AN ABSTRACT OF A THESIS

DIFFERENTIAL TOPOLOGY AND THE H-COBORDISM THEOREM

Quinton Westrich

Master of Science in Mathematics

Differential topology is the study of smooth manifolds and smooth maps be-tween manifolds. The h-Cobordism Theorem provides a condition for determiningwhether two manifolds are diffeomorphic. This thesis presents some basic elementsof differential topology and a proof and discussion of the h-Cobordism Theorem.

DIFFERENTIAL TOPOLOGY AND THE H-COBORDISM THEOREM

A Thesis

Presented to

the Faculty of the Graduate School

Tennessee Technological University

by

Quinton Westrich

In Partial Fulfillment

of the Requirements for the Degree

MASTER OF SCIENCE

Mathematics

May 2010

Copyright c© Quinton Westrich, 2010

All rights reserved

CERTIFICATE OF APPROVAL OF THESIS

DIFFERENTIAL TOPOLOGY AND THE H-COBORDISM THEOREM

by

Quinton Westrich

Graduate Advisory Committee:

Alexander Shibakov, Chairperson date

Andrzej Gutek date

Jeffrey Norden date

Richard Savage date

Approved for the Faculty:

Francis OtuonyeAssociate Vice-President forResearch and Graduate Studies

Date

iii

DEDICATION

This thesis is dedicated to Indranu Suhendro.

iv

ACKNOWLEDGMENTS

I would like to thank Alexander Shibakov, my thesis advisor, and Connie Hood,

my undergraduate mentor.

v

TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Chapter

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Preliminary Definitions and Notation . . . . . . . . . . . . . 3

2. SMOOTH MANIFOLDS . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Differentiable Structures . . . . . . . . . . . . . . . . . . . . 8

2.2 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Whitney’s Imbedding Theorem . . . . . . . . . . . . . . . . 14

2.4 Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3. ALGEBRAIC TOPOLOGY . . . . . . . . . . . . . . . . . . . . . 20

3.1 The Fundamental Group . . . . . . . . . . . . . . . . . . . . 20

3.2 Singular Homology and Cohomology . . . . . . . . . . . . . 22

3.3 de Rham Cohomology . . . . . . . . . . . . . . . . . . . . . 28

4. CRITICAL POINTS . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Morse’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 39

vi

vii

Chapter Page

4.3 Sard’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4 Approximation Lemmas in Rn . . . . . . . . . . . . . . . . . 54

5. COBORDISMS AND ANALYSIS . . . . . . . . . . . . . . . . . . 58

5.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2 Morse Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.3 Elementary Cobordisms . . . . . . . . . . . . . . . . . . . . 64

5.4 Rearrangement of Cobordisms . . . . . . . . . . . . . . . . . 78

6. THE h-COBORDISM THEOREM . . . . . . . . . . . . . . . . . . 80

6.1 Cancellation Theorems . . . . . . . . . . . . . . . . . . . . . 80

6.2 Proof of the h-Cobordism Theorem . . . . . . . . . . . . . . 85

6.3 Applications of the h-Cobordism Theorem . . . . . . . . . . 86

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

APPENDIX: A THEOREM ON QUADRATIC FORMS . . . . . . . . . . . . 89

VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

LIST OF TABLES

Table Page

1.1 Results for the (Smooth) h-Cobordism Theorem in Each Dimension 3

viii

LIST OF FIGURES

Figure Page

2.1 The case n = 3, λ = 2. Here ϕ : S1 × B1 → S2. . . . . . . . . . . . 18

2.2 Surgery of type (2, 1) on S2. . . . . . . . . . . . . . . . . . . . . . 18

2.3 The case n = 3, λ = 1. Here ϕ : S0 × B2 → S2. . . . . . . . . . . . 19

2.4 Surgery of type (1, 2) on S2. . . . . . . . . . . . . . . . . . . . . . 19

5.1 The Manifold L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Level Surfaces of L1 . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.3 A gradient-like vector field on a triad (W ;V, V ′). . . . . . . . . . . 65

5.4 The neighborhoods U,U0, U1, U2 in the existence construction aredrawn schematically for the triad in Example 6. . . . . . . . . . 66

5.5 Integral curves for X. See [11] p. 147. . . . . . . . . . . . . . . . . 67

5.6 V0, V1, Vε, and V−ε are drawn schematically for a 2-manifold withµ = λ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.7 The left-hand sphere SL and the left-hand disk DL are shownschematically for a 2-manifold with µ = λ = 1. Here ϕL :−1, 1 × (−1, 1)→ V0. . . . . . . . . . . . . . . . . . . . . . . . 72

5.8 The right-hand sphere SR and the right-hand disk DR are shownschematically for a 2-manifold with µ = λ = 1. Here ϕR :(−1, 1)× −1, 1 → V1. . . . . . . . . . . . . . . . . . . . . . . . 73

5.9 Bounding Curves of L1 ⊂ R2 . . . . . . . . . . . . . . . . . . . . . 74

5.10 Bounding Surfaces of L2 ⊂ R3 . . . . . . . . . . . . . . . . . . . . 74

ix

CHAPTER 1

INTRODUCTION

1.1 Overview

The purpose of this thesis is to sketch a proof of the h-Cobordism Theorem.

Along the way, many diverse tools from analysis, algebra, topology, and geometry are

implemented.

In Chapter 1, preliminary definitions for multidimensional calculus are given.

Also some standard notations for special subsets of Euclidean space which appear

frequently in topology are established.

Chapter 2 provides an outline of some of the foundational results in differential

topology. The definition of a manifold is given, followed by the defintion for vector

bundles on manifolds. Of fundamental importance in differential topology is Whit-

ney’s Imbedding Theorem. A simple version is proved. The chapter concludes with

a section on surgery on manifolds. It will be shown later (in Chapter 5) that certain

cobordisms correspond to surgery on the boundary manifolds.

In Chapter 3, an outline of some major ideas and results in algebraic topology

are presented. In the first section, homotopy and homotopy groups are defined. In

the next section, singular homology is presented as well as some of the more general

ideas from homological algebra. The Eilenberg-Steenrod Axioms for Homology are

presented as a succinct presentation of some foundational properties of homology

theories. Chapter 3 closes with the development of de Rham cohomology as an

alternative example of a cohomology theory. The definition of orientation is given

here, and is used in the final chapter to prove the h-Cobordism Theorem.

1

2

Chapter 4 covers the essential facts about critical points needed in the proof of

the h-Cobordism Theorem. Since Morse Theory is the driving force for the analysis of

cobordisms, certain facts about critical points are paramount for the proof of the h-

Cobordism Theorem. Morse’s Lemma gives a coordinate system in the neighborhood

of a critical point of a map from a manifold to R such that the function is represented

in a particularly simple way. Sard’s Theorem says that the set of critical values has

measure zero in the codomain. Finally, some approximation lemmas are given which

will be used to show the existence of Morse functions in Chapter 5.

The main work of this thesis is contained in Chapter 5. Here, the basic def-

initions of cobordisms are given. Some results of Morse Theory are given and then

applied to cobordisms as a means of analyzing them. The notion of a gradient-like

vector field is introduced. The idea of the proof of the h-Cobordism Theorem is to

alter the gradient-like vector field on the cobordism so as to cancel irrelevant critical

points. The analytical setting for this procedure is developed here. In particular, cer-

tain imbedded spheres and disks are identified which are later used to cancel critical

points. The chapter concludes with a theorem which provides for rearranging critical

points so that a cobordism can be decomposed into a composition of cobordisms, each

with all its critical points on the same level and with the same index.

Chapter 6 presents the h-Cobordism Theorem, a sketch of the proof, and

some applications of the theorem. An outline of the proof is as follows. First, it is

shown that if the intersections of the embedded spheres are suitable, critical points on

particular indices can be cancelled or traded. Next, it is shown that, given particular

dimensional, homological, and homotopy requirements, it is always possible to make

the intersections suitable. Thus, one can always cancel critical points in the middle

dimensions. This is where the dimension requirement in the h-Cobordism Theorem

3

Dimension True/False Credit

0, 1, 2 True (trivial or vacuous)3 True Perelman (2003)4 ? ?5 False Donaldson (1987)≥ 6 True S.Smale (1962)

Table 1.1. Results for the (Smooth) h-Cobordism Theorem in Each Dimension

enters. Finally, critical points of indices 0 and 1 are cancelled. Once all the critical

points are cancelled, the Morse number of the cobordism is zero, and so it is a product

cobordism, i.e. diffeomorphic to a boundary manifold times the unit interval.

Results of the smooth h-Cobordism Theorem are presented in Table 1.1.

1.2 Preliminary Definitions and Notation

The purpose of this section is to establish notational conventions and to provide

some fundamental definitions for later reference.

The following set notations will be adopted for this thesis:

Rn = x = (x1, . . . , xn) : xi ∈ R, i = 1, . . . , n (Euclidean n-space)

Rn+ = (x1, . . . , xn) ∈ Rn : xn ≥ 0 (Euclidean half-space)

I = [0, 1] (unit interval)

Sn = x ∈ Rn+1 : ‖x‖ = 1 (n-sphere)

Dn = x ∈ Rn : ‖x‖ ≤ 1 (n-disk)

Bnε (x) = y ∈ Rn : ‖x− y‖ < ε (ε-ball centered at x).

