hand sphere K n Vo and one near the sphere K ,111 Vo P , P , then p extends uniquely to a smooth function I;: W -+ [o, 11 that is constant on each trajectory, zero near K and one near K P P' . Define a new Morse function g: W + [0,1] by g(q) = ~(f(q),;(q)) where ~(x,y) is any smooth function [0,1] x [o,I] + [0,1] with the properties: (see Figure 4.2) ac For all x end y , dx,y) > 0 ad ~(x,y) increaees from 0 to 1 aa x increases frm 0 to 1 . G(~(P),o)= a ~(f(p'),l) = a'

G(X,Y) = X for x near 0 or 1 and for all y ,

%r,0) = 1 for x in a neighborhood of f (p) ,

for x in a neighborhood of f (p*) .

Figure 4.2

The reader can easily check that g has the desired properties

4.2. Extension: If more generally the Moree function f of 4.1 is allowed two -sets of critical points p {pl,. .. ,pn) , P' = (pi,. .. ,p*) B with all points of p at a single level f (p) and all points of pr at a single level f(p' ) , then the theorem remains valid. In fect the proof may be repeated verbatim. Still using the notation of page 37 let A = index(c) , A' = index(c ' ) , and n = dim H . If

dim SR + dim SJ: < dim V i.e., (n - - 1) + (A* - 1) C n-1 or A >- A' then, roughly stated, there is room enough to move out of the way S~

Theorem 4.4. If A >- X1 , then it is possible to alter the gradient-like vector field for f on a prescribed small neighborhood of

V so that the corresponding new spheres SR and in V do not intersect. More generally if c is a cobordism with several index critical points ply.. .,pk of f , and c1 a cobordiem with several index A1 critical points pipof f , then it is poseible to alter the gradient like vector field for f on a prescribed small neighborhood of V so that the corresponding new spheres in V are pairwise disJoint.

Definition 4.5. An open neighborhood U of e. submanifold v-m C vV , which is diffemorphic to b? x R in such s way that I? corresponds to &?x 0 , is called a product neighborhood of in vV .

Lmima 4.6. Suppose M and N are tub submanifolda of dimension m and n in a manifold V of dimension v . If M haa a product neighborhood in V , and m+n < v , then there exiete a differnorphiem h of

V onto itself smoothly isotopic to the identity, such that h(~)is disjoint from M . Remark: The assurrption that M bas a product neighborhood is not necessary, but it simplifies the proof.

v-m Proof of 4.6: Let k: M x R d U C V be a diffecnnorphism onto a product neighborhood U of M in V such that k(M x @) = M . Let 1 N = U Cl N and consider the composed map g = nok - 0 lNo where v-m n:MxR dRV-m is the natural projection.

The manifold k(~x i?)r V will intersect N if and only if

2 E g(~~). If No is not empty, dim No = n < v-m ; consequently the theorem of Sard (see de Rham [l,p.10]) shows that g(No) has measure v -m zero in R .. Thus we may choose a point i? E R"-~ - dN0) . We will construct a diffemorphism of V onto itself that carries

M to k(M x i?) and is isotopic to the identity. One can easily construct a smooth vector field ((3 on R~'~such that ((3= d for lq 5 Iq v-m and ((9= 0 for [a >- 214 . Since f has ccmpact support, and R has no boundary, the integral curves ~r(t,?) are defined for all real values of t . (compare Milnor [4,p.10]. ) Then @(o,?) is the identity on R~'~,@(l,x) is a diffeanorphism carrying 0 to ?, and @(t,$ , 0 <- t <- 1 , gives a smooth isotopy frm jr(0,x) to $(l,a . Since this isotopy leaves all points fixed outside a bounded set in R~-~we can use it to define an isotopy

Then h = hl is the desired diffeomorphiam V +V . Proof of Theorem 4.4s To simplify notation we prove only the first statement of 4.4. The general statement is proved similarly.

Since the sphere hcrs a product neighborhood in V (cf. 3.9), S~ Lenun~4.6 prwides a dlffemrphism h: V +V smoothly isotopic to the identity, for which h(sR) n SL = $ . The isotopy is used ae follows to alter 5 . 1 1 Let a < p be so large that il[a,$ lies in the prescribed neighborhood of V . The integral curves of $ = ~/g(f) determine a dif fernorphiam

1 such that f(cp(t,q)) = t , and cp(pq) = q E V . Defbe a diffeomorphiam 1 H of x V onto itself by setting ~(t,q)= (t,ht(q)) , where 1 ht(9) is a ~00thisotopy [ayF] X V 4 V fiaan the identity to h adjusted so that ht is the identity for t near a and ht = h for t near 3- Then one readily checks that

is a smouth vector field defined on f'l[a,H] 1 which coincides with 2 -1 -1 1 near f (a) and f (5) = V , and satisfies ~c(f)= 1 identically. Thus - 1 the vector field on W which coincides with ((f)~' on ~~[a,~]and with 5 elsewhere is a new smooth gradient-like vector field for f .

Figure- 4.4. Now for each fixed q E V , cp(t,ht(q)) describes an integral - -1 -1 1 mrve of (p(a,q) in f (a) to d$,h(q)) = h(q) in f (F) = v It follows that the right-hand sphere ~(ax S ) of p in fl(a) is R - carried to h(sR) in V . Thus h(~) is the new right-head sphere - R S~ of p Clearly = S So SR fl = h(sR) t? SL = f as required. . L . This completes the proof of Theorem 4.4.

In the argument above we have proved the following lemma which is frequently needed in later sections.

Lemma 4.7. Given axe a triad (w; Vo, V1) with Morse function f

and gradient-like vector field 5 , a non-critical level V = f'l(b) and a diffemorphism h: V +V that is isotopic to the identity. If

f-l[a,b] , a < b , contains no critical points, then it is possible to construct a new gradient-like vector field for f such that

(a) coincides with 6 outside f-l(a,b) (b) 6 = h cpt where cp and 6 are the diffemorphisms - f-'(a) +V determined by following the traject&ies of and 6 , respectively.

Replacing f by -f one deduces a similar propositian in which 1

is altered on f-l(b,c), b < c , a neighborhood to the right rather than to the left of V .

Recall that any cobordism c may be expressed as a camposition of

a finite number of elementary cobordisme (corollary 2.11). Applying the

Preliminary Rearrangement Theorem 4.1, 4.2 in canbination with Theorem 4.4

we obtain Theorem 4.8. Final Rearrangement Theorem. Any cobordism c may b~ expressed as a composition

where each cobordism c admits a Morse function with just one critical k level and with all critical points of index k . fiternate version of 4.8. Without using the notion of cobordism, we have the following prop-

osition about Moree functions: Given any Morse function on a triad

(Wj VO, V1) , there exists a new Morse function f , which has the same critical. points each with the same index, and which hae the properties:

(2) f(p) = index(p) , at each critical point p of f .

Definition 4.9. Such a Morse function will be called self-indexing (or nice)- .

Theorem 4.8 Is due to Smale [8] and Wallace [9]. 5 5.. A Cancellation Theorem

In view of the Final Rearrangement Theorem another question ariaes naturally. Uhen is a camposition cc' of an elementary cobordism of index A with an elementary cobordism of index

A + 1 equivalent to a product cobordism? Ngure 5.1 shows how this may occur in dimension 2.

Figure 5.1

Let f be a Morse function on the tried (wn; Vo, V1) for ccl, having critical points p, p1 of index A, A + 1 such that f (p ) < 112 < f (pt ). A gradient-like vector field for f determines in V = f-l(1/2) a right-hand sphere

SR of p and a left-hand sphere SL of pl. Note that I dimsR +dimsL=(n-~-~)+~=~-1=dirnv.

Definition 5.1

!No submanifolds #, PTn C vv are said to have transverse intersection (or to intersect transversely) if at each point q e M fl N the tangent space to V at q is spanned by the vectors tangent to M and the vectors tangent to N. (1f m + n < v this is impossible, so transverae intersection simply As a preliminary to the wor Tbeorem 5.4 we prove:

Theorem 5.2

The gradient-like vector field 5 may be so chosen that SR

I has transverse intersection with 8L in V.

Fbr the proof we use a lema stated with the notation of

Definition 5.1:

Lema 5.3 If M has a product neighborhood in V, then there is a diffeomorphism h of V onto itself smoothly isotopic to the identity such that h(~)has transverse intersection with N.

Remark: This lema apparently includes Lema 4.6; in fact the . proof is virtually the same. The product neighborhood assumed for M is actually unnecessary.

Proof:- As in Lemma 4.6 let k : M x R~-~-> u c v be a diffeomorphism onto a product neighborhood U of M in V such that k(~X 8) = M. Let No = U n N, and consider the v-m ccanposed map g = nD k'llI'? 'where n : M x R~'~-> R is 0 the natural projection. The manifold k(~x 2) wufail- to have transverse v-m intersection with N if and only if ?E R is the image under g of some critical- point q E: No at which g fails to have maximal rank v - m. But according to the theorem of Sard (see Milnor [LO, p. 101 and deRham [l, p.101) the image g(c) of the set C N of all critical points of g has c 0 v-m v-m measure zero in R . Hence we can choose a point i?& R - g(c), and, as in Lemna 4.6, con&ruct an isotopy of the identity map

of V to a diffeomorphism h of V onto itself that carries

M to k(M X if). Since k(M X 3) meets N transversely, the proof is ccmplete.

Proof of 'Iheorem 5.2:

'Ihe above lemma provides a diffeomorphism h : V -> V smoothly

1 isotopic to the identity, such that h(SR) intersects SL

transversely. Using Lema 4.7 we can alter the gradient field

so that the new right-hand sphere is h(sR), and the left- hand sphere is unchanged. Pris completes the proof.

In the remainder of 55 it will be assumed that SR has tram-

f 1 verse intersection with S Since dim SR dim SL = dim V, the L* + intersection will consist of a finite number of isolated points.

For if qo is in SR fl SL there exist local coordinate functions

x1.. X on a neighborhood U of q in V such i IAat X (q = 0 i = 1 n and fl SR the locus 0 , , . - 1, U is 1 f x (9) = .. . = xA(q) = 0 while U fl SL is the locus xAtl(q) - n-1 I - ,. . = x (q) = 0. Clearly the only point in SR fl SL n U is As 90 . a consequence there are just a finite number of traject- 1 ories going from p to p' , one through each point of SR fl SL . Still using the notations introduced on page 45 we now state

the major theorem of this section. Theorem 5.4 I If the intersection of SR with SL is transverse and consists of a single point, then the cobordism is a product cobordism. In fact it is possible to alter the gradient-like vector field f on an arbitrarily small neighborhood of the single trajectory T from p to p' producing a nowhere zero vector field t' whose trajectories all proceed from Vo to V1. Fhther g1 is a gradient-like vector field for a Morse function f1 without critical points that agrees with f near Vo U V1.