4

Further, Bn1 (x) = Bn(x) and Bn

ε (0) = Bnε so that, for example,

Bn = x ∈ Rn : ‖x‖ < 1 (n-ball).

The general linear group of n× n nonsingular matrices will be denoted GL(n).

The definitions and theorems below are standard.

Definition 1. Let U ⊆ Rn be an open set. A function f : U → Rm is differentiable

at x ∈ U if there is a linear transformation λ : Rn → Rm such that

limh→0

∥∥∥f(x + h)− f(x)− λ(h)∥∥∥∥∥∥h∥∥∥ = 0.

It can be shown that if such a transformation exists, it is unique. This unique linear

transformation λ is denoted Df(x) and called the derivative of f at x. In the

canonical bases of Rn and Rm, the m × n matrix of Df(x) is called the Jacobian

matrix of f at x, and is denoted by J(f(x)).

Theorem 1. Let U ⊆ Rn be an open set. If f : U → Rm is differentiable at x ∈ U ,

then∂f i

∂xj

∣∣∣∣x

exists for all 1 ≤ i ≤ m and all 1 ≤ j ≤ n. Moreover,

J(f(x)) =

∂f 1

∂x1

∣∣∣∣x

∂f 1

∂x2

∣∣∣∣x

· · · ∂f 1

∂xn

∣∣∣∣x

∂f 2

∂x1

∣∣∣∣x

∂f 2

∂x2

∣∣∣∣x

· · · ∂f 2

∂xn

∣∣∣∣x

......

. . ....

∂fm

∂x1

∣∣∣∣x

∂fm

∂x2

∣∣∣∣x

· · · ∂fm

∂xn

∣∣∣∣x

.

Conversely, f is differentiable at x if each partial derivative∂f i

∂xj

∣∣∣∣x

exists and is

continuous in a neighborhood of x.

5

Definition 2. Let A ⊆ Rn be any subset of Rn. A function f : A → Rm is said to

be differentiable if there is a neighborhood U of A and an extension f of f such

that f is differentiable at each point in A. If there is a neighborhood of A for which

every partial derivative of order k exists and is continuous, then f is said to be of

class Ck(A), and one writes f ∈ Ck(A). If f ∈ Ck(A) for all k ∈ N, f is said to be

smooth, and one writes f ∈ C∞(A). (Often one omits the A and simply says that f

is Ck, etc.) If f is smooth and has a smooth inverse, it is called a diffeomorphism.

The following is a generalization of Taylor’s Theorem for functions of a single

variable. It provides a power series representation of a Ck function f at a point x,

given the value of f and its derivatives up to order k − 1 at some point x0 nearby.

This form of Taylor’s Theorem is used to prove Sard’s Theorem below.

Theorem 2 (Taylor’s Theorem). Suppose D ⊆ Rn is an open subset of Rn, f : D → R

is in Ck(D), the points x,x0 ∈ D, and the line segment connecting x and x0 is

contained in D, i.e., tx + (1− t)x0 : t ∈ I ⊆ D. Then

f(x) = f(x0) +n∑i=1

∂f

∂xi

∣∣∣∣x0

(xi − xi0) +1

2!

n∑i,j=1

∂2f

∂xi∂xj

∣∣∣∣x0

(xi − xi0)(xj − xj0) + · · ·+

+1

(k − 1)!

n∑i1,...,ik−1=1

∂k−1f

∂xi1 · · · ∂xik−1

∣∣∣∣x0

(xi1 − xi10 ) · · · (xik−1 − xik−1

0 ) +Rk(x),

(1.1)

where

Rk(x) =1

k!

n∑i1,...,ik=1

∂kf

∂xi1 · · · ∂xik

∣∣∣∣tx+(1−t)x0

(xi1 − xi10 ) · · · (xik − xik0 ), (1.2)

for some t ∈ (0, 1).

6

Proof. The idea of the proof is to simply draw a ray from x0 to x and let this ray

act as the domain of f , so that f is now a function R → R and the single variable

version of Taylor’s Theorem can be applied. Here are the details1.

Define h = x− x0 so that h points from x0 to x. Define the curve γ : I → Rn

by γ(t) = x0 + th, where I ⊆ R is the open subset I = t ∈ R : x0 + th ∈ D. Note

that [0, 1] ⊆ I. Let φ = f γ : I → R. Since γ is C∞ and f is Ck, it follows that φ

is Ck. Applying the chain rule to φ gives

φ′(t) =n∑i=1

∂f

∂xi

∣∣∣∣γ(t)

· γ′i(t) =n∑i=1

∂f

∂xi

∣∣∣∣γ(t)

hi

φ′′(t) =d

dt

(n∑i=1

∂f

∂xi

∣∣∣∣γ(t)

hi

)=

n∑i=1

∂

∂xi

(d(f γ)

dt

∣∣∣∣t

)hi =

n∑i,j=1

∂2f

∂xi∂xj

∣∣∣∣γ(t)

hihj

...

φ(k)(t) =n∑i=1

∂

∂xi(φ(k−1)(t)

)hi =

n∑i1,...,ik=1

∂kf

∂xi1 · · · ∂xik

∣∣∣∣γ(t)

hi1 · · ·hik .

Now, by Taylor’s Theorem for a single variable, there exists s ∈ (0, 1) such that

φ(1) = φ(0) + φ′(0) +1

2!φ′′(0) + · · ·+ 1

(k − 1)!φ(k−1)(0) +Rk(1),

where

Rk(1) =1

k!φ(k)(s).

Noting that φ(1) = f(x), φ(0) = f(x0), and γ(0) = x0, we obtain the desired formulas

(1.1) and (1.2).

1c.f. [3] pp. 86 and 94

7

The Implicit Function Theorem has been framed in a variety of ways. That

herein is similar to the one in [5] pp. 223—225. First, some definitions are introduced

which are needed for the statement of the theorem.

Definition 3. Let U ⊆ Rm be an open set, x ∈ U , and f : U → Rm. If there is

a neighborhood V of x such that f |V has a smooth inverse, then f is called a local

diffeomorphism at x. A local diffeomorphism at 0 is a called a local coordinate

system at f(0).

Theorem 3 (Implicit Function Theorem). Let U ⊆ Rm be an open set, x ∈ U , and

f : U → Rm be a smooth map. If rank J(f(x)) = m, then f is a local diffeomorphism

at x.

The theorem below is an immediate consequence of the Implicit Function The-

orem.

Theorem 4. Let U ⊆ Rm be an open set, x ∈ U , and f : U → Rm be a smooth map.

If rank J(f(x)) is constant on a neighborhood of x, then there is a local coordinate

system g at x and a local coordinate system h at f(x) such that

h−1fg(x1, . . . , xm) = (x1, . . . , xk,0).

This simply says that, given the hypothesis, there are coordinate systems for

which f is just a projection Rm → Rk followed by the inclusion Rk → Rn.

CHAPTER 2

SMOOTH MANIFOLDS

Smooth manifolds and smooth maps between manifolds are the objects of study

in differential topology. A manifold is a topological space which locally “looks like”

Rn. In Section 2.1 manifolds and smooth maps will be formally introduced along with

several other definitions which provide a foundation for the differential topology in

this thesis. In Section 2.2 an introduction to vector bundles is given. Vector bundles

provide additional structure on manifolds and are used here as a tool for the study

of the manifolds themselves.

2.1 Differentiable Structures

Definition 4. Let V ⊆ Rm and f : V → Rm. We say that f is smooth or differ-

entiable of class C∞ if f can be extended to a map g : U → Rm, where V ⊆ U

and U is open in Rn, such that the partial derivatives of g of all orders exist and are

continuous.

Definition 5. A (topological) manifold is a metric space M for which there is

an integer n ≥ 0 such that if x ∈ M , there is a neighborhood U of x such that U is

homeomorphic to Rn or Rn+. The boundary of M , denoted ∂M , is the set of all

points in M which do not have neighborhoods homeomorphic to Rn.

Definition 6. Suppose U and V are two open subsets of a manifold M and

x : U → x(U) ⊂ Rn and y : V → y(V ) ⊂ Rn

8

9

are two homeomorphisms. Then x and y are called C∞-related if the maps

y x−1 : x(U ∩ V )→ y(U ∩ V )

x y−1 : y(U ∩ V )→ x(U ∩ V )

are C∞.

Definition 7. An atlas for a manifold M is a family of mutually C∞-related home-

omorphisms whose domains cover M . A particular member (x, U) of an atlas A is

called a chart for the atlas A, or a coordinate system on U .

Lemma 1. If A is an atlas of C∞-related charts on a manifold M , then A is contained

in a unique maximal atlas A′ for M .

Definition 8. A C∞ manifold, or differentiable manifold, or smooth mani-

fold, is a pair (M,A), where A is a maximal atlas for M .

Definition 9. Let (M,A) and (N,B) be two C∞ manifolds. We say that (M,A)

and (N,B) are diffeomorphic if there is a bijective function f : M → N , called a

diffeomorphism, such that

x ∈ B ⇐⇒ x f ∈ A.

Definition 10. Let (M,A) be a differentiable manifold and N be an open submanifold

of M . Then we can define a differentiable manifold (N,A′), called a C∞ submani-

fold of M , where the atlas A′ consists of all (x, U) in A with U ⊂ N .

Definition 11. Let M be a smooth n-manifold and N be a smooth m-manifold. A

function f : M → N is called differentiable if for every coordinate system (x, U)

for M and (y, V ) for N , the map y f x−1 : x(U) ⊆ Rn → Rm is differentiable.

10

Definition 12. Let M be a smooth n-manifold and N be a smooth m-manifold. A

function f : M → N is called differentiable at p ∈ M if y f x−1 : Rn → Rm

is differentiable at x(p) for coordinate systems (x, U) and (y, V ) with p ∈ U and

f(p) ∈ V .

Definition 13. A function f : M → R is called differentiable iff f x−1 is differ-

entiable for each chart x, i.e. iff it is differentiable as a map between manifolds with

the usual atlas on R (the maximal atlas generated by the identity on R).

Lemma 2. We have the following:

1. A function f : Rn → Rm is differentiable as a map between C∞ manifolds

iff it is differentiable in the usual sense.

2. A function f : M → Rm is differentiable iff each f i : M → R is differen-

tiable.

3. A coordinate system (x, U) is a diffeomorphism from U to x(U).

4. A function f : M → N is differentiable iff each yi f is differentiable for

each coordinate system y of N .

5. A differentiable function f : M → N is a diffeomorphism iff f is bijective

and f−1 : N →M is differentiable.

2.2 Vector Bundles

Definition 14. An n-dimensional vector bundle (or n-plane bundle) is a 5-

tuple ξ = (E, π,B,⊕,), such that

(i) E and B are topological spaces

(ii) π : E → B is continuous and surjective

(iii) ⊕ :⋃p∈B

π−1(p)× π−1(p)→ E is such that ⊕ (π−1(p)× π−1(p)) ⊂ π−1(p)

(iv) : R× E → E is such that (R× π−1(p)) ⊂ π−1(p)

11

(v) ⊕ and make each fibre π−1(p) into an n-dimensional vector space over R

(vi) For each p ∈ B, there is a neighborhood U of p and a homeomorphism

t : π−1(U)→ U × Rn

which is also a vector space isomorphism for each π−1(q) onto q×Rn, for all

q ∈ U .

E is called the total space and B is called the base space. Condition (vi) is called

local triviality.

Definition 15. Two vector bundles ξ1 = π1 : E1 → B and ξ2 = π2 : E2 → B

are called equivalent, written ξ1 ' ξ2, if there is a homeomorphism h : E1 → E2

which takes each fibre π−11 (p) isomorphically onto π−1

2 (p). The map h is called an

equivalence.

Definition 16. A bundle map from a vector bundle ξ1 to a vector bundle ξ2 is a

pair of continuous maps (f , f), with f : E1 → E2 and f : B1 → B2, such that

(i) the following diagram commutes

B1 B2f//

E1

B1

π1

E1 E2f // E2

B2

π2

(ii) for every p ∈ B1, the map

f |π−11 (p) : π−1

1 (p)→ π−12 (p)

is linear.

12

Definition 17. A section of a vector bundle π : E → B is a continuous function

s : B → E such that π s = idB.

Definition 18. Let M be an n-manifold with p ∈M . A tangent vector at p is an

operation Xp which assigns to each differentiable function f : U → R, where U ⊆ M

is a neighborhood of p, a real number, subject to

1. If g is a restriction of f , Xp(g) = Xp(f).

2. For all α, β ∈ R, Xp(αf + βg) = αXp(f) + βXp(g).

3. Xp(f · g) = Xp(f) · g(p) + f(p) ·Xp(g), where the dot denotes multiplication

in R.

Lemma 3. Let (x1, . . . , xn) be a coordinate system about p ∈M and Xp be a tangent

vector at p. Then Xp may be written uniquely as a linear combination of the operators

∂

∂xi

∣∣∣∣p

:

Xp =n∑i=1

αi∂

∂xi

∣∣∣∣p

.

Definition 19. For each p ∈ M , the tangent vectors at p form an n-dimensional

vector space TMp (and

∂

∂xi

∣∣∣∣p

n

i=1

form a basis for TMp, by Lemma 3). Let

TM =⋃p∈M

TMp.

Define π : TM → M as mapping a tangent vector Xp at p to p. Local triviality is

provided by the map tU : π−1(U) → U × Rn defined as follows. If Xp ∈ π−1(U),

then p ∈ U and Xp =n∑i=1

αi∂

∂xi

∣∣∣∣p

for some αi ∈ R and some coordinate system

13

(x1, . . . , xn) on U . Set

tU(Xp) = (p, α1, . . . , αn).

A topology on TM is induced by requiring that tU is a homeomorphism. Since

t−1V tU : (U ∩ V )× Rn → (U ∩ V )× Rn

is a homeomorphism, this topology is unambiguously determined. Also, t is a vector

space isomorphism for each fibre π−1(p) 7→ p × Rn. This forms a vector space

bundle on M called the tangent bundle.

Definition 20. A section of TM is called a vector field on M .

(The tangent bundle can be given a smooth manifold structure and so one can

also define smooth vector fields. See e.g. [11] Ch.3.)

Definition 21. If f : M → N is a smooth map of smooth manifolds, then the

differential of f at p ∈M is the function

f∗ : TMp → TNf(p) defined by (f∗Xp)(g) = Xp(g f),

for each g : N → R. (The differential f∗ is often written df .)

Definition 22. A 1-parameter group of diffeomorphisms of a manifold M is

a C∞ map ϕ : R×M →M such that

1. for each t ∈ R, the map ϕt : M → M defined by ϕt(q) = ϕ(t, q) is a

diffeomorphism of M onto itself,

2. for all t, s ∈ R, one has ϕt+s = ϕt ϕs.

14

Given a 1-parameter group ϕ of diffeomorphisms of M , a vector field X on M gen-

erates the group ϕ if every smooth map f : M → R obeys

Xq(f) = limh→0

f(ϕh(q))− f(q)

h.

Lemma 4. A smooth vector field on M which vanishes outside of a compact set

K ⊆M generates a unique 1-parameter group of diffeomorphisms on M .

A proof is given in [7] on pp.10-11. In particular, if M is compact, a smooth

vector field on M generates a unique 1-parameter group ϕ of diffeomorphisms on M .

Given q ∈M , a 1-dimensional submanifold of M , called an integral curve, is given

by the set p ∈M : p = ϕt(q) for some t ∈ R.

2.3 Whitney’s Imbedding Theorem

Definition 23. An immersion is a differentiable map f : M → N such that

rank (f) = dim (M) at each p ∈M .

Definition 24. An imbedding is an injective immersion which is a homeomorphism

onto its image.

Theorem 5. Any compact manifold can be imbedded smoothly into a Euclidean space.

Proof. Let M be a compact n-dimensional manifold. Since M is compact, there is

a finite cover Uiki=1 which gives a coordinate chart (Ui, xi)ki=1 for M . There is

a refinement U ′iki=1 furnishing a partition of unity ψiki=1 subordinate to Uiki=1.

Define the map f : M → Rnk+k by

f = (ψ1 · x1, ψ1 · x2, . . . , ψk · xk, ψ1, . . . , ψk).

15

Then rank (f) = n since for each p ∈M , p is in some Ui so that

(∂fα

∂xβi

)=

∂

∂x1i

(ψ1 · x1) · · ·∂

∂xni(ψ1 · x1)

∂

∂x1i

(ψ1 · x2) · · ·∂

∂xni(ψ1 · x2)

.... . .

...

∂

∂x1i

(ψ1 · xn) · · · ∂

∂xni(ψ1 · xn)

......

...

∂

∂x1i

(ψi · x1) · · ·∂

∂xni(ψi · x1)

∂

∂x1i

(ψi · x2) · · ·∂

∂xni(ψi · x2)

.... . .

...

∂

∂x1i

(ψi · xn) · · · ∂

∂xni(ψi · xn)

......

...

∂

∂x1i

(ψk · xk) · · ·∂

∂xni(ψk · xk)

∂

∂x1i

(ψ1) · · · ∂

∂xni(ψ1)

.... . .

...

∂

∂x1i

(ψk) · · · ∂

∂xni(ψk)

=

∂

∂x1i

(ψ1 · x1) · · · ∂

∂xni(ψ1 · x1)

∂

∂x1i

(ψ1 · x2) · · · ∂

∂xni(ψ1 · x2)

.... . .

...

∂

∂x1i

(ψ1 · xn) · · · ∂

∂xni(ψ1 · xn)

......

...

1n×n

......

...

∂

∂x1i

(ψk · xk) · · · ∂

∂xni(ψk · xk)

∂

∂x1i

(ψ1) · · · ∂

∂xni(ψ1)

.... . .

...

∂

∂x1i

(ψk) · · · ∂

∂xni(ψk)

.

So for each p ∈ M , one of the n × n blocks of the Jacobian is the n × n identity

matrix. This shows that f is an immersion of M into Rnk+k.

It is also easy to see that f is injective. Suppose f(p) = f(q). Then there is a

U ′i such that p ∈ U ′i . Thus ψi(p) = 1. Since f(p) = f(q), one has by the definition of

16

f that ψi(q) = 1. Thus q ∈ Ui. Now, since f(p) = f(q) it must be the case that

ψi · xi(p) = ψi · xi(q)

=⇒ ψi(p)xi(p) = ψi(q)xi(q)

=⇒ xi(p) = xi(q)

=⇒ p = q,

since xi is injective on Ui. It is therefore seen that f is an imbedding of M into the

Euclidean space Rnk+k.