(see Rgure 5.2 below. )

Remark: The proof, due to M. Morse [~][32], is quite formidable. Not including the technical. theorem 5.6 it occupies the following

10 pages. Figure 5.2

After First we prove the theorem making an assumption about the

behavior of 6 neaz T.

Preliminary Eypothesie 5.5 mere is a neighborhood UT of the traJectory T from p to pl, and a coordinate chart g : UT -> R such that:

1) p and p1 correepond to the points (0,. 0 and (1, 0, ..., 0).

where g(q) = 3 and where: 3) v(xl) is a smooth function of 5, positive on (0, 11, zero a% 0 and 1, and negative elsewhere. Also,

Assertion 1)

Given an open neighborhood U of !I! one can always find in U

a smaller neighborhood U1 of T so that no trajectory leads

from U1 outside of U and back again into U1.

Proof: If this were not so, there would exist a sequence of (partial) trajectories TI, T2,. .. , Tk, ... where % goes from a point rk, through a point sk outside U to a point tk,

and both sequences (rk) and (tk) approach T. Since W Bba - is compact we may assume that sk converges to s E W - U. The integral curve $(t, s ) through s must came from Vo or go

to or do both, else it would be a second trajectory joining V1 p to p'. Suppose for definiteness that it comes from Vo.

Then using the continuous dependence of ~(t,s ' ) on the init-

ial value st, we find that the trajectories through all points

near s originate at Vo. 'Ihe partial trajectory Ts, from

V to any point st near a is compactj hence the least die- 0 tame d(s ) from T to Ts , (in any metric) depends continuously

on st and will be bounded away frcon 0 for all 8' in some

neighborhood ~f s. Since rk E Te the points rk cannot k approach T as k -> CO, a contradiction. - Let U be any open neighborhood of T such that U C UT

and let U' be a 'safe' neighborhood, T C U' C U, provided by Assertion 1).

Assertion 2)

It is possible to alter on a compact subset of U'

producing a nowhere zero vector field El, such that every

integral curve of through a point in U was outside U at

some t%e t' < 0 and will again be outside U at some time

t" > 0.

Proof: 7 Replace ifi2) = (v(%), -x2, .. ., xn) by a smooth vector field 2 = (vt(xl, p), -x2, ..., x n ) where p = [x2 + ... * and

(i) V' (xl, ~(2))5 v(xl) outside a compact neighborhood of g(T) in g(Uf).

(11) vl(x1 , 0) is everywhere negative. (see Flgure 5.4)

%is determines a nowhere zero vector field g' on W. In our local coordinates, the differential equations satisfied by the integral curves of 5' on UT are

Consider the integral curve t) = (%(t), x (t)) with 2( .. . , n initial due (<, ..., xn),0 es t increasee. 0 0 then (a) If one of x~+~,.. . , xn is nonzero, say x: f 0, ot I xn(t) 1 = lxn e 1 increases. exponentially and ?(t) event- leaves g(U) (g(c) is compact, therefore bounded).

0 0 2 (b) If x:+~ = .. . = xn = 0, then ) = [(x2) + .. . + 0 Suppose 4 (xw1)2~1/2 e-t decreases exponentially. x(t) remains in g(~). Since v'(xl, ~(2))Ls negative on the %-axis, there exists 6 > 0 so small that vf(5, ~(2)) is negative on the ccaapact set % = (2E g(ij)1 p(x3) 5 6) . Then vt(xl, p(9) ) has a negative upperbound -c~< 3 on Ks . Eventually p(d (t) ) ( 6, and thereafter

%(t < -01 dt - . Thus d(t) must eventually leave the bounded set g(U) after all.

A similar argument wlll show that ?(t) goes outside g(~) as t decreases.

Assertion 3) hrery trajectory of the vector field 5' goes frm Vo to V1.

-Proof: If an integral curve of 5' is ever in Ut it eventually gets outside U, by Assertion 2). Leaving U' it follow trajectories of 5; so once out of U it will remain out of U1 permanently by Assertion 1). Consequently it must follow a trajectory of to V1. A parallel argument shows that it comes from Vo. On the other hand if an integral curve of 5' is never in U* it is an integral curve of 5 that goes from

Assertion 4) In a natural way, e determines a diffeamolphism

: ([O, 11 X Vo j OX Vo, 1XV o )-> (Wj Vo, V1)

Proof: Let $(t, q) be the family of integral curves for 5'.

Since is nowhere tangent to MW, an application of the implicit function theorem shows that the function ql(q)

[respectively 'Co(q)] that aeaigns to each point q E W the time at which $(t, q) reaches V1 [respectively minus the time when it reaches Vo] depends smoothly on q. Then the

also smooth. Clearly the smooth vector field ~~("(9)) (9) has integral curves that go from Vo to V1 in unit time. To

simplify notation assume that had this property from the

outset. men the required differnorphiem maps

(t, Qo) -> *(t, qO) and its inverse is the smooth map s -> (T0(9),. dd). Assertion 5l- Ihe vector field EL is a gradient-like vector field for a Morse function g on W (with no critical points)

that agrees with f on a neighborhood of Vo U V1.

-Proof: In view of Assertion 4) it will suffice to exhibit a a Mrse function g : [O, 11 X Vo + [o, 11 such that -& > 0 and g agreeewith fl=fa$d near OxV UIXVo (w 0 w msume that Vo = f-l(o) and V = fl()af. Clearly there exists 6 > o such that, for ~LI. q e vor $ (t, q) > o if t < 5 or t > 1 - 6. Let A : (0, 11 -> [O, 11 be a smooth function zero for t E [6, 1. - 81 and one for t near 0 and 1. Consider the function

Choosing 6 sufficiently emu we may assume that k(q) > 0

for all q E Vo. Ben g apparently has the required properties. Granting the Preliminary Hypothesis, this completes the proof of the Flrst Cancellation 'Beorem 5.5. lb establish Theorem 5.5 in general it remains to prove:

I Assertion 6) When SR and SL have a single, transverse intersection it is alwayn possible to choose a new gradient- like vector field €' so that the Preliminary Hypothesis 5.5 is satisfied.

Remark: The proof, which occupies the last 12..'~agesof thla eection, has two parts -c the reduction of the problem to a technical lemma (Theorem 5.6), and the proof of the lennna.

-Proof: Let w) be a vector field on FIn that is of the form described in the Preliminary E&pothesis, with sin&arities at the origin 0 and the unit point e of the %-axis. The function

F@) = f(p) + ie a Morse function on R~ for which qe) is a cadient- like vector field. By a suitable choice of +Ae function v(5) we may arrange that F(e) = f (p' ) , i.e. 21: v(t)dt = f(pl) - f(p) . Recall that according to the definition 3.1 of a gradient-like vector field for f, there exists a co-ordinate

system (xl, ..., xn ) about each of the critical points p 2 2 and p1 in which f corresponds to a function + xl +. .. . + xn of suitable index, and E ha8 coordinates (2X ,. . . , 2 xn

'Ben one readily checks that there exist levels bl and b2, a 1 = f(p)

(a) The diffeomorphisms carry to 5, F to f, and points on the segment oe to points on T.

(b) Let pi denote T 0 (b, 1 = 1 2. The image of LI is a neighborhood in [al, b of the ppl of fW1 1] segment T, while the image of L;, is a neighborhood in f-I [bgr a*] of the segment p2pt of T (see Figure 5.5). Figure 5.2 S-;F Observe that the trajectories of T@) with initial points in a small neighborhood U1 of ql(%) in g1-1 f -1(bl) proceed to points in g2-1 f -1(b) that form a diffemorphic image u2 of U1 and in doing so sweep out a set Lo diffeomorphic to U1 x [0, 11 such that L1 U Lo U L2 is a neighborhood of oe. There is a unique extension of g to a smooth imbedding - 1 of L1 Lo into W determined by the condition that gl U $ trajectories go to trajectories and F levels go to f levels. Now let us suppose for the moment that the two imbeddings of U, into fm1(b2) given by 4 and g2 coincide at least on some small neighborhood of il(pg) in U2. Then pL and g2 together give a diffeomorphism g of a small neighborhood V of oe onto a neighborhood of T in W that preserves trajectories and levels. !@is implies that there is a smmthg positive, real-valued Function k defined on E(v) such that for dll points in E(v) - g* $" k &!. Choosing the neighborhood V of oe sufficiently small we may assume that the kction k is defined, smooth and positive on all. of W. Then 5' = ke is a gradient-like vector field satisfying the Preliminary Hypothesis 5.5. So when the above supposition holds the proof of Assertion 6) is complete. In the general case, the vector field e determines a diffeomorphism h : f-l(bl) -> f-'(b2) and the vector field + q determines a diffeomorphism h ' : U1 -> U2. Clearly the supposition made in the previous paragraph holds if and only if h coincides with ho = g2h'q1 near pl. Ibw, by Lemma 4.7, any diffeomorphism isotopic to h corresponds to a new gradient-like vector field that differs from only on f'l(bl, b2). Thus Assertion 6) will be established if E can be deformed to a diffeomorphism which coincides with ho near pl and for which the new right-hand sphere L(sR(bl)) in level b2 still has the single transverse intersection p2

I with sL(b2). (The bl or b2 here indicates the level in which the sphere lies. ) For convenience we will specify the required deformation of h by giving a suitable isotopy of h'lh that deforms h-%I 0 0 to coincide with the on a very small neighborhood of p 1 identity map on a still smaller neighborhood of pl. Observe that, after a preliminary alteration of g2 if necessary, hi's is orientation preserving at pl = hilh(pl) and both hi% SR(bl) and "(al) have the same intersection number (both +1 or both -1) with' sL(bl) at pl. (For a defln- ition of intersection number see 6.) Then the folldwing local theorem provides the required Isotopy.

Let n = a + b. A point x E R~ may be written x = (u, v), u E Ra, v E Rb . We identify u E Ra with (u, o) E: R~ and v E R~ with (0, v) a R~.

'Iheorem 5.6 Suppose that h is an orientation-preserving imbedding of R~ into R~ such that 1 1) h(0) = 0 (where 0 denotee the origin in Rn) I 2) h(Ra) meets R~ only at the origin. 'Ihe intersection is I transverse and the intersection number is +1 (where we I agree that Ra meets Rb with intersection number +1)- I Then given any neighborhood N of the origin, there exists t n t a smooth isotopy h Iln -> R , 0 < t < 1, with ho = h t ' - - such that I t (I) ht(x) = h(x) for x = 0 and for x E Rn - N, 0 <- t <- 1. t (11) hl(x) = x for x in sane small neighborhood N1

of 0.

1 h'SR") . I !