Theorem 6 (H. Whitney, 1936). Let f : M → N be a smooth map which is an

embedding on a closed subset F ⊆ M and let ε : M → R be a positive continuous

function. If dimN ≥ 2dimM + 1, then there exists an embedding f : M → N

ε-approximating f and such that f |F = f |F .

2.4 Surgery

One way of inducing a topological change on a manifold is achieved by per-

forming “surgery.” In the definition below and in the rest of this thesis, q denotes

disjoint union.

Definition 25. Given a manifold V of dimension n− 1 and an embedding

ϕ : Sλ−1 × Bn−λ → V

let χ(V, ϕ) denote the quotient manifold obtained from the disjoint sum

(V r ϕ(Sλ−1 × 0))q (Bλ × Sn−λ−1)

17

by identifying ϕ(u, θv) with (θu,v) for each u ∈ Sλ−1, v ∈ Sn−λ−1, and θ ∈ (0, 1).

If V ′ denotes any manifold diffeomorphic to χ(V, ϕ) then we will say that V ′ can be

obtained from V by surgery of type (λ, n− λ).

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

sphere of dimension λ − 1 and replacing it by an embedded sphere of dimension

n− λ− 1.

Example 1 (Surgery of type (2, 1) on S2). From the definition, n = 3, λ = 2, V = S2.

Thus ϕ : S1 × B1 → S2 (see Figure 2.1) and

χ(S2, ϕ) = (S2 r ϕ(S1 × 0))q (B2 × S0)/ ϕ(u, θv) = (θu, v) ,

where u ∈ S1, v ∈ S0, and θ ∈ (0, 1). The effect of the surgery then is to first cut out

a 1-sphere from S2 leaving two components, each diffeomorphic to B2. It then glues

a different pair of B2’s to each component so as to create two S2’s. See Figure 2.2.

Therefore, χ(S2, ϕ) is diffeomorphic to S2 q S2.

Example 2 (Surgery of type (1, 2) on S2). We have n = 3, λ = 1, and V = S2. So

ϕ : S0 × B2 → S2 and

χ(S2, ϕ) = (S2 r ϕ(S0 × 0))q (B1 × S1)/ ϕ(u, θv) = (θu,v) ,

where u ∈ S0, v ∈ S1, and θ ∈ (0, 1). χ(S2, ϕ) is diffeomorphic to a 2-torus T2. See

Figures 2.3 and 2.4.

18

Figure 2.1. The case n = 3, λ = 2. Here ϕ : S1 × B1 → S2.

Figure 2.2. Surgery of type (2, 1) on S2.

19

Figure 2.3. The case n = 3,λ = 1. Here ϕ : S0 × B2 → S2.

Figure 2.4. Surgery of type (1, 2) on S2.

CHAPTER 3

ALGEBRAIC TOPOLOGY

3.1 The Fundamental Group

Definition 26. If X and Y are topological spaces, then a homotopy of maps from X

to Y is a map F : X×I → Y . Two maps f0, f1 : X → Y are said to be homotopic if

there exists a homotopy F : X×I → Y such that F (x, 0) = f0(x) and F (x, 1) = f1(x)

for all x ∈ X.

The relation “f is homotopic to g” is an equivalence relation on the set of

maps from X to Y and is denoted f ' g.

Definition 27. A map f : X → Y is said to be a homotopy equivalence with

homotopy inverse g if there is a map g : Y → X such that g f ' idX and

f g ' idY . In this case X and Y are said to have the same homotopy type and

the notation X ' Y is used.

Definition 28. A space is said to be contractible if it is homotopy equivalent to the

one-point space.

Definition 29. A topological subspace A ⊆ X is called a strong deformation

retract of X if there is a homotopy F : X × I → Y (called a deformation) such

that:

1. F (x, 0) = x,

2. F (x, 1) ∈ A,

3. F (a, t) = a for all a ∈ A and all t ∈ I.

It is just a deformation retract if the last equation is required only for t = 1.

20

21

Definition 30. If A ⊆ X then a homotopy F : X × I → Y is said to be relative to

A (or relA) if F (a, t) is independent of t for a ∈ A. A homotopy that is relX is said

to be a constant homotopy.

Definition 31. If F : X × I → Y and G : X × I → Y are two homotopies such that

F (x, 1) = G(x, 0) for all x, then define a homotopy F ∗ G : X × I → Y , which is

called the concatenation of F and G, by

(F ∗G)(x, t) =

F (x, 2t), if t ≤ 12

G(x, 2t− 1), if t ≥ 12

Definition 32. If F : X × I → Y is a homotopy, then the inverse homotopy of

F is F−1 : X × I → Y given by F−1(x, t) = F (x, 1− t).

If A ⊆ X and B ⊆ Y then maps which carry A into B are denoted (X,A)→

(Y,B). Let [X;Y ] denote the set of homotopy classes of mapsX → Y , and [X,A;Y,B]

denote the set of homotopy classes of maps (X,A)→ (Y,B) such that A goes into B

during the entire homotopy. If A = x0 and B = y0 and the points x0 and y0 are

understood, one writes simply [X;Y ]∗ instead of [X, x0 ;Y, y0].

Definition 33. Let (X,A) = (S1, ∗), where ∗ ∈ S1. Then if [f ], [g] ∈ [S1, Y ]∗,

concatenation induces a well-defined product on homotopy classes by

[f ] · [g] = [f ∗ g].

It can be shown that [S1, Y ]∗ forms a group under the induced product, where inverses

are provided by homotopy classes of homotopy inverses, and the identity is given by

the homotopy class of the constant homotopy. This group is called the fundamental

22

group, or Poincare group or first homotopy group, and is denoted

π1(Y, y0) = [S1;X]∗.

Definition 34. A topological space X is said to be arcwise connected if for any

two points p, q ∈ X, there exists a map ϕ : I → X with ϕ(0) = p and ϕ(1) = q.

Definition 35. An arcwise connected space X with π1(X, x0) = 1 is called simply

connected.

A simply connected space is actually independent of the choice of x0.

3.2 Singular Homology and Cohomology

Definition 36. The standard n-simplex is the convex set ∆n ⊆ Rn+1 consisting

of all (n+ 1)-tuples (t0, . . . , tn) of real numbers with

ti ≥ 0, t0 + t1 + · · ·+ tn = 1.

Any continuous map σ : ∆n → X, where X is a topological space, is called a singular

n-simplex in X. The ith face of a singular n-simplex σ : ∆n → X is the singular

(n− 1)-simplex

σ φi : ∆n−1 → X,

where the linear imbedding φi : ∆n−1 → ∆n is defined by

φi(t0, . . . , ti−1, ti+1, . . . , tn) = (t0, . . . , ti−1, 0, ti+1, . . . , tn).

23

Definition 37. A left R-module RG consists of an abelian group G, a ring R, and

a mapping R × G → G, denoted by juxtaposition, such that for all a, b ∈ G and

r, s ∈ R,

1. (a+ b)r = ar + br

2. a(r + s) = ar + as

3. a(r · s) = (ar)s

4. a1 = a.

(A right R-module GR is defined symmetrically. See [6] p. 14.) A left R-module

RG is called free if it has a basis gi : i ∈ I ⊆ G, such that every element g ∈ G

can be written uniquely in the form

g =∑i∈I

rigi,

where ri ∈ R and all but a finite number of the ri are 0. (See [6] p. 81.) A homo-

morphism (R-homomorphism) φ : RG → RH is a group homomorphism of G

into H which satisfies the extra condition

φ(rg) = rφ(g),

for all g ∈ G and r ∈ R. (See [6] p. 15–16.)

Definition 38. For each n ≥ 0, the singular chain group Cn(X;R) with coeffi-

cients in the commutative ring R is the free (left) R-module having one generator [σ]

for each singular n-simplex σ in X. For n < 0, the group Cn(X;R) is defined to be

zero. The boundary homomorphism

∂ : Cn(X;R)→ Cn−1(X;R)

24

is defined by

∂[σ] = [σ φ0]− [σ φ1] +− · · ·+ (−1)n[σ φn].

Lemma 5. ∂ ∂ = 0.

Definition 39. The nth singular homology group Hn(X;R) is the quotient mod-

ule Zn(X;R)/Bn(X;R), where Zn(X;R) is the kernel of ∂ : Cn(X;R)→ Cn−1(X;R)

and Bn(X,R) is the image of ∂ : Cn+1(X;R)→ Cn(X;R). (The “groups” are really

left R-modules.)

It is useful here to insert a few more purely algebraic constructs in order to

define relative homology later.

Definition 40. A graded group is a collection of abelian groups Ci indexed by

the integers. A chain complex is a graded group Ci together with a sequence of

homomorphisms ∂ : Ci → Ci−1 such that ∂2 : Ci → Ci−2 is zero. The operator ∂ is

called a boundary operator or differential.

The singular chain groups Cn(X;R) along with the boundary homomorphisms

form a chain complex.

Definition 41. If C∗ = (Ci, ∂) is a chain complex, then its homology is defined

to be the graded group

Hn(C∗) =ker ∂ : Cn → Cn−1

im ∂ : Cn+1 → Cn.

Thus, Hn(X) = Hn(C∗(X)).

Definition 42. If A∗ and B∗ are chain complexes, then a chain map f : A∗ → B∗

is a collection of homomorphisms f : Ai → Bi such that f ∂ = ∂ f .

25

Definition 43. A sequence of groups Ai−→ B

j−→ C is called exact if im(i) =

ker(j).

Theorem 7. A “short” exact sequence 0 → A∗i−→ B∗

j−→ C∗ → 0 of chain com-

plexes and chain maps induces a “long” exact sequence

· · · ∂∗−→ Hn(A∗)i∗−→ Hn(B∗)

j∗−→ Hn(C∗)∂∗−→ Hn−1(A∗)

i∗−→ · · · ,

where ∂∗JcK = Ji−1 ∂ j−1(c)K and is called the connecting homomorphism.

Definition 44. Let A ⊆ X be a pair of topological spaces. Then Cn(A;R) is

a submodule of Cn(X;R) and the inclusion is a chain map. Let Cn(X,A;R) =

Cn(X;R)/Cn(A;R). Then

0→ C∗(A)i−→ C∗(X)

j−→ C∗(X,A)→ 0

is an exact sequence of chain complexes and chain maps. The relative homology

of (X,A) is defined to be

Hn(X,A;R) = Hn(C∗(X,A;R)).

By the theorem, there is an induced “exact homology sequence of the pair

(X,A)”:

· · · ∂∗−→ Hn(A)i∗−→ Hn(X)

j∗−→ Hn(X,A)∂∗−→ Hn−1(A)

i∗−→ · · · .

Definition 45. A homology theory (on the category of all pairs of topological

spaces and continuous maps) is a functor H assigning to each pair (X,A) of spaces,

26

a graded (abelian) group Hp(X,A), and to each map f : (X,A) → (Y,B), ho-

momorphisms f∗ : Hp(X,A) → Hp(Y,B), together with a natural transformation of

functors ∂∗ : Hp(X,A)→ Hp−1(A), called the connecting homomorphism (where

H∗(A) is used to denote H∗(A,∅), etc.), such that the following five axioms are sat-

isfied:

1. (Homotopy Axiom.)

f ' g : (X,A)→ (Y,B) =⇒ f∗ = g∗ : H∗(X,A)→ H∗(Y,B).

2. (Exactness Axiom.) For the inclusions i : A → X and j : X → (X,A)

the sequence

· · · ∂∗−→ Hp(A)i∗−→ Hp(X)

j∗−→ Hp(X,A)∂∗−→ Hp−1(A)

i∗−→ · · ·

is exact.

3. (Excision Axiom.) Given the pair (X,A) and an open set U ⊆ X such

that U ⊆ int(A) then the inclusion k : (X r U,A r U) → (X,A) induces

an isomorphism

k∗ : H∗(X r U,Ar U)≈−→ H∗(X,A).

4. (Dimension Axiom.) For a one-point space P , Hi(P ) = 0 for all i 6= 0.

5. (Additivity Axiom.) For a topological sum X =∐

αXα the homomor-

phism

⊕(iα)∗ :

⊕Hn(Xα)→ Hn(X)

27

is an isomorphism, where iα : Xα → X is the inclusion.

Not all important “homology theories” satisfy every axiom above. For exam-

ple, Cech homology fails exactness, and bordism and K-theories fail the dimension

axiom [2]. However, singular homology provides a nice example of a homology theory.

Theorem 8. Singular homology is a homology theory.

The following theorem will be needed in Chapter 6.

Theorem 9. If B ⊂ A ⊂ X and ∂∗ : Hi(X,A) → Hi−1(A,B) is the composition

of ∂∗ : Hi(X,A) → Hi−1(A) with the map Hi−1(A) → Hi−1(A,B) induced by inclu-

sion, then the following sequence is exact, where the maps other than ∂∗ come from

inclusions:

· · · ∂∗−→ Hp(A,B)i∗−→ Hp(X,B)

j∗−→ Hp(X,A)∂∗−→ Hp−1(A,B)

i∗−→ · · · .

One can also define the dual theory to homology: cohomology.

Definition 46. The cochain group Cn(X;R) is defined to be the dual module

HomR(Cn(X;R), R) consisting of all R-linear maps from Cn(X;R) to R. The value

of a cochain c on a chain γ will be denoted by 〈c, γ〉 ∈ R. The coboundary of a

cochain c ∈ Cn(X;R) is defined to be the cochain δc ∈ Cn+1(X;R) whose value on

each (n+ 1)-chain α is determined by the identity

〈δc, α〉+ (−1)n〈c, ∂α〉 = 0.

Lemma 6. δ δ = 0.

Definition 47. The nth singular cohomology group Hn(X;R) is the quotient

module Zn(X;R)/Bn(X;R), where Zn(X;R) is the kernel of δ : Cn(X;R)→ Cn+1(X;R)

and Bn(X,R) is the image of δ : Cn−1(X;R)→ Cn(X;R).

28

3.3 de Rham Cohomology

The convention is here adopted that all vector spaces are finite dimensional

over R.

Definition 48. A map

ϕ :k∏i=1

Vi → R,

where the Vi are vector spaces, is called k-linear if it is linear in each argument, i.e.,

if

v 7→ ϕ(v1, . . . , vi−1, v, vi+1, . . . , vk)

is linear for each choice of v1, . . . , vi−1, vi+1, . . . , vk.

Remark 1. The set of all k-linear maps from V k to R forms a vector space and is

denoted

T k(V ) = ϕ : V k → R : ϕ is k-linear.

So T 1(V ) = V ∗ and we set T 0(V ) = R.

Definition 49. An element ϕ ∈ T k(V ) is called alternating if

ϕ(v1, . . . , vi, . . . , vj, . . . , vk) = −ϕ(v1, . . . , vj, . . . , vi, . . . , vk).

29

Definition 50. The vector space ΛkV of all k-linear alternating forms with

real values is

ΛkV = ϕ ∈ T k(V ) : ϕ is alternating.

Definition 51. The wedge product, or exterior product, of a k-form and a

`-form is a (k + `)-form:

∧ : ΛkV × Λ`V → Λk+`V

∧ : (ϕ, ψ) 7→ ϕ ∧ ψ

(ϕ ∧ ψ)(v1, . . . , vk+`) =1

k!`!

∑σ∈Sn

sgn (σ)ϕ(vσ(1), . . . , vσ(k))ψ(vσ(k+1), . . . , vσ(k+`))

1. If β1, . . . , βn is a basis for V ∗, then a basis of ΛkV is given by

βi1 ∧ βi2 ∧ · · · ∧ βik , 1 ≤ i1 < i2 < · · · < ik ≤ n,

and therefore

dim ΛkV =

nk

.

Thus, any p-form can be written in this basis as

ϕ =∑

i1<i2<···<ik

ϕi1···ikβi1 ∧ · · · ∧ βik , ϕi1···ik ∈ R,

30

or

ϕ =1

k!

∑i1,...,ik

ϕi1···ikβi1 ∧ · · · ∧ βik , ϕi1···ik ∈ R.

2. dim ΛnV = 1 so any n-form can be written as

ϕ = kβ1 ∧ · · · ∧ βn.

An orientation of V is the choice of one of the two equivalence classes of

ΛnV r 0, i.e., some non-zero n-form ϕ modulo a positive factor.

Definition 52. A section of a vector bundle π : E → B is a continuous function

s : B → E such that π s = idB. If M is a manifold, a section of TM is called a

vector field on M .

Definition 53. If f : M → R, then the differential of f is the section of T ∗M

defined by

df(p)(X) = Xp(f), for Xp ∈ TpM.

Remark 2. If p ∈ U , where U is an open subset of a manifold M , and x : U → Rn

is a coordinate chart for M , then a basis for Tp(U) is given by

∂

∂x1

∣∣∣∣p

, . . . ,∂

∂xn

∣∣∣∣p

.

A vector field for TU is given by

X(p) =∂

∂x1

∣∣∣∣p

.

31

In this case, for any f : M → R, we have

df(p)(X) =∂f

∂x1

∣∣∣∣p

.

In particular,

dx1(p)(X) =∂x1

∂x1

∣∣∣∣p

= 1.

In general,

dxi(p)

(∂

∂xj

∣∣∣∣p

)=

∂xi

∂xj

∣∣∣∣p

= δij,

so that dx1p, . . . , dx

np is a basis for T ∗pU . In particular, we have

df =n∑i=1

∂f

∂xidxi.

Definition 54. Let k ∈ 0, 1, . . . , n and M be a smooth manifold. A differential

form of degree k, or k-form for short, is a mapping

ω : M → Λk(TM), ϕ : p 7→ ϕp ∈ Λk(TpM).

Remark 3. If p ∈ U , where U is an open subset of a manifold M , and x : U → Rn

is a coordinate chart for M , dx1p, . . . , dx

np , is a basis for Λ1(TpU). Thus, a basis for

Λk(TpU) is

dxi1 ∧ dxi2 ∧ · · · ∧ dxik , 1 ≤ i1 < i2 < · · · < ik ≤ n.

32

Definition 55. Let ω give an orientation of M , g be a metric on M , and dxi be

an oriented orthonormal basis of T ∗M = Λ1(TM). The Hodge star is the linear

isomorphism

∗ : Λk(TM)→ Λn−k(TM)

∗(dxi1 ∧ · · · ∧ dxik) =1

(n− k)!εi1···inη

i1i1 · · · ηikikdxik+1 ∧ · · · ∧ dxin .