Figure 5.6 F Ra

i I -a 5.7 i Let h : Rn R be the map in the hypothesis of Theorem n 5.6. 'Ihere exists a smooth isotopy ht : R~ -> R , 0 <- t <- 1, such that (1) ho = h and 5 is the identity map of Rn. b (11) for each t E [O, 11, ht(Fta) fl R = 0, and the intersection is transverse. Proof of Lem 5.7: since h(0) = 0, h(x) may be expressed in the form 1 n h(x) = qh + + xnh (x) x = x ) where (x) ... , (5,..., 11 , i h (x) is a smooth vector function of x and (consequently) i ah h (0) = a;;- (0) , i = 1, ..., n. (see ~11nor [4, p.61). LP i we define ht by 1 1 4-t(~)= h(tx) = 5h (tx) + ... + xnhn(tx), 0 <- t <- 1, then ht(x) is clearly a smooth isotopy of h to the linear map 1 hl(x) = yh (0) + ... + xnhn(0) Since h(Ra) and ht(Ra) have precisely the same orienting basis hl(0), .. . , ha(0) of tangent vectors at 0 E IIn, it follows that for eU t, 0 -< t <- 1, ht(Ra) has transverse b b positive intersection with B at 0. Clearly ht(Ra) n R = 0. Thus if hl is the identity linear map we are through.

If not, consider the family A C GL(n, R) consisting of all urientation-preserving non-singular linear transformations

L of Rn such that L(R~)has transverse positive inter- b section with R , i.e. all transformations with matrices of A + the form L = (&) where A is an a X a matrix and

det L > 0 , det A > 0.

Assertion: For any L E .A there is a smooth isotopy Lt,

0 <- t <- 1, deforming L into the identity, such that Lt E A for all ti or, equivalently, there ie a smooth path in A from L to the identity. -Proof: Addition of a scalar multiple of one of the first a rows [columns] to one of the laet b rows [respectively,

columns] clearly may be realized by a smooth deformation

(= path) in A. A finite number of such operations will reduce

the matrix L to the form

where B is a b X b matrix and (necessarily) det B > 0. As

is well known a finite number of elementary operations on the

matrix A, each realizable by a deformation in GL(~,R),

serve to reduce A to the identity matrix. A similar state-

ment holds for B. Plus there are smooth deformations At, Bt, 0 <- t <- 1, of A and B to identity matrices with det At > 0 and det Bt > 0. 'Ihey provide a deformation in A of L' to

the identity. This completes the proof of the aseertion and

also the proof of Lenrma 5.7.

Proof of lheorem 5.6: Let ht, 0 <- t <- 1, be the isotopy of 5.7.

Let E C N be an open ball about 0 and let d be the distance from 0 to R~ - h(E). Since ht(0) = 0 and the time interval

0 -< t <- 1 is compact, there exists a small open baU El about 0 with C E so that iht(x)( < d for all x E El.

Now define

) for x E El

for x E - E As it stands, this is an isotopy of h Asan 1% U (# - E) initial step we will extend it to an isotopy of h thcrt satisfies at least the conditions (I) and (IT) of 'Iheorem 5.6. rn Flrst observe that to 9 isotopy ht, O

H : [o, 11 X Ftn -> [O, 11 X R~ and conversely. The relation is simply

lhe imbedding H determines on its image a vector field

where (t, Y) = H(t, x) , i.e. y = ht(x). 'Ihis vector field, together with the imbedding ho, completely determines ht and

hence H. In fact

of integral curvee with initial values (0, y) e 0 x ho(Ftn).

lhese observations suggest a device due to R. Ihm. We

will extend the ieotopy Kt to all of [0, 11 x lIn, by first extending the vector field - -

to a vector field on [0, 11 x Ftn of the form (1, (t, Y)).

'Ihe Vector Field Clearly admits an extension to a small open neSghhorhoaii of its closed damain [o, I] X [% U (Rn - E)). This gives an extension of ?(t, y) to a neighborhood U of its closed domain. Multiplication by a smooth function identically one on the original closed domain ana zero outside U produces an extension to [O, 11 X Rn. Finally, setting the first co-ordinate equal to 1 we get a smooth extension ?"(t, Y) = 1, a,Y)). Notice that a family of integral curves $(t, y) is defined for y e Rn and for -all t E [O, 11. For y E Rn - h(E) this is trivial. For y E h(~)it follows frw the fact that the integral curve must remain in the compact set [o, L] X h(E). 'Ibe family g gives a smooth level preserving imbedding

Jr : [O, 11 X Rn -> [O, 11 X Rn Then the equation

*(t, Y 1 = (t, ~~h-l(y)) servcs to define the required extension of gt to a smooth isotopy of h that satisfies at least conditions (I) and (11) of lheorem 5.6.

Using a similar argument one can prove the foUowing theorem of R. Tham which we will use in Section 8. (fir a full proof see Milnor 112, p.51 or Thom [13] ). lheorem 5.8 Isotopy Exbemion meorem.

Let M be a smooth compact submanifold of the smooth manifold H without boundary. If ht,O -< t -< 1, is a smooth isotopy of i : M C H, then ht is the reetriction of a smooth isotopy I I 69 ht,O -< t <- 1, of the identity map N -> N such that ht fixes points outside a compact subset of N. Returning to the proof of Theorem 5.6, let $ denote the extended isotopy. The Last condition (111) of Theorem F 5.6 will be violated if gt Introduces new intersections of the image of R~ with R~ as indicated in Flgure 5.9.

Hence we can use Et only for small values of t, say t <- t', where no new intersection can occur. We will apply the above process to construct a further deformation of , which alters Kt, only at points in El, where coincides with

htl. After a finite number of steps we will obtain the isotopy required. The details follow.

Note that we can write the isotopy ht of Lenrma 5.7 in

the form 1 n (*1 ht(x) = x,-h (t, x) + ... + xnh (t, x) i where h (t, x) is a smooth function of t and x, i % i = 1 , u and (consequently) h (t, 0) = q (O).(%e proof given in Milnor (4, p.61 is unaffected by the pea- meter t. ) L?aauLu There exist positive constants K, k such that for all x

e) ()kx for re#, where ra:Itn->R a

is the natursl projection.

Proof- : The first inequality comes Fmm differentiating (* ) . The second follows frm the fact that ht(IiB) is transverse to R~ for 811 t in the compact interval [O, 11.

We now complete the proof of lheorem 5.6 with an inductive step as follows. Suppose we have smehon obtained an Imbedding h" : # + isotopic to h such that

1) For some t 0 < t < 1, g(x) coincides with 0' - 0- ht (x) for ell x near 0 and with h(x) for all 0 x outside N.

2) G(R~)n R~ = 0. We perform the construction for gt given on pages 61 to 63 taking in place of h and [to,11 in place of [O, 11 and making the following two special choices (a) and (b ).

(a) Choose the ball E C N so small that, for aU. points

x E E, g(x) = ht (r) and the inequalities of Lemma 5.9 0 hold.

Note that on the set [to&]X (3U (R~- E)) where Et is initially defined we have +ai;,x 66 Now is the component of (t, y). so it is clear from the construction on page (62) that we can

(b ) choose the extended F?-cornponent T(t, y) of q(t, y ) to have modulus everywhere less thaa 5r. Then Kt will satisfy (9) everywhere in [to, 11 x R'.

We assert that gt wiJl introduce no new intersection of the image of R~ witb R~ for to -< t <- to + if'k In fact if x E Ila n (E - %), the distance of iit (x) frm 0 R~ is

k %US ($1 shows that for to <- t <- to + g we have

Finally, to make possible composition with similar isotopies, we may adjust the parameter t so that the isotopy 1 k Lt9 to < t < t satisfies - - 0 = min (1, to + g ), Kt(x) = h(x)- for t near to r I Et1(x) for t near to

Since the constant k depends only on ht, the required smooth isotopy is a composition of a finite number of isotopiea constructed in the above manner. So Theorem 5.6 is complete.

Thi8 means that Assertion 6) (page 55) is established, and hence the Flrst Cancellation 'Ibeorem is proved in general. $6 A Stronger CanceUation lheorem

Throughout these notes singular homology with integer coefficients will be used unless otherwise specified.

Let M and M1 be smooth submanifolds of dimenslone r and s in a smooth manifold V of dimension r + s that intersect in points pl, ...,pk , transversely. Suppose that M is oriented and that the normal bundle v'(M1 ) of M1 in V is oriented. At pi choose a positively oriented r-frame

El, ...,gr of linearly independent vectors spanning the tangent of M at pi. Since the intersection at pi is transverse, the vectors El, ...,gr represent a basis for the fiber at p of the normal bundle v(M* ). i

Definition 6.1 The intersection number of M -and M1 -at pi is defined to be +1 or -1 according as the vectors El, ...,Er represent a positively or negatively oriented basis for the fiber at pi of v(M1). Ihe intersection number Mt.M -of M -and M1 is the sum of the intersection numbers at the points

Remark 1) In an expression MI-M we agree to write the mani-

fold with oriented normal bundle first.

Remark 2) If V is oraented, any submanifold N is orientable

if and only if its no& bundle is orientable. In fact given

an orientation for N there is a natural way to give an orien-

tation to v(N) and conversely. Nanoely we require that at any point in N a positively oriented frame tangent to N followed vo 68. by a frame positively oriented in V(N) is a frame positively oriented in V.

Hence if V is orietited there is a natural way to orient v(M) and M1, The reader can check that with these orientations.

M.M1 = (-I)~' M' .M.

If the orientation and orientation of nonnal. bundle are not related by the above convention we clearly still have

M.M1 = +- Mt*M provided V is orientable.

Now assume that M, M' and V are compact connected manifolds without boundary. We prove a lemma which implies that the intersection number M-Ma does not change under deformations of M or ambient isotopy of Mt and which provides a definition of the intersection number of two closed connected submanifolds of V of complementary dimensions, but not necessarily intersecting transversely. ?he lemma is based on the following corollary of the Thom Isomorphism Theorem (see the appendix of Milnor [lg]) and the Tubular Neighborhood lheorem (see

Munkres [5, p.461 and kng [3, p.731 or Wlnor 112, p.191).

Lemma 6.2 (without proof )

With M1 and V as above, there is a natural isomcrphism : H~(M*)-> H~(v,v - MI).

Let a be the canonical generator of H~(M')= 2, and let [MI E H~(M)be the orientation generator. Ihe announced lewa is: ' Lennna 6.3 In the sequence - I L9, H H (V) V r M'.), r (fl) -6> r L>-Hr (v, where g and g' are induced by inclusion, we have glog([M]) = M1.M $(a). Proof: Choose disjoint open r-cella U1, ..., Uk in M con- taining pl, ..., pk respectively. The naturality of the Thm isomorphism implies that the inclusion induced map

is an isomorphism given by y E~ $(a) where yi is the i -> orientation generator of H~(u~,Ui - pi) and ei is the intersection number of M and MI at pi. Ihe following commutative diagram, in which the indicated isomorphism comes from excision and the other homomorphisms are induced by inclusion, cmpletes the proof.

Ne can now reinforce the First Cancellation Theorem 5.4. Let us return to the situation of Theorem 5.4 as set out on page 45. l~unel~(wnj vo, V1) is a triad with Moree function f having a gradient-like vector field 5, and p, p' with f (p) < 112 < f (pl) are the two critical points of f, of index A, A + 1 respectively. Suppose that an orientation has been 1 given to the left-hand sphere SL in v = f-l(1/2) and also to the normal bundle in V of the right-hand sphere SR.

Theorem 6.4 Second Cancellation lheorem SUpPose W, V and V1 are simply connected, and h 2, 0 >- 70 I h+l

lheorem 5.4 then apply.

Remrk 1) Observe that V = f-l(1/2) is also simply connected.

In fact, applying Van Kampen's theorem (crowell and Fox [17, p.631) n-A twice we find n (v) :n (D 1 1 R (p) U V U DF1(q)). his uses A 2 2, n - A 2 3 1. But by 3.4 the inclusion ~~(p)u v u ~~(q)c w is a homotopy equivalence. Combining these two statements we see

that nl(V) = 1.

Remark 2) Notice that conclusion of the theorem is obviously true whenever A = 0 or A = n - 1. Also the reader can verify with the help of 6.6 below that the theorem holds. even with the single dimension restriction n >- 6 ! (The cases we will not check are A = 1 and A = n - 2.) The one extension of use to us comes from turning the triad around:

Corollary 6.5 lheorem 6.4 is also valid if the dimension conditions are h 2 3, (A + 1) <- n - 2.

Proof of Corollaq: Orient SR and the normal bundle VSL 1 of SL In V. Now W is simply connected hence orientable.

So V is orientable and it follows from Remark 2) page 67 that t 1 SL.SR =+S-R asL =+I.- If we now apply lheorem 6.4 to the triad (H"; V1, Vo) with Morse function -f and gradient-like vector fiela -t we clearly get corollary 6.5.

The proof of 6.4 will be based on the following delicate theorem which is essentially due to Whitney 171.

Theorem 6.6

Let M and M' be smooth closed, transversely intersecting submanifolds of dimensions r and s in the smooth (r + 6)-manifold V (without boundary). Suppose that M is oriented and that the normal bundle of M' in V is oriented. Further suppose that r + s >- 5, s >- 3, and, in caae r = 1 or r = 2, suppose that the inclusion induced mag nl(V - M') -> nl(V) ie 1-1 into.

Let p, q E M Il M1 be points with opposite intersection numbers such that there exists a loop L contractible in V that

consists of a smoothly imbedded arc from p to q in M followed by a smoothly imbedded arc from q to p in M1 where both arcs miss M tl M1 - [p, q). With these aesumptions there exists an isotopy ht90 <- t <- 1, of the identity i : V -> V such that

(i) 'Ihe isotopy fixes i near M ll M' - (p, q)

(ii) hl(~)n M' = M n MI - [P, ql Figure 6.1 Remark: If M and Mt are connected, r >- 2 and V is simply connected no explicit assumption about a loop L need be made. For applying the Hopf-Rinow theorem (see Mlnor [4, p.62 1 ), with complete Riemannian metrics on M - S and Mt - S, where

S = M tl M' - (p, q), we can find a smoothly imbedded arc p to Q in M and similarly q to p in Mv giving a loop L that misses

S. The loop L is certainly contractible if V is simply connected.

Proof of lheorem 6.4

According to 5.2 we can make a preliminary adjustment of 5 1 1 near V so that SR and SL intersect transversely. If SR fl SL

1 is not a single point, then SR SL = 2 1 implies that there

1 exists pair SL a of points pl1 91 in SRn with opposite intersection numbers. If we can show that Theorem 6.6 applies to this situation, then after we adjust 5 near V, using Lemna 4.7, SR t and will have two fewer intersection points.. lhus if we repeat SL 4 t this procese finitely many times SR and SL will intersect transversely in a single point and the proof will be complete.

Since V is 13impl.y connected (~e~a~.k1 page 76) it is clear that in case A -> 3 all the conditions of Theorem 6.6 are satis- If X = 2, rr (V n = 1 fied. it remains to show that 1 - sR) -> 1(v) is 1-1, i.e. that nl(v - sR) = 1. N:ow the trajectories of 5 determine a diffeomorphism of Vo - SL onto V - SR, where SL

denotes the left-haad 1-sphere of p in Vo. Let N be a pro-

duct neighborhood of s,., in Vo. Since n - A - 1 = n - 3 2 3,

4 we have nl(~ - R~)Z, and the diagram of fundamental groups corresponding to Van Kampenls theorem bow implies that q(vo - SL)= 1. .!5is completes the proof of Pleorem 6.4 modulo proving Theorem 6.6.

Proof of 6.6

Suppose that the intersection numbers at p and q axe +1 and -1 respectively. Let C and C1 be the smoothly

imbedded arcs in M and M1 from p to q extended a little

1 way at both ends. Let C and Co be open arcs in the plane 0 intersecting traneversely in points a and b, and enclosing a disk D (with two corners) ae in Hgure 6.2 below. Choose an t embedding rpl : C U Co -> M U M1 so that rp1(co) and 0 I rpl(Co) &re the arcs C and C', with a and b corresponding

to p .snd q. 'Ihe theorem will follow quickly from the next

lemma, which embeds a standard model.

Lema 6.7 For some neighborhocd U of D we can extend t r-1 rpl(u n (C U co) to an embedding, rp : U x R - 0 r -1 such that c'(M) = (U n Co)x R x 0 and I s -1 <~(Mw)= (U n c,) x o x R . Fiaure 6.2 The St~ndara Assuming kma 6.7 for the moment, we will construct an isotopy Ft : V -Z V such that Fo is the identity,

F~(M)n M' = M n M' - Lp, qJ, and FJ~it3 the identity outside the image of cp, 0 <- t <- 1. Let W denote p(U X RrW1 X R*-') and define Ft to be the identity on V - W. Define Ft on W as follows.

Choose an isotopy Gt : U -> U of our plane model such that

1. ) Go is the identity map,

2. ) Gt is the identity in a neighborhood of the boundary a-u of u, O

F'igure 6.3

Let P such that w

s-1 Define an iootopy F$ : U x A'-' x R'-~ -> U x R'V' x R by

the requluind isot4py on W. 'Ib.ls fludbes aeproof bf 'iheorem

6.6, modulo proving 6.7.

Lemma 6.8 mere exists a Riemannian metric on V such that 1. ) in the associated connection (see Milnor [4, p.441) M and M1 are totally geodesic submanifolds of V (i.e. if a

geodesic in V ie tangent to M or to Mt at any point

tben it Ides entirely fn M m Mt, respeetlrely. )

2. ) there exist coordinate neighborhoods N and N about P Q * p and q in which the metric is the euclidean metric and

so that N fl C, N fl Ct, Nq fl C, and N fl Ct are straight P P 9 line segments,

-Proof (due to E. Feldman): We know that M intersects Mt transversely in points pk with p and pl, ..., = pl q = P2' Cover M U Mt by coordinate neighborhoods W, W in V .. . m Pt8 with coordinate diffeumorphisms hi : Wi --+ R , i 1, ,m ,

such tihat a. ) there are disjoint coordinate neighborhoods N1, .. . , %

with pi E Ni C Ei Wi and Ni fl W = for i = 1, k c J ..., and J = k + 1, ...,me be) hi (wi flM) C R~ x 0

hi (wi flMt) C 0 x R' i = 1, ...,k.

6.) hi (wi n C) and hi (wi fl C*) are strdght line segments

Comtruct a Riemann metric < ? , z> on the open set

W = W1 U U W by piecing together the metrics on the 0 .. . m Wi induced by the h i = 1, , using a partition of unity.

Note that because of a*) this metric is eucllaean in +he Ni, i =1, ...,k. With this metric construct open tubular neighborhoods T and Tt of M and M1 in W using the exponential map (see 0 hng [3, P- 731) By choosing them thin enough we may essume that T flT' C N1 U .. . U % and that hi(^ n T* n N,) = x OD:, R~ x R~ = R** oare c i = 1, k , for some E, E' > 0 depending on i. The situation is represented schematically in Figure 6.4.

2 Let A : T -> T be the smooth involution (A = A A = identity) which is the antipodal map on each fiber of T. Define a new Riemarrt! metric < ? i? >A on T by = , A l (c ?, i?> + < A*?, A*$>). P \

Assertion: With respect to this new metric, M is a totally geodesic submanifold of Tm To see this, let w be a geodesic in T tangent to M at some point z E M. It is easy to see that A is an isometry of T in the new metric and hence sends geodesics to geodesics. Since M is the fixed point set of A, it follow6 that ~(w) and w are geodesics with the same tangent vector at A(Z) = 2. By uniqueness of geodesics, A is the identity on w. merefore wC M, which proves the assertion. a

Similarly define a new metric < ?, on Ttm It follows from property b. ) and the form of T fl T1 that these two new metrics agree with the old metric on T n Tt and hence together define a metric on T U Tt. mending to all of V the restriction of this metric to an open eet 0, with M U M' C 0 C 5 c T U Tt , ccrmpletee the construction of a metric on V satisfying conditions 1. ) and 2. ).

Proof of Lenrma 6.7 (The proof occupies the rest of Section 6)

Choose a Riemannian metric on V prcvlded by Lemma 6.8.

Let 7)(q) (p), (q) be the unit vectors tangent to the arc8 C and Ct (oriented from p to q) at p and Q.

Since C Is a contractible space, the bundle over it of vectors orthogonal to M is trivial. Using this fact conetruct a field of unit vectors along C orthogonal to M and equal to the parallel translates of T (p ) and of -.r (q) along v"' and N fl C respectively. P Construct erne corresponding vector field in the model.

(see Figure 6.5)

F'igure 6.5 Using the exponential map, we see that there exists a neighborhood of Co in the plane and an extension of plICo to an

imbedding of this neighborhood into V. Actually, the exponential map gives an imbedding locally, and then one uses the following

lema, whose elementary proof may be found in Munkres 15, p.49 Lemma 5.7 (which is incorrectly stated)]

Lenrma 6.9 Let A be a closed subset of a compact metric space 0 A. Let f : A -> B be a local hnaeomorpbism such that f l~~

is 1-1. men there is a neighborhood of W of A. such that

t Similarly extend Cq 1 co to an imbedding of a neighborhood t of C3 using 8 field of unit vectors along Ct orthogonal to

Mt which along N n Ct and N fl C1 coneists of the parallel P Q translates of ~(p)and -7(q) respectively. When r = 1

this is possible only because the intersection numbers at p

and q are opposite.

Using property 2) of the metric on V (see 6.8) we see t that the imbeddings agree in a neighborhood of Co U C and 0 hence define an Imbedding

of a closed annular neighborhood N of BdD such that 1 9i1(M) = N fl Co and cpil(~?) = IV n Coo Let 8 denote the inner boundary of N and let Do C D be the disc bounded by

S in the plane. (see Mgure 6.5)

Since the given loop L is homotopic to the loop q2(s),

the latter is cclltractible in V. Actually q2(s) ie contractible in V - (hl U M' ) as the following lemma will show.