Definition 56. The exterior derivative d is the linear map

d : Λk(TM)→ Λk+1(TM)

dω =1

k!

∑i1,...,ik

dωi1···ik ∧ dxi1 ∧ · · · ∧ dxik

=1

k!

∑i,i1,...,ik

∂

∂xiωi1···ikdx

i ∧ dxi1 ∧ · · · ∧ dxik .

1. Although the Hodge star and the exterior derivative were both defined in

terms of a coordinate system, they are in fact both independent of the

choice of coordinates.

2. Let f : R3 → R. Then

df =∂f

∂xdx+

∂f

∂ydy +

∂f

∂zdz.

The 1-form df can be identified with the vector

∇f =

(∂f

∂x,∂f

∂y,∂f

∂z

).

33

3. Let ω = ω1dx+ ω2dy + ω3dz be a 1-form. Then

dω =

(∂ω3

∂y− ∂ω2

∂z

)dy∧dz+

(∂ω3

∂x− ∂ω1

∂z

)dx∧dz+

(∂ω2

∂x− ∂ω1

∂y

)dx∧dy.

Identifying

dy ∧ dz ↔ x, dx ∧ dz ↔ y, dx ∧ dy ↔ z,

we obtain

dω = ∇× (ω1, ω2, ω3).

4. Let ω = ω3dx ∧ dy − ω2dx ∧ dz + ω1dy ∧ dz. Then

dω = ∇ · (ω1, ω2, ω3) dx ∧ dy ∧ dz.

Theorem 10 (Stoke’s Theorem). If M is an oriented n-manifold with boundary ∂M

and ω is an (n− 1)-form on M with compact support, then

∫M

dω =

∫∂M

ω.

Definition 57. The de Rham complex on a manifold M is the graded algebra

Ω∗(M) =n⊕k=0

Λk(TM)

34

together with the operator d. It gives rise to the exact sequence

· · · −→ Λk−1(TM)d−→ Λk(TM)

d−→ Λk+1(TM) −→ · · · .

Definition 58. Forms ω which satisfy dω = 0 are called closed. For each k, we

have the vector space

Zk(M) = ω ∈ Λk(M) : ω is closed = kerd : Λk(TM)→ Λk+1(TM).

If there exists a (k− 1)-form ϕ such that ω = dϕ, then ω is called exact. The vector

space

Bk(M) = ω ∈ Λk(M) : ω is exact = imd : Λk−1(TM)→ Λk(TM).

Further, we set B0(M) = 0.

Theorem 11. d2 = 0.

Proof. Note that d2 = (d : Λk(TM) → Λk+1(TM)) (d : Λk−1(TM) → Λk(TM)).

Now,

ddω =1

k!(k + 1)!

∑i,j,i1,...,ik

∂2

∂xj∂xiωi1···ikdx

j ∧ dxi ∧ dxi1 ∧ · · · ∧ dxip = 0

since the partial derivatives commute whereas dxi ∧ dxj = −dxj ∧ dxi.

Corollary 1. Every exact form is closed.

Corollary 2. Bk(M) is a vector subspace of Zk(M).

35

Definition 59. The quotient vector space

Hk(M) = Zk(M)/Bk(M) =kerd : Λk(TM)→ Λk+1(TM)imd : Λk−1(TM)→ Λk(TM)

is called the k-dimensional de Rham cohomology vector space of M .

Theorem 12 (Poincare Lemma). If M is smoothly contractible to a point p0 ∈ M ,

then every closed form ω on M is exact.

Corollary 3. Hk(Rn) =

0 for k > 0

R for k = 0.

CHAPTER 4

CRITICAL POINTS

The analytical proof of the h-Cobordism Theorem relies heavily on the results

of Morse Theory, which provides analytical techniques to study topological properties

of manifolds. These techniques are based on the study of critical points of certain

maps. These critical points provide information about the local behavior of the

manifold.

Morse Theory and Smale’s proof of the h-Cobordism Theorem hinge on Sard’s

Theorem and Morse’s Lemma. Sard’s Theorem says roughly that the image of the set

of critical points is small. Morse’s Lemma provides a nice coordinate system around

critical points of certain maps.

4.1 Basic Definitions

The presentation below is similar to that in [7].

Definition 60. Let M and N be smooth manifolds and f : M → N a smooth map.

A point p ∈ M is a critical point of f if the induced map f∗ : TMp → TNf(p) has

rank < n. The set of critical points of f is denoted Σf . In the special case N = R,

the real number f(p) is called a critical value of f and a number c ∈ R which is

not a critical value of f is called a regular value of f .

In terms of coordinates (x1, . . . , xn) in a neighborhood U of a critical point p

of f : M → R,

∂f

∂x1

∣∣∣∣p

=∂f

∂x2

∣∣∣∣p

= · · · = ∂f

∂xn

∣∣∣∣p

= 0.

36

37

Definition 61. Let p ∈M be a critical point of the map f : M → R. If X, Y ∈ TMp,

then X and Y have extensions to smooth vector fields X and Y on M . The Hessian

of f at p is the symmetric bilinear functional f∗∗ : TMp × TMp → R defined by

f∗∗(X, Y ) = Xp(Y (f)).

Lemma 7. The Hessian f∗∗ is well-defined and symmetric.

Proof. To see that f∗∗ is symmetric, notice that

f∗∗(X, Y )− f∗∗(Y,X) = Xp(Y (f))− Yp(X(f)) = [X, Y ]p(f) = 0,

since the Poisson bracket [X, Y ]p ∈ TMp and f∗ : TMp → TRf(p) is zero.

To see well-definedness, notice that f∗∗(X, Y ) = Xp(Y (f)) = X(Y (f)) so

that f∗∗(X, Y ) is independent of the extension X, and by symmetry f∗∗(X, Y ) =

f∗∗(Y,X) = Y (X(f)) so that f∗∗(X, Y ) is independent of the extension Y .

Let (x1, . . . , xn) be a coordinate system on an open set U 3 p, and X, Y ∈ TMp

have coordinates

X =n∑i=1

ai∂

∂xi

∣∣∣∣p

and Y =n∑i=1

bi∂

∂xi

∣∣∣∣p

.

Then Y has an extension to a vector field Y on U given by

Yq =n∑i=1

bi(q)∂

∂xi

∣∣∣∣q

, (q ∈ U)

38

where now bi : U → R is a constant function. In these coordinates, one has

f∗∗(X, Y ) = X(Y (f)) =n∑i=1

ai∂

∂xi

∣∣∣∣p

(n∑j=1

bj∂f

∂xj

)=

n∑i,j=1

aibj(p)∂f

∂xi∂xj

∣∣∣∣p

=

[a1 · · · an

]

∂f

∂x1∂x1

∣∣∣∣p

· · · ∂f

∂x1∂xn

∣∣∣∣p

.... . .

...

∂f

∂xn∂x1

∣∣∣∣p

· · · ∂f

∂xn∂xn

∣∣∣∣p

b1...

bn

Thus, the matrix

(∂2f

∂xi∂xj

∣∣∣∣p

)represents f∗∗ with respect to the basis

∂

∂x1

∣∣∣∣p

, . . . ,∂

∂xn

∣∣∣∣p

for TMp.

Definition 62. Let V be a finite-dimensional vector space and H : V × V → R a

bilinear functional on V . The index of H is the maximal dimension of a subspace of

V on which H is negative-definite. The nullspace of H is the subspace null (H) ⊂ V

given by

null (H) = v ∈ V : H(v, w) = 0 for all w ∈ V .

Definition 63. A critical point p of f : M → R is said to be non-degenerate if

dim null f∗∗ = 0.

In terms of coordinates, a critical point is non-degenerate if

det

(∂2f

∂xi∂xj

∣∣∣∣p

)6= 0.

39

Definition 64. If p is a critical point for f : M → R, the index of f at p is the

index of f∗∗ on TMp.

4.2 Morse’s Lemma

There is now a suitable background to state Morse’s Lemma. However, it is

desirable at this stage to present some lemmas and then refer to these in the proof of

Morse’s Lemma.

Lemma 8. Let V be a convex neighborhood of 0 ∈ Rn and f : V → R be a C∞

function with f(0) = 0. Then

f(x) =n∑i=1

xigi(x)

for some suitable C∞ functions gi : V → R with gi(0) =∂f

∂xi

∣∣∣∣0

.

Proof. Let h : Rn+1 → Rn be defined by h(t,x) = tx so that hi(t,x) = txi. Then

f(x) = f(tx)∣∣1t=0

=

∫ 1

0

∂(f h)

∂t

∣∣∣∣tx

dt =

∫ 1

0

n∑i=1

∂f

∂xi

∣∣∣∣h(t,x)

∂hi∂t

∣∣∣∣(t,x)

dt

=

∫ 1

0

n∑i=1

∂f

∂xi

∣∣∣∣tx

· xi dt =n∑i=1

xi∫ 1

0

∂f

∂xi

∣∣∣∣tx

dt =n∑i=1

xigi(x).

In the first equality, f(0) = 0 is used, and, in the second, the Fundamental Theorem

of Calculus. In the third, the chain rule has been employed. In the last equality, gi is

defined as gi(x) =

∫ 1

0

∂f

∂xi

∣∣∣∣tx

dt. Therefore,

gi(0) =

∫ 1

0

∂f

∂xi

∣∣∣∣0

dt =∂f

∂xi

∣∣∣∣0

∫ 1

0

dt =∂f

∂xi

∣∣∣∣0

.