Lemma 6.10

If 3 , n 2 5 , ie a emooth manifold, Y a smooth submanifold of codimension at least 3, then a loop in V1 - Y that is contractible in Vl is alsj contractible in V 1 - Yo Before proving 6.10 we recall two theorem of Wbiimey.

Lemma 6.n (see Milnor [15, p.62 and p.631) at f : Y -> Y be a continuous map of smooth manifolds which is smooth on a closed subset A of Y. Then there exists a smooth map

63 : 5 -> y BU& that g 2 f (g is homotopic to f) and

Lemma 6.12 (see Whitney I161 and Milnor (15, p.633)

Let f : y -> % be a smooth map of smooth manifolds which is an imbedding on the closed subset A of Y. Assume that dimM2>2dimY+le- men there exists an imbedding g : -> approximating f such that g 3 f and gl~= f 1~.

Proof of 6.10: 2 1 Let g : (D , S ) -> (V1, V1 - Y) give a contraction in Vl of a loop in V1 - Y. Because dim (vl - Y) 2 9 the above lemmas give a smooth imbedding 2 1 h : (D , S ) -> (V1, V1 - 5) such that g 1 is homotopic to h 1 in V1 - Ye I3. 1s 2 Ihe no& bundle of h(D2) 3.3 trivial since h(D ) is 2 n-2 contractible. Hence there exists an imbedding H of D X R 2 into V1 such that ~(u,o) = h(u) for u E D . Take E > 0

so small that 121 < E, ?E R'-*, implies H(S? X 2) C V1 - Y. n Since codimension 5 >_ 3 , there exists (cf. 4.6) xo E R -2 , IgI < E, such that H(D* x s)fl M,. = Now in Vl - 5 we have g 1 2 2 H 1 .cI- constant. This com- Is hl~l= =is1 x 0 Is xo pletes the proof of Lemma 6.10.

Now we can show that V(S) is contractible in V - M U Ms. For it is contractible in V - Mt by 6.10 if r >- 3, and if is r = 2 by the hypothesis that (V- M ) - 1(v) 1-1.

We now choose a continuous extension of cp to U = N U D 2 0

that maps Int D into V - (M U Mt). Applying &mas 6.11 1 and 6.12 to (p2 114t D we can obtain a smooth imbedding

: U -> V coinciding with on a neighborhood of 3 ? U bt D, M' - and such that cp 3 (u) $ M U for u Co U Coo r-1 It remains now to extend cp to U x R x R~~~ as 3. desired.

We let U1 denote cp (u), and for convenience in nota- 3 ' tion we shall write C, C1, Co, and Co in place of U1 flC, 1 U' fl C', U fl Co, and U fl Co, respectively.

* Lemma 6.13 mere exist smooth vector fields El, .. . , Erel,

'119 along U' which satisfy condition 1. ) below

satisfy 3.) 1. ) are orthonormal and are orthoaonal to Ut

2.) along C are tangent M 3. ) along C1 are tangent to Mt .

-Proof: The idea is to construct el, . , Er-l in steps, first along C by parallel t.ranslation, then extending to C U Ct by a bundle argument, and then to Ut by another bundle argument.

The details follow.

Let T and 7' be the normalized velocity vectors along

C and Ct, and let Vt be the field of unit vectors along Ct which are tangent to U1 and are inward orthogonal to Ct. Then

~'(p)= ~(p)and vt(.q) = -~(q)(see Mgure 6.6) j'iwxe 6.6

Choose r - 1 vectors (p), .. (p)which are tangent to M at p, are othogonal to UU', and are such that the r-frame *(PI, s~(P), YE^-^ (P) is positively oriented in TM Parallel translating these r - 1 vectors along C P . gives r - 1 smooth vector fields E,Y *a*, kr4 along C. These vectors fields satisfy 1.) because parallel translation preserves inner products (see Mii30r [4, p.4bI). They satisfy

2.) because parallel translation along a curve in a totally geodesic sub~~.~nifoldM sends tangent vectors to M into tangent vectors to M (see Helgason, Differential Oeometly acld Symoletric

maces, P. 80). Actually, given the conetruction of the Rieman-

nian metric in 6.8, condition 2. ) easily follows from the exist-

ence of the "antipodal isometry1' A on a tubular neighborhood

of M (compare the argument on p.76). Finally, by continuity

the r-frame T, El? Er-l is positively oriented in TM (= tangent bundle of M) at every point of C. Nov pa.raJ-lel translate ~~(p), . ., (p) along NP n C1 k1w, -,!& (9) along Nq fl Ct . Ry hypothesis the intersection numbers of M and Mt at p and q are +1 and -1. %is means that (p, (p.. .(p ia positively oriented in v(Mt) at p while ~(q),~~(q), ..., Er-l(q) is

negatively oriented in v(M*) at q. Since V1(p ) = ~(p)and

V1(q) = -r(q), we can conclude that at all points of both

N ll Ct and N fl Ct, the frames vtY Ely are posi- P q .. tively oriented in v (MI).

Ihe bwdle over C1 of (r - 1)-frames [r -1 orthogonal to Mt and to U', and such that v *,cl, .. ,cr-l is positively oriented in v (M' ) is trivial with fiber SO(^ - 1 ), which is connected. Hence we may extend El, .,Erwl to a smooth field of (r - 1)-frames on C U C' that satisfy conditions 1) and 2).

Ihe bundle over Ut of orthonormal (r-1)-frames orthogonal e to Ut is 3 trivial bundle with fiber ~(r+ B - 2)/0(s - 1) - - F-1(fl+s-2 ), the Stiefel manifold of orthonormal (r - 1)- +e -2 Pramea in R' . So far we have constructed a smooth croes- section El, ..a,kr-l of this bundle over C U Ct. Composing

El, El, ***Ar-l with the proJection into the fiber, we get a smooth map of C U C1 into ~(r+ s - 2)/0(s - 1) which is simply connected since s -> 3. (see Steenrod [18, p.1031). Hence there is a continuous extension to U1 and by Lennna 6.n there exists a smooth extension. Thus we can define El, 4*.,Er-l over &U of UI to eetiefy 1.) and 2.).

To define the remaing desired vector fields, observe that the bundle over U' of orthonormal frames 'lp in

TV such that each qi is orthogonal to U1 and to El, *,kr-l is a trivial bundle because U1 is contractible. Let the desired field of frames ql, ...,qs-l on Ut be a smooth cross-section of this bundle. lhen El, .a.,Cr-l, ql, ...,Q~-~ satisfy 1. ). Fkthermore, since El, .. e,k,-l are orthogonal to MI along Ct, it follow8 that q, .. .,v~-~ sat- ism 3 ) This finishes the proof of Lemma 6.13.

Completion of Proof of knma 6.1 s -1 Defineamap UxFIr-lXR ----3V by

It follows from Lemma 6,9 and the fact that Wemap is a local diffeomorphism that there exists an open E-neighborhood \ r+s-2 = Rr-l about the origin in R. such that if q+ : U x % -> V denotes this map restricted to U X NE then % is an embedding. (U may have to be replaced by a slightly smaller neighborhood, which we still denote by U. )

Define an embedding cp : U X X R'-' -> V by

t s -1 cp(c0 X 0 X R ) C M1 because M and M1 are totally geodesic submanifolds of V. Moreover, since ~I(Ux 0) = Us intersects

M and M1 precisely in C and C1, transversely, it follows that, for E > 0 sufficiently smaJ.1, hage(p) intersects M and MI precisely in the above product neighborhoods of C and

C1. This means {'(M) = C x R~-~x 0 and <'(M') = 0 1 s -1 Co x 0 X R . Thus 9 is the required embedding. This ends the proof of Lemma 6.7. §7 Cancellation of Critical Points in the Middle Dimensions

Definition 7.1 Suppose W is a compact oriented smooth n-dimensional manifold, and set X = BdW. It is easy to check that X is given a well defined orientation, called the induced oriantation, by saying that an (n - 1)-frame T~,. ..,qn-l of vectors tangent to X at same point x E X is positively oriented if the n-frame w..,~ is positively oriented in V, T~, n-1 TCJ where V is any vector at x tangent to W but not to X and pointing out of W (i.e. y is outward normal to x).

Alternatively, one specifies [XI E Hn-l (x) as the induced orientation generator for X, where [x] is the image of the orientation generator [W] E H n (w, X) for W under the boundary homomorphism H~(w, X) -> Hn-l (x) of the exact sequence for the pair (w, x).

Remark: Ihe reader can easily give a natural correspondence between an orientation of a compact manifold 8 specified by an orientation of the tangent bundle (in terms of ordered frames) and an orie~tation M specified by a generator [M] of H~(M; 2)

(cf.~lnor [lg, p.211). It is not difficult to see that the two ways given above to orient BdW are equivalent under this natural correspondence. Since we will aLways use the eecond way to orient MW, the proof is omitted.

Suppose now that we are given n-dimensional triads

(w; V, v'), (w'; V1, v"), and (W U WT; V, v"). Suppose also that f is a Morse f'unctioh on W U Wt with critical points

t ? are all on one level and are of index A, while ql, ...,(+,, are on another level and are of index + 1, and Vt is a non-critical level between them. Choose a gradient-like vector field for f and orient the left-hand dish D (q ), D (q ) L 1 ..., L a t ? t in W and l+,(ql), ...,I+,($) in W'.

Orientation for the normal bundle of a right-hand VDR (qi ) disk in W is then determined by the condition that ~~(ql)have intersection number +1 with %(ql) at the point qi. The normal bundle YS~((~I)of sR(qi) in V1 is naturally isomorphic to the restriction of VD~(~)to S ( ). Hence the orien- R PI tation of VD~$ ) deternines an orientation for vsR(qi ) . Combining Definition 7.1 and the above paragraph we conclude that once orientations have been chosen for the. left-hand disks in W and W', there is a natural way to orient the left-hand spheres in Vt and the normal bundles of the right-hand spheres 11 Vf in . Consequently the intersection number sR(qi) SL (q3 ) of left-hand spheres with right-hand spheres in V' are well defined . horn Section 3 we know that H (w,V) and %+1(~ U Wt, W) A 2 are free abexian with generators - =~+1(w* , Vt )

ively, represented by the oriented left-hand disks. 8

Lemma 7.2 Let M be an oriented closed smooth manifold of dimension embedded in Vt with [MI E $(M) the orientation generator, and let h : %(M) -> %(w, V) be the map induced by inclusion* %en h( [MI ) = SRiqL) -14 [D~(PL)1 + .. . + sR(ql)

* M [IIL(ql) 1 where sR(q1) M denotes the intersection number of sR(qi) and M in V1.

Corollary 7.3 With respect to the baseg represented by the oriented left-hand disks, the boundary map a : S+~(W U W1, W) -> %(w, V) for the triple W U Wt 2W 2 V is given by the f I matrix (a ) of intersection numbers id atl = sR(~i) SIL (qj ) in

V', naturally determined by the orientations assigned to the left-hand disks.

Proof of CoroU~py: Consider one of the basis elements 11 [DL(qj )] E %+1(~ U W ', W). We can factor the map a into the composition

U- \+,@U Wt, W) e > %+p',Vr) > H (vt) bound* A I

Here e is the inverse of the excision isomorphism, and i,

is induced by inclusion. 1 t we have By definition of the orinntation for SL (q3 ), t 11 boundary O e( rDL(qj)I = I*( [SL(qj )I 1. t 1 The result follows by setting M = S (q ) in Lemma 7.2. L3 Proof of 7.2: We assume 1 = 1, the proof in the general case

being similar. Set q = ql, DL= DL(q1), DR = ~~(9~)'and

S, = sR(ql). We must show that h([~]) = SRbM.[DL].

Consider the following diagram:

The deformation retraction r : W. ->,V '. U DL constructed in Theorem 3.14 maps V' - SR to VU(D q), sothe fi homomorphism hl induced by r Iv' is well-defined. Ihe ob- vious deformation retraction of V U (D~- q) to V induces

the isomorphism he. All the other homomorphisms are induced by inclusion.

The diagram comutes because i = (r v ) (since the maps i r 1 V : V' -> W are homotopic) and the corresponding diagram of topological spaces and continuous maps commutes pointwise,

with rlvt in place of i.

Fhwm -Lemma 6.3 ve know that h ( [M ] ) = SRe y *(a) where 0 a E Ho(sR) is the canonical generator and \k : HO(sR) -> Hence in order to prove v V - SR ) is the mom isomorphism. that h( [MI ) = SR M [DL-), by commutativity of the diagram it suffices to show that

(*I h3 hg O hl(t(d) = [%la 'Ihe class *(a) is represented by any oriented disk DA which intersects SR Pn 6Qe pbnt x, transversely, with in%er- section number SR DA = +1. Referring to the standard form for an elementary cobordism given in Theorem 3.13, and the conventions by which D (S ) is oriented, one can see that the image R r(DA) of Dh under the retraction r represents

l = SR l = +1 D~ $ times the orientation generator h-'ho1([DL]) for H (V U DL, 2 3 A V U (D~- (1)). It follows that -1 -1 hlt(d = hg hj ([DL]) or h3h2\ *(a) = [DL] as required. Thus the proof of 7.2 is complete.

Given any cobordism c represented by the triad (w; V, V1), according to 4.8 we can factor c = c c c so that c o 1 .. . n A admits a Morse function all of whose critical points &re on the

same level and have index A. Let cocl c be represented .. . A by the manifold W C W, A = 0, 1, n, and set W = V, 60 A ..., -1 that

= v w -1 cwocw,c a*. cwn = w Define C = H (W A A A' WA-l) -H*(w~,WA-l) and let a : CA -' Ck-l be the boundary homo~llorphismfor the exact sequence of the triple -90 Theorem 7.4 C, = (c~,a) is a chain complex (i.e. a2 = 0) and qc*) H (w, V) for all A.

Proof: (~otethat we do not use the fact that C is free abe- - A lian, but only that H+(IJ~,blh-l) is concentrated in degree A. )

That a2 = 0 is clear from the definition. To prove the isomorphism, consider the following diagram.

Ihe horizontal is the exact sequence of the triple (w~+~,Wk, W ) and the vertical is the exact sequence"of (W A-2 h' Wh-l' W ) One checks easily that the diagram commutes. %en clearly A-2 H&c,) z HA (wwp Whe2 ). ~utH~(w~, w A-2 ) H~(w,v). Leav- ing the reader to verify this last statement (see Milnor [lg, peg]), we have the desired isomorphism $(c,) E %(w, v).

Theorem 7.5 (~oincareDuality. )

If (1.l; V, V1 ) is a smooth manifold triad of dimension n and

W is oriented, then %(W, V) is isomorphic to $IoA(w, V' ) for all A.

Proof: Let c = c c c and C, = (c~, be defined with - o 1 .. . n a) respect to a Morse function f as above, and fix a gradient-like vector field 5 for f. Given fixed orientations, the left-hand disks of c form a basis for C = H (W W ). From 7.3 we A h A A' A-1 know that with respect to this basis the boundary map 8 : CA -> C is given by the matrix of intersection numbers of A-1 oriented left-hand sph~resof c with right-hand spheres of 'h C having oriented normal bundles. A-1 Similarly let W' C W represent c c ... c for P n-p n-p+l n t p = 0 1, , n and set W = V. Define C' = H (w', Wlfml 1 I' P CI and at : C' -> C' as before. For any right-hand disk DR, Cr CI -1 disk) together with the orientation of W give a natcrally defined orientation for DR. Tihell a : C' -> C1 is gduen IJ P-1 by a matrix of intersection number of oriented right-hand spheres with left-hand spheres having oriented normal bundles, * Let C1 = (Ctp, 81) be the cochaln complex dual to ths chain complex Ci = (C1, at) (mus c'~= Hom(C1, 2)). Choose v v as basis for c*~%he basis dual to the basis of C1 which is k' determined by the oriented right-hQaul dim of c n-Cr* An isomorphism C -> C tn-A is induced by assigning to h each oriented left-hand disk, the dual of the oriented right- hand disk of the same critical point. Now, as we have stated, t t : cA -> c~-~is given by a matrix (a ) = (sR(pi) ' sL(pj)). a ij

It is easy to see that 6' : can-' -> C 1n-A+l is given by the 11 matrix (bi j) = (sL(pj) sR(pi) ). But since W is oriented, b =+a the sign depending only on A. (cf. 6.1 Remark 2. ij - ijp

The sign turns out to be (-1 ) Thus a corresponds to -+ 6 l, and it follows that the isomorphism of chain groups induces an isomorphism %(c*) 8-'(ct*). Now 7.1 implies H~(C*) 2 H~(w; V) and H (c:) " H (W, V1) i' i' i' for each h and . Wreover, the latter isamorphism implies u that H~(C' ) 2 #(w, Vt) for each . Fbr if two chain complexes have isomorphic homology then the duaL cochain I complexes have isomorphic cohomology. mefollows from the Uni- I versal Coefficient Theorem. I Combining the last two paragraphs we obtain the desired

isomorphism H (w, V) 8-h(~,Vt ). h

'Iheorem 7.6 Basis Theorem

Suppose (w; V, V' ) is a triad of dimension n possessing a Morse function f with all critical points of index h and I on the same level; and let be a gradient-like vector field I for f. Assume that 2 <- A -< n - 2 and that W ie connected. lhen given any basis for H~(w, v), there exist a Morse function ft and a gradient-like vector field for f' which agree I with f and in a neighborhood of V U V' and are such that I f' has the same critical points as f, all on the same level, I and the left-hand disks for , when suitably oriented, I determine the given basis. I -of: Let pl, pk be the critical points of f and let I_ ...,

bl* -*, bk be the basis of \(w, V) Z @ ... @Z (k-sumands) represented by the left-hand dish ~~(p~),.. . , ~~(p~) with any fixed orientations. Let the normal bundle6 of the right- hand disks D~(P=), .. ., D~(P~)be oriented so that the matrix I (DR(pI) DL(pJ)) of intersection numbers is the identity k x k matrix.

Consider first any oriented A-disk D smoothly imbedded in

W with BdDC V. D represents an element

qbl + ..* + cpk E B~(w,v) for some integers , , ; that is, D is homolgous to qDL(pl) + .. . + g,l+,(pk). It follows from an eaeily proved relative version of Lemma 6.3 that, for each j = 1, .. . , k

TZlus D represents the element

D~(P~)*D bl + ... + ~~(p~).D bk. We shall construct f1 and kt so that the new oriented

for j = 3, 4, ..., k. It follows from the previous paragraph that b bg, bg, One can also the new basis is then 1 + ..., bk. cause a basis element to be replaced by its negative simply by reversing the orientation of the corresponding left-hand disk.

Since a composition of such elementary operations yields any desired basis, this will complete the proof.

Ihe steps involved are roughly as follows: increase f in a neighborhood of pl, alter the vector field so that the left-

11 hand disk of pl sweeps across" P2 with positive sign, and then readjust the f'unction so that there ia only one critical value. 734

More precisely, using 4.1 find a Morse function f, which L agreee with f outside a small neighborhood of p, such that .L fl(~l) > f (p1 ) and fi has the same critical points and gradient- like vector field as f. Choose to so that fl(pl) > t > f(p) 0 and set Vo = fiL(to).

The left-hand (A 1 )-sphere SL of p1 in V and the right- - 0 hand (n - A - 1)-spheres sR(pi) of the pi, 2 <- i <- k, lying in V are disjoint. Choose pointe a c SL and b E sR(p2). 0 Since W, and hence Vo, is connected, there is an enbedding

n-A-1 Lemma 7.7 There exists an embedding p : (0, 3) x R'-' X R

-> V such that 0

3. ) the image of cp misses the other spheres. Moreover,, cp can

positive orientation and so that cp((0, 3) x Ith-' X 0)

intersects sR(p2) at cp(2, 0, O) = b with intersection

number +1. Proof:- Choose a Riemann metric for V so that the arc A = %(0, 0 3) SL so is orthoqonal to and to SR (p2 ) and that these spheres are totaly geodesic submanifalds of V (cf, Lema 6.7). 0 Let ~(a)and v(b) be orthonormal (A - 1)-frames at a and b such that p(a) is tangent to SL at a with positive orientation and p(b) is orthogonal to S (p ) at b R 2 with intersection number +l. The bundle over A of orthonormal

(A- 1)-frames of vectors orthogonal to A is a trivial bundle with fiber the Stiefel manifold V (Rn-2 , which is connected A-1 since A - 1 < n - 2. Hence we may extend to a smooth cross- section p along all of A.

The bundle over A of orthonormal (n - A - 1)-frames of vectors orthogonal to A and to p is a trivial bundle with fiber Vn - - (Rn-A-l ) Let 9 be a smooth cross-section.

Now use the exponential map associated to the metric to define the desired embedding with the help of the (n - 2)- frames pq. The details are similar to those in the completion of the proof of Lemma 6.7, page 83 . This finishes the proof of Lemma 7.7.

Completion of hoof ofthe meis Theorem 7.6

Using cp we construct an isotopy of Vo which sweeps s~ across sR(p2), as follows. (see Mgure 7.2) 1 Fix a number 6 > 0 and let a : R -> [l, 25 ] be a smooth function such that a(u) = 1 for u >- 26 and a(u) > 2 for As in the last paragraph in the proof of meorem 6.6, page

74$ construct an isotopy H A-1 #-A4 t of (0, 3) x R such that 1.) Ht is the identity outside some compact set, 0 <- t <- 1 . 4 A-1 2.) ~~(1,3 O) = (t a (~21~)+ (1 - t), y. 0) for F'E R .