40

Example 3. Let f : R2 → R be f(x, y) = x3 + y3 + x2 + xy. Then

g1(x, y) =

∫ 1

0

∂f

∂x

∣∣∣∣(tx,ty)

dt =

∫ 1

0

3(tx)2 + 2(tx) + (ty) dt

=

[t3x2 + t2x+

1

2t2y

]∣∣∣∣1t=0

= x2 + x+1

2y

and

g2(x, y) =

∫ 1

0

∂f

∂y

∣∣∣∣(tx,ty)

dt =

∫ 1

0

3(ty)2 + (tx) dt =

[t3y2 +

1

2t2x

]∣∣∣∣1t=0

= y2 +1

2x.

Therefore, we can write

f(x, y) = xg1(x, y) + yg2(x, y) = x

(x2 + x+

1

2y

)+ y

(y2 +

1

2x

).

Lemma 9 (Morse). If p is a non-degenerate critical point of f , there is a neighborhood

U of p and a coordinate system x : U → Rn such that, for q ∈ U ,

f(q) = f(p)− (x1(q))2 − · · · − (xλ(q))2 + (xλ+1(q))2 + · · ·+ (xn(q))2, (4.1)

for some λ between 0 and n. Moreover, λ is the index of f at p.

Proof. Let f(q) = f(q)− f(p). Then f(p) = 0 and p is a nondegenerate critical point

of f .

Let y : U1 → Rn be a coordinate chart in a neighborhood U1 of p. Take

y : U2 → Rn to be the coordinate chart on U2 ⊆ U1, U2 3 p, given by y(q) = y(q)− y(p).

So y(p) = 0. Now f y−1 : Rn → R and f y−1(0) = f(p) = 0. By Lemma 8, it

41

follows that there is a neighborhood V1 3 0 such that, for x ∈ V1,

f y−1(x) =n∑i=1

gi(x)xi and gi(0) =∂(f y−1)

∂xi

∣∣∣∣∣0

=∂f

∂xi

∣∣∣∣∣p

= 0,

where gi : V1 → R is C∞. Applying Lemma 8 again, this time to the gi, there is a

neighborhood V2 ⊆ V1 of 0 such that, for x ∈ V2,

gi(x) =n∑j=1

hij(x)xj and hij(0) =∂gi∂xj

∣∣∣∣0

=∂2(f y−1)

∂xi∂xj

∣∣∣∣∣0

for some smooth functions hij on V2. Altogether, one has

(f y−1

)(x) =

n∑i,j=1

hij(x)xixj.

Non-degeneracy of p gives

det (hij(0)) = det

∂2f

∂xi∂xj

∣∣∣∣∣p

6= 0.

Define hij(x) = 12

(hij(x) + hji(x)). Then hij = hji and

det(hij(0)

)=

1

2(det (hij(x)) + det (hji(x))) = det (hij(0)) 6= 0.

Modulo a linear change in coordinates (see Appendix A), one has h11(0) 6= 0.

Then by continuity of h11, there is a neighborhood W1 of 0 such that h11(x) 6= 0.

42

Thus, on W1,

(f y−1

)(x) =

n∑i,j=1

hij(x)xixj

=n∑

i,j=1

hij(x)xixj

= h11(x)(x1)2 + 2n∑i=2

h1i(x)x1xi +n∑

i,j=2

hij(x)xixj

= h11(x)

[(x1)2 + 2

n∑i=2

h1i(x)

h11(x)x1xi

]+

n∑i,j=2

hij(x)xixj

= h11(x)

(x1 +n∑i=2

h1i(x)

h11(x)xi

)2

−

(n∑i=2

h1i(x)

h11(x)xi

)2+

n∑i,j=2

hij(x)xixj

= ±

[√|h11(x)|

(x1 +

n∑i=2

h1i(x)

h11(x)xi

)]2

− h11(x)

(n∑i=2

h1i(x)

h11(x)xi

)2

+n∑

i,j=2

hij(x)xixj.

The last two sums are

n∑i,j=2

hij(x)xixj − h11(x)

(n∑i=2

h1i(x)

h11(x)xi

)2

=n∑

i,j=2

hij(x)xixj − h11(x)n∑

i,j=2

h1i(x)h1j(x)

h211(x)

xixj

=n∑

i,j=2

[hij(x)− h1i(x)h1j(x)

h11(x)

]xixj

=n∑

i,j=2

[h11(x)hij(x)− h1i(x)h1j(x)

h11(x)

]xixj

=n∑

i,j=2

[C(1,1),(i,j)

h11(x)

]xixj,

43

where C(1,1),(i,j) is the cofactor

C(1,1),(i,j) = det

h11(x) h1j(x)

hi1(x) hij(x)

.Since det(hij(0)) 6= 0, there is a neighborhood where det(hij)(x) 6= 0. Thus, not

every C(1,1),(i,j) = 0.

Set

ψ(x) =

(√|h11(x)|

(x1 +

n∑i=2

h1i(x)

h11(x)xi

), x2, . . . , xn

).

Then ψ : V2 → Rn, ψ(0) = 0, and

J(ψ(0)) =

∂ψ1

∂x1

∣∣∣∣0

∂ψ1

∂x2

∣∣∣∣0

· · · ∂ψ1

∂xn

∣∣∣∣0

0 1 · · · 0

......

. . ....

0 0 · · · 1

.

So det J(ψ(0)) 6= 0 iff∂ψ1

∂x1

∣∣∣∣0

6= 0. But

∂ψ1

∂x1=

∂

∂x1

(√|h11(x)|

)(x1 +

n∑i=2

h1i(x)

h11(x)xi

)+

√|h11(x)|

(1 +

∂

∂x1

(h1i(x)

h11(x)

)xi

)

so that∂ψ1

∂x1

∣∣∣∣0

=√|h11(0)| 6= 0. Therefore det J(ψ(0)) 6= 0 and so, by the Inverse

Function Theorem, ψ−1 exists and is C∞ in a neighborhood W of 0.

44

Now, if q ∈ U2 and (ψ y)(q) = x ∈ W , then

f(q) = f(p) + f(q) = f(p) +(f y−1 ψ−1

)(x)

= f(p)± [ψ1(ψ−1(x))]2 + · · · = f(p)± (x1)2 + · · · ,

where the tail of the sum is nonzero for q 6= p and consists of terms not containing

x1.

Continuing this process on each remaining variable (induction) and then re-

ordering coordinates (composing with a non-singular permutation matrix), one ob-

tains a coordinate system around p such that Equation (4.1) holds.

It remains to show that λ is the index of f at p. By Equation (4.1),

(∂2f

∂xi∂xj

∣∣∣∣p

)=

−2 0

. . .

−2

2

. . .

0 2

.

But this is just the matrix of f∗∗ in the basis∂

∂x1

∣∣∣∣p

, . . . ,∂

∂xn

∣∣∣∣p

. This gives a negative

definite subspace of TMp of dimension λ and a positive definite subspace of dimension

n − λ. Then λ is the maximal dimension since otherwise the negative definite and

positive definite subspaces would intersect. Therefore λ is the index of f∗∗.

45

4.3 Sard’s Theorem

Definition 65. An n-cube C ⊂ Rn of edge ` > 0 is a product

C = I1 × I2 × · · · × In ⊂ Rn

of closed intervals of length `; thus Ij = [aj, aj + `] ⊂ R. The measure (or n-

measure) of C is

µ(C) = µn(C) = `n.

Definition 66. A subset A ⊆ Rn has measure zero if for every ε > 0 there exists

a family of n-cubes Cα such that

1. A ⊆⋃Cα,

2.∑µ(Cα) < ε.

A subset A ⊂ M of an n-manifold is said to have measure zero if for every chart

(ϕ,U), the set ϕ(U ∩A) ⊆ Rn has measure zero. In both cases the notation µ(A) = 0

is used.

A fundamental fact about critical points is the following.

Theorem 13 (Sard’s Theorem). Let f : M → N be a smooth map of smooth mani-

folds. Then

µ(f(Σf )) = 0 (in N).

Proof. Let M be an m-manfold and N an n-manifold. Since manifolds are second

countable spaces, one merely needs to show that f(U ∩ Σf ) has measure zero in N ,

46

where (x, U) is a coordinate chart for M . This is the case if maps f : Rm → Rn have

µ(f(Σf )) = 0 since diffeomorphisms preserve the measure zero property.

If m < n, then f(Rm) has measure zero in Rn and, thus, so does f(Σf ).

Suppose m ≥ n. Divide Σf into three sets

Σf = Σ1 ∪ Σ2 ∪ Σ3

in order to show that each of these has measure zero in Rn. This division occurs as

follows:

Σ1 =

p ∈ Σf : (∀i ∈ 1, . . . , n), (∀r ≤ m

n),

∂rfi∂xj1 · · · ∂xjr

(p) = 0

Σ2 =

p ∈ Σf : (∃i ∈ 1, . . . , n), (∃r ≥ 2),

∂rfi∂xj1 · · · ∂xjr

(p) 6= 0

Σ3 =

p ∈ Σf : (∃i ∈ 1, . . . , n), (∃j ∈ 1, . . . ,m), ∂fi

∂xj(p) 6= 0

.