Rgure 7.2

Define an isotopy Ft bf V by F~~V)= 0 0 Ht 6 f1(v) for v & Image (g) and F~(v)= v otherwise. fiom property

1.) of Ht is well-defined. we see that Ft NOW u~ingLemma 3.5 find a product neighborhood Vo X [0, 11 embedded in W on the right side of Vo such that it contains no critical points and Vo X 0 = Vo . Using the isotopy Ft, alter the vector field 5 on this neighborhood as in Lemma 4.7, obtain- * ing a new vector field E' on W.

Since and E1 agree to the left of Vo (that is, on -1 fi (- CO, to]), it follows that the right-hand spheres in Vo associated to kt are still sR(pp), .. . , sR(pk). The left-hand

1 sphere of p associated to 5' is SL = F~(s~).morn property 1 I 2.) of H we know that SL misses sR(p3), sR(pk). Hence 0 ..., by 4.2 we can find a brse function f' agreeing in a neighborhood of BdW with fl (and so with f), having pt as associated gradient-like vector field, and having only one critical value.

This completes the construction of f and 5 I. It remains to show that the new left-hand disks represent the desired basis. The left-hand disks of p2, ..., pk associated to are still DL(p2), .. ., ~~(p~)since k1 = 5 to the left of the -1 neighborhood Vo x [0, 11, that is, on fl (-w, to]. Since

I' = also to the right of Vo x [0, 11, the new left-hand

1 1 disk ~~(p~)intersects ~~(p~)at pl = DL(pl) fl~~(p~) w ith intersection number D (p ) D' (p ) = +1. It follows from R 1 L 1 t property 2.) of Ht that D (p ) intersects D*(P*) in a single L 1 I point, transversely, with intersection number D*(P~) DL(pl) = +l* I Finally, property 3. ) of cp implies that ~~(p~)is disjoint 8 from D (p ), D p ) and hence that DR(pi) DL(pl) = 0 R 3 .. . , Rk for i = 3, k. Thus the basis for H (w, V) represented ..., h by the left-hand disks associated to t1 is indeed b 1 + b2, b29

*-*tbk, which completes the proof of 7.6.

Theorem 7.8 suppose (w; V, V1) is a triad of dimension n >- 6 possessing a Morse function with no critical points of indices

0, 1 or n - 1, n. Futhermore, assume that W, V and V' are ell skply connected (hence orientable) and that H*(w, V) = 0.

Then (w; V, V' ) is a product cobordism. Let c denote the cobordism (; V, V). It follows from lheorem 4.8 that we can factor c = c c c so that c 2 3 .. . n-2 admits a Morse function f whose restriction to each c is a A Morse function all of whose critical points are on the same level and have index A. With the notation as in Theorem 7.4 we have a a the sequence of free abelian groups -> C -> .so -h> %-2 n-3 ->Ca -> a ->Ca For each h, choose a basis %+l A ... 2' A+l X+1 z1 , so., Z for the kernel of a : CA+l -> C Since kA+l A-

H,(w, V) = 0 it follows from Theorem 7.4 that the above sequence A+1 it3 exact and hence that we may choose bl , ..., ~+1 L+1 \ X+l such that bi a z for i = 1, kh. men z1 -> 1 ..., , -.., k+l h+l, bkA+l zk bl . is a basis for C A+l A h+1

Since 2 <- A <- A + 1 <- n - 2, using Theorem 7.6 we can find \ . a Morse function ft and gradient-like vector field 4' on c

so that the left-hand disks of c and c represent the h A+l chosen bases for C and CA+l. h Let p and q be the critical points in c and c A A+l corresponding to zt and by1.. By increasing f in a neighborhood of p an3 decreasing f ' in a neighborhood of q (see

4.1, 4.2)yeobtain c~c~+~= clcXPqk+lj C C' where cP has exactly

one critical. point p and c has exactly one critical point 4- Q Let Y be the level manifold between c and c 0 P 4- It is easy to verify that c c and its two end manifolds are P Q all simply connected (compare Remark 4 page 70). Since a by1 = z the spheres sR(p) and sL(q) in V have ioter- 0 section nwnber +l.- Hence the Second Cancellation Theorem 6.4 or Corollary 6.5 implies that c c is a product cobordism and P P that ft and its gradient-like vector field can be altered on the interior of c c so that f' haa no critical points there. P Q Repeating this process as often as possible we clearly eliminate all critical points. Then, in view of Theorem 3.4, the proof of

Theorem 7.8 is complete. 8 Elimination of Critical Points of Index 0 and 1.

Consider a smooth triad (wn; V, Vt). We will always assume that it carries a 'self indexingt Morse function f (see

4.9) and an associated gradient-like vector field 5. Let

Theorem 8.1

Index 0) If H (\q, V) = 0, the critical points of index 0 can 0 be cancelled against an equal number of critical points of index 1.

Index 1) Suppose W and V are simply connected and n >- 5.

If there are no critical points of index 0 one can insert for

each index 1 critical point a pair of auxiliary index 2 and index 3 critical points and cancel the index 1 critical

points against the auxiliary index 2 critical points. ('Thus

one 'tradee' the critical points of index 1 for an equal u . number of critical points of index 3. )

Remark: The method we used to cancel critical points of index

2 <- A <- n - 2 in Theorem 7.8 fails at index 1 for the follow-

ing reason. Ihe Second Cancellation Theorem 6.4 holds for A = 1, n >- 6. (see page 70) , but we woad want to apply it where the simple connectivity assumption of 6.4 is spoiled by the presence

of several index 1 critical points.

Proof for Index 0;

0 If in ' V we can always find SRn-l and SL intersecting o+ in a single point, then the proof will follow from 4.2, 5.4 (me

Nrst Cancellation he or em) and a finite induction (cf. proof for index 1 below). Consider homology with coefficients in z2 = 2/22. Since H~(w,V; Z2) = 0, by Theorem 7.4, a qwl? Wo; z2) --> H(W~,V; z2) is onto. BU~a ia ci.early given by the matrix of intersection numbers modulo 2 of the right- hand (n - 1)-spheres and left-hand 0-spheres in vo+* Hence for any SR there is at least one S: with SR- s; + 0 n-1 mod 2. lllis says SR n SOL consists of an odd number of points which con only be 1. This completes the proof for index 1.

To construct auxlllaay critical points we will need

Given 0 <- < n, there exists a smooth map f : Rn -> R SO that f(5,..., xn) = 5 outside of a compact set; and so that f has just two critical points pl, pp, non-degenerate? of indices A, A + 1 respectively with f (pl) < f (pp ).

n-A-1. Proof: We identify R~ with R x R~ x R , and denote a general point by (x, y, z). Let y2 be the square of the length of y E R~.

Cboose a function s(x) with compact support so that

x + S(X) has two non-degenerate critical points, say xo, xl.

Mgure 8 .1 Mrst consider the function x + B(X)- y 2 + z2 on R~. 'Ihis has two non-degenerate critical points (xo, 0, 0) and (xl, 0, 0) with the correct indices,

Now "taper" this function off as follows. Choose three smooth f'unctionc a, p, y : R -> R+ with compact support so that i

2) bl(t)1 < l/Max (~(x)( for all t (Primes denote derivatives. ) X 3) p(t) = 1 whenever a(t) # 0. 4) y(x) = 1 whenever el(x) # 0.

Ngure 8.2

Now let 2 2 2 2 2 f = x + e(x) a(y + z ) + y(x)(-y + z ) @(y2 + z 1. Note that

(a) f - x has compact support

(b) Within the interior of the region where a = 1 (hence

$ = 1) and y = 1 this is our old function, with the old critical points.

Ihe third term has absolute value < 1 by (5). Hence if 2 2 a sl(x) = 0 or a(y + z ) = 0 we have # 0. Thus we must

only look at the region where s '(x) # 0 (hence Y = 1) and 2 2 a(y + z ) # 0 (hence f3 = 1) to look for critical points.

(d) Within the region y = 1, p = 1 we have grad (f) =

+ z2) + 1)). But s(x) at(y2 + z2) -+ 1 # 0 by (2). Hence the gradient can vanish only when y = 0, z = 0, and therefore a = 1. But this case has already been described in (b).

Proof 8.1 for Index 1: 'Ihe given situation may be represented schematically

v V1+ v2+ etc.

index 1 2 3

The first step of the proof is to conetruct, for any right-hand

(n - 2)-sphere in V1+ of a critical point p, a suitable 1-sphere to be the left-hand sphere of the index 2 critical point that 'will cancel p.

Lemma 8.3

If SRn-2 is a right-hand sphere in V1+, there exists a 1-sphere n-2 imbedded in V1+ that has one treneverae intersection with SR and meets no other right-hand aphere.

Proof:

Certainly there exists a small imbedded 1-disc D C Vl+ , n-2 which, at its midpoint go, transversely intersects SR , and which has no other intersection ~ithright-hand spheres. Pans- late the end points of D left along the trajectories of to a pair of points in V. Since V is connected, and of dimension n - 1 >- 2, these points may be joined by a smooth path in V which avoids the left hand 0-sphere8 in V. %is path may be translated back to a smooth path that joins the end points of D

in V1+ and avoids all right-hand spheres. NOW one can easily

construct a smooth map g : $ -> vl+ such that

neighborhood A of a onto a neighborhood of qo in D. (b) g(~l - a) meets no right-hand (n - 2)-sphere. Since dim V = n - 1 2 3, Whitneyls theorem 6.12 provides a smooth imbedding with these properties. This completes the proof

We will need the following corollary of 'Iheorems 6.11, 6.12.

If two smooth imbeddings of a smooth manifold -dinto a smooth

manifold N~ are homotopic, then they are smoothly isotopic

provided n >- 2m + 3.

Remark: Actually 8.4 holds with n >- 2m + 2 (see Whitney [16] )

Proof of Theorem 8.1 for Index ~&~~ti~~~d:

Notice that V2+ is always simply connected. In fact the

inclusion V2+C W factors into a sequence of inclusions that are alternately inclusions associated with cell attachments and

inclueions that are homotopy equivalences. (see 3.14 ). Be cells

I attached are of dimension n - 2 and n - 1 goiog to the left and of dimension 3, 4, .. . going to the right. lhus V2+ is (09 connected since W is, and n (V ) = nl(w) = 1 (cf. ~~~~k 1 1 2t page 70). Given any critical point p of index 1, we con-

struct an 'idealt 1-sphere S in V1+ as in Lema 8.3. After

adjusting E if necessary to the right of V2+ we may assume

that S meets no left hand 1-snheres in v~+. (see 4.6, 4.7).

!Ben we can translate S right to a 1-sphere S1 in V2+.

In a collar neighborhood extending to the right of V2+, we can choose co-ordinate functions embedding an ~1,.. . , x n open set U into R~ so that f 1 u = xn (cf . proof of 2.9). Use Lemma 8.2 to alter f on a cowact subset of U inserting

a pair q, r, with f (q) < f (r), of 'auxiliary1 critical points

of index 2 and 3. (see Mgure 8.3).

Figure 8.3

l Let S2 be the left-hand 1-sphere of q in V2+ . Since

is simply connected, and 5.8 imply that th~reis an V2+ 8.4 Sg isotopy of the identity V2+ -' V2+ that carries to S1 . Thus after an adjustment of 5 to the right of V2+ (see 4.7), the left-hand sphere of q i~ V2+ will be S1 . Then the

left-hand sphere of q in "1+ is S, which, by construction

intersects the right-hand sphere of p in a single point, transversely,

Without changing alter f (by 4.2) on the interiors 1 1 of f-l[$, 1 p] and of f-l[l g k], k = (f(q) + f(r))/2, increasing the level of p and lowering the level of q so that for some 6 > 0 1 1 +s

Now use the First Cancellation Theorem to alter f and 5 on

f-' [l + 6, 2 - 61 eliminating the two critical points p and

Q. Mnally move the critical level of r right to 3 (using

4.2).

We have now 'traded' p for r, and the process may be

repeated until no critical points of index 1 remain. This

completes the proof of lheorem 8.1. 59. The h-Cobordism lheorem and Some Applications.

Here is the theorem we have been striving to prove.

Theorem 9.1 The h-Cobordism Theorem

Suppose the triad (wn; V, V1) has the properties

1) W, V and V' are simply connected.

2) H,(w, V) = 0 3) dim w=n>6 -. '.&en W is diffemorphic to V X [O, 11

Remark: The condition 2) is equivalent to 2)' H*(w, V' ) = 0. * For H*(w, V) = 0 implies H (w, V' ) = 0 by ~oincargduality. * But H (w, V1) = 0 implies H*(w, V1) = 0. Similsrly 2)' implies 2).

-Proof : Choose a self-indexing Morse function f for (w; V, V1 ). lheorem 8.1 provides for the elimination of critical points of index 0 and 1. If we replace the Morse function f by -f the triad is 'turned about' and critical points of index A become critical points of index n - A. Thus critical points of (original) index n and n - 1 may also be eliminated. Now Theorem 7.8 gives the desired conclusion,

Definition 9.2 A triad (w; V, vI) = 0 is an h-cobordism and

V is said to be h-cobordant to V' if both V and V1 are

deformation retracts of W,

--Remaxk: It is an interesting fact (which we will not use) that an equivalent version of Theorem 9.1 is obtained if we substitute

for 2 ) the apparently stronger condition that (W; V1 V' ) be an h-cobordism. Actually 1) and 2) together imply that

(w; V, Vt ) is an h-cobordism. In fact

(i) %(v) = 0 , n1(w, V) = 0 , H*(w, V) = 0 together imply (ii) ni(w, V) = 0 i = 0, 1, 2, ... by the (relative ) Hurewicz isomorphism theorem (HU, [20, p. 1661; Hilton [21, p.1031). In view of the fact that (w, V) is a triangulable pair (Mmkres [ 5, p.101) ) (ii) implies that a strong deformation retraction W -> V can be constructed.

(see Hilton [21, p.98 llnn 1.71;) Since 2) implies H*(w, v') = 0,

Vt is, by the same argument, a (strong) deformation retract of

An important corollary of Theorem 9.1 is

Theorem 9.2

Tho simply connected closed smooth manifolds of dimension 4 >- 5 that are h-cobordant are diffeomorphic.

A Few Applications (see also Smale 1221 [6] ) Proposition A) Characterizations of the smooth n-disc D~,n >- 6. Suppose wn is a compact simply connected smooth n-manifold, n >- 6, with a simply connected boundary. men the following four assertions are equivalent.

1). wn is diffeomorphic to Dn.

2). wn is homeomorphic to Duo

3). wn is contractible.. 4). wn has the (integral) homology of a point. -Proof: Clearly 1) => 2) => 3) => 4). So we prove 4) => 1). If D is a smooth n-disc imbedded in Int W, then (W Int Do; 0 - BdDo, V) satisfies the conditions of the h-Cobordism iheorem.

In particular, (by excision) H,(W - Int Do, BdDo) H*(w, Do) = 0. Consequently the cobordism (wn; $, V) is a composition of

(D~;$, B~D~)with a product cobordism (W - Int Do; BdDo, V). It follows from 1.4 that W is diffeomorphic to Do.

Proposition B) The Generalized ~oincar; Conjecture in dimensions 2 5. (see Smale [21]. ) If I?, n >- 5, is a closed simply-connected smooth manifold with the (integral) homology of the n-sphere sn, then

8 is homeomorphic to 9". If n = 5 or 6, 8 is diffeomorphic to sn.

Corollary If a closed smooth manifold I8, n >_ 5, is a homotopy n-sphere (i.e. is of the hornotopy type of sn) then is homeomorphic to sn.

Remark: mere exist smooth 7-manifolds M~ that are homeo- 7 morphic to s7 but are not diffeomorphic to S . (See Milnor [241-

Proof of B Suppose first that n >- 6. If Do C M is a smooth

n-disc, M Int D satisfies the conditions of A). - 0 In particular

$-i(~, D~)(excision )

if i>0 (exact sequence ) Z if i=O (\o Consequently M = (M - Int D ) U D is diffeomorphic to a union 0 0 of two copies D: , D: of the n-disc with the boundaries identified under a diffeomorphism h : E~D: -> BdD2.n

Remark: Such a manifold is called a twisted sphere. Clearly every twisted sphere is a closed manifold with Morse number 2, and conversely.

?he proof is completed by showing that any twisted sphere n M = D: L& D: is homeomorphic to sn. Let gl : D; -> s be an imbedding onto the aouthern hemisphere of 5" C R~+' 1.e. the set (2 = 1, x < 0). Each point of D: may be n+l - I n written tv, 0 <- t <- 1, v E BdD2. Define g : M -> S by

rrt -1 nt (ii) g(tv) = sin g (h (v)) cos en+l vhere en+l -2 1 + n+l n = (0, .. . , 0, 1) E R , fur all points tv in D2. Then g is a well defined 1-1 continuous map,onto sn , and hence is a homeomorphism. This completes the proof for n >- 6.

If n = 'j we use:

Theorem 9.1 (~ervaireand Milnor [25], Wail [26 ] )

Suppose M" is a closed, simply connected, smooth manifold with the homology of the n-sphere SO. ?hen if n = 4, 5, or 6, # bounds a smuoth, compact, contractible manifold.

Then A) implies that for n = 5 or 6 8 is actually diffeomorphic to sn.

Proposition C Characterization of the 5-disc

Suppose W' is a compact simp1y connected smooth manifold that has the (integral ) homology of a point. Let V = EdW. 4 5 1) If V is diffeomorphic to S then W is diffeomorphic to D . 2) If V is homeomorphic to S4 then W is homeomorphic to

Proof of 1) Form a smooth 5-manifold M = W % D~ where h 4 is a diffeomorphism V ->B~D~ = S . Then M is a simply connected manifold with the homology of a sphere. In B) we proved that M is actually diffeomorphic to S 5 . Now we use

Theorem 9.6 (palais [271, Cerf [28L Milnor [12, p.U] )

Any two smooth orientation-preserving imbeddings of an n-disc into a connected oriented n-manifold are ambient isotopic.

Tbus there is a diffeomorphism g : M -> M that maps 5 D C M onto a disc Dl5 such that D: = M - ~nt D: is also a disc. lhen g maps W C M diffeomorphically onto D:.

Proof of 2) Consider the double D(W) of W (i .e. two copies of W with the boundaries identified - see Munkres [5, p. 54 1 1. Ihe submanifold V C D(W) has a bicollar neighborhood in D(w), and D(W) is homeomorphic to s5 by B). Brown [23] has proved :

Theorem 9.7 If an (n - 1 )-sphere C, topologically imbedded in sn, has a bicollar neighborhood, then there exists a hwe- n omorphism h : 5" -> S that maps C onto snelC sn. lhus sn - E has two components and the closure of each is an n-disc with boundary C.

It follows that W is homeomorphic to D5 . Wscompletes the proof of C). Proposition D) The Differentiable Schoenfliess Theorem in Dimensions >- -5. Suppose z is a smoothly imbedded (n - 1)-sphere in sn. If

n 2 5, there is a smooth ambient isotopy that carries C onto the equator snw1C sn.

Proof : sn - C has two components (by Alexander duality ) and

hence 3.6 shows that C is bicollared in sn. Ihe closure in

S" of a component of sn - C is a smooth simply connected manifold Do with boundary C. and with the (integral) homo-

logy of a point. Fbr n 2 5, Do ia actually diffeomorphic

to D' by A) and C). lhen the theorem of Palais and Cerf

(9.6) provides an ambient isotopy that carries Do to the

lower hemisphere and hence BdD = C to the equator. 0

n Remark: This shows that if f : sn-' ->-S is a smooth 4 imbedding, then f is smoothly isotopic to a map onto

C snj but it is not in general true that f is smoothly n-1 "n isotopic to the inclusion i : S .- - . It is false if n-l f = i 0 g, where g : S -, sn-l is a diffeomorphism n which does not extend to a diffeomorphism Dn -> D . (The

reader can easily show that g extends to Dn if am3 only if

the twisted sphere D" U D~ is diffeomorphic to sn.) In 1 g 2 fact if f ie smoothly isotopic to i, by the Isotopy Exten- n sion meorem 5.8, there exists a diffeomorphism d : sn -> S

such thag d 0 i = f = i 0 g. This gives two extensions of g n to a diffemrphism Dn -> D . Concluding Remarks :

It is an open question whether the h-Cobordism Theorem is true for dimensions n < 6. Let (w"; V, V1) be an h- cobordism where $ is simply connected and n < 6. n = 0, 1, 2: The theorem is trivial (or vacuous ). -n = 3: V and Vt must be 2-spheres. Then the theorem is easily deduced from the classical Poincar6 Conjecture : Every compact smooth 3-manifold which is homotopy equivalent to S3 is diffeomorphic to S3 . Since every twisted +sphere (see page 110) is diffeomorphic to El3 (see Smale [30], Mnkres [31] ) the theorem is actually equivalent to this conjecture. -n = 4: If the classical Poincard ConJecture is true V and V1 must be 3-spheres. Then the theorem is readily seen to be equivalent to the '4-Disk Conjecturet: Every compact contract-

ible smooth 4-manifold with boundary s3 is diffeomorphic to 4 -D . Now a difficult theorem of Cerf [29] says that every twisted 4 4-sphere is diffeomorphic to S . It follows that this conjecture is equivalent to: Every compact smooth 4-manifold which 4 is homotopy equivalent to s4 is diffe-rphic to S . n = 5: Proposition C) implies that the theorem does hold when 4 V and Vt are diffeomorphic to S . However there exist many types of closed simpl~connected 4-manifolds. Barden (unpublished )

showed that if there exists a diffeomorphism f : V1 -> V homotopic to r lV, where r : W -> V is a deformation

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