To see that Σf = Σ1 ∪ Σ2 ∪ Σ3, suppose p ∈ Σf r Σ3. Then∂fi∂xj

(p) = 0 for all i, j.

Suppose p ∈ Σf r (Σ2 ∪Σ3). Then all derivatives of all orders are zero at p. Clearly,

then, p ∈ Σ1.

I. We first show that µ(f(Σ1)) = 0. To show that µ(f(Σ2 ∪ Σ3)) = 0, we then

proceed by induction on m, showing that for each m, f(Σ2 ∪ Σ3) has measure

zero in Rn for all n ∈ 1, . . . ,m.

To see that µ(f(Σ1)) = 0, notice that we can do a Taylor expansion around each

p0 ∈ Σ1 in a neighborhood U ⊆ Rm containing p0. Let q be the greatest integer

less than or equal to mn

. The idea is to use the fact that the first q− 1 terms are

zero since p0 ∈ Σ1 to get a condition on f for points in a neighborhood of the

critical point. Here are the details. By Taylor’s theorem, for all p ∈ U and for

47

each component function fi : Rm → R,

fi(p) = fi(p0) +m∑j=1

∂fi∂xj

(p0)(pj − pj0)+

+1

2!

m∑

j1,j2=1

∂2fi∂xj1∂xj2

(p0)(pj1 − pj10 )(pj2 − pj20 )

+ · · ·+

+1

q!

m∑

j1,...,jq=1

∂qfi∂xj1 · · · ∂xjq

(p0)(pj1 − pj10 ) · · · (pjq − pjq0 )

+Rq+1(p),

where

Rq+1(p) =1

(q + 1)!

m∑i1,...,iq+1=1

∂q+1fi∂xi1 · · · ∂xiq+1

(tp+(1−t)p0)(pi1−pi10 ) · · · (piq+1−piq+1

0 ),

for some t ∈ (0, 1). Thus, for i ∈ 1, . . . , n, we have

|fi(p)− fi(p0)| = |Rq+1(p)|

=

∣∣∣∣∣∣ 1

(q + 1)!

m∑i1,...,iq+1=1

∂q+1fi∂xi1 · · · ∂xiq+1

(tp+ (1− t)p0)(pi1 − pi10 ) · · · (piq+1 − piq+1

0 )

∣∣∣∣∣∣=

1

(q + 1)!

∣∣∣∣∣∣m∑

i1,...,iq+1=1

∂q+1fi∂xi1 · · · ∂xiq+1

(pi)(pi1 − pi10 ) · · · (piq+1 − piq+1

0 )

∣∣∣∣∣∣ ,

48

where we have set pi = tp+ (1− t)p0. By the Cauchy-Schwarz inequality on the

iq+1 sum, we have

|fi(p)− fi(p0)|

=1

(q + 1)!

∣∣∣∣∣∣m∑

iq+1=1

m∑i1,...,iq=1

∂q+1fi∂xi1 · · · ∂xiq+1

(pi)(pi1 − pi10 ) · · · (piq − piq0 )

(piq+1 − piq+1

0

)∣∣∣∣∣∣≤ 1

(q + 1)!

√√√√√ m∑kq+1=1

∣∣∣∣∣∣m∑

i1,...,iq=1

∂q+1fi∂xi1 · · · ∂xiq∂xik+1

(pi)(pi1 − pi10 ) · · · (piq − piq0 )

∣∣∣∣∣∣2

×

×

√√√√ m∑`q+1=1

∣∣∣p`q+1 − p`q+1

0

∣∣∣2

=‖p− p0‖(q + 1)!

√√√√√ m∑kq+1=1

∣∣∣∣∣∣m∑

i1,...,iq=1

∂q+1fi∂xi1 · · · ∂xiq∂xik+1

(pi)(pi1 − pi10 ) · · · (piq − piq0 )

∣∣∣∣∣∣2

=‖p− p0‖(q + 1)!

√√√√√∣∣∣∣∣∣

m∑i1,...,iq+1=1

∂q+1fi∂xi1 · · · ∂xiq+1

(pi)(pi1 − pi10 ) · · · (piq+1 − piq+1

0 )

∣∣∣∣∣∣2

=‖p− p0‖(q + 1)!

∣∣∣∣∣∣m∑

i1,...,iq+1=1

∂q+1fi∂xi1 · · · ∂xiq+1

(pi)(pi1 − pi10 ) · · · (piq+1 − piq+1

0 )

∣∣∣∣∣∣ .

Repeating this process on the other indices, we obtain

|fi(p)− fi(p0)| ≤‖p− p0‖q+1

(q + 1)!

∣∣∣∣∣∣m∑

i1,...,iq+1=1

∂q+1fi∂xi1 · · · ∂xiq+1

(pi)

∣∣∣∣∣∣ .

49

It follows now that

‖f(p)− f(p0)‖ =

√√√√ n∑i=1

|fi(p)− fi(p0)|2

≤ ‖p− p0‖q+1

(q + 1)!

√√√√√ n∑i=1

∣∣∣∣∣∣m∑

i1,...,iq+1=1

∂q+1fi∂xi1 · · · ∂xiq+1

(pi)

∣∣∣∣∣∣2

.

We write this as

‖f(p)− f(p0)‖ ≤ B ‖p− p0‖q+1 , (4.2)

where

B =1

(q + 1)!

√√√√√ n∑i=1

∣∣∣∣∣∣m∑

i1,...,iq+1=1

∂q+1fi∂xi1 · · · ∂xiq+1

(pi)

∣∣∣∣∣∣2

≥ 0.

Because of the relation (4.2), when we subdivide U into smaller regions, the

volume of the smaller regions shrinks faster than the number of regions grows.

Formally, take U ⊆ U to be an m-cube of length λ ∈ R. Let s ∈ N (think

s 1). Divide U into sm m-cubes of length λs. Denote those smaller m-cubes

which contain points from Σ1 by Ci, where i ∈ 1, . . . , t. Clearly, t ≤ sm.

Further, each Ci is contained in a closed ball Bi of radius λs

√m centered at a

point p0 ∈ Ci ∩ Σ1. This radius is the distance between the opposite corners of

an m-cube of length λs, computed using the Euclidean norm.

Now, for any p ∈ Bi, we have (4.2) so f(p) is in a ball centered at f(p0) of radius

maxp∈Bi

(B ‖p− p0‖q+1) ≤ B

(λ

s

√m

)q+1

.

50

1 2 . . . s

1

2

s

.

.

.

λ

λ/s

p0

Ci

Bi

U~

f(Bi)f(p

0)

Ci'

f

n-space

m-space

This ball is then contained in a cube C ′i of length twice the radius of the ball

(centered at f(p0)). Thus, the volume of f (⋃Ci) is no greater than

t︸︷︷︸no. of Ci

[2B

(λ

s

√m

)q+1]n

︸ ︷︷ ︸maxvol of a cubeC′i

≤ sm

[2B

(λ

s

√m

)q+1]n

= sm−n(q+1)(2Bλ√m)n(q+1)

.

If m− n(q + 1) < 0, then the volume of f (⋃Ci) goes to zero as s→∞. Thus,

we have shown that f(Σ1) has measure zero in Rm provided q > mn− 1.

II. We now show that µ(f(Σ2 ∪ Σ3)) = 0 by induction on m.

(i) Let m = 1. Since we’ve assumed that m ≥ n, we have to show that

µ(f(Σ2∪Σ3)) = 0 for all n ≤ 1, i.e., for just n = 1. Then mn−1 = 1−1 = 0.

By I, we have that µ(f(Σ1)) = 0 in Rn for all q > 0, i.e., all q ∈ 1, . . . ,m.

This means that

µ

(f

(m⋃q=1

p ∈ Σf : (∀r ≤ q)

(∂rf

∂tr(p) = 0

)))= 0.

51

But

m⋃q=1

p ∈ Σf : (∀r ≤ q)

(∂rf

∂tr(p) = 0

)=

p ∈ Σf : (∃q ∈ 1, . . . ,m)(∀r ≤ q)

(∂rf

∂tr(p) = 0

)

and clearly

(p ∈ Σf ) =⇒[∃q ∈ 1, . . . ,m)(∀r ≤ q)

(∂rf

∂tr(p) = 0

)]

(take q = 1). So

Σf =

p ∈ Σf : (∃q ∈ 1, . . . ,m)(∀r ≤ q)

(∂rf

∂tr(p) = 0

)

and therefore, µ(f(Σf )) = 0. But Σ2 ∪ Σ3 ⊆ Σf and therefore

µ(f(Σ2 ∪ Σ3)) = 0.

(ii) Suppose now that µ(f(Σ2 ∪Σ3)) = 0 in Rj for all f : Rk → Rj, where k ∈

1, . . . ,m−1 and j ∈ 1, . . . , k. We’ll show that µ(f(Σ2∪Σ3)) = 0 in Rn

for f : Rm → Rn, where n ∈ 1, . . . ,m. We do this by first showing that

µ(f(Σ2rΣ3)) = 0 and then that µ(f(Σ3)) = 0 since Σ2∪Σ3 = (Σ2rΣ3)∪Σ3

implies f(Σ2 ∪ Σ3) = f((Σ2 r Σ3) ∪ Σ3) = f(Σ2 r Σ3) ∪ f(Σ3).

(a) µ(f(Σ2 r Σ3)) = 0:

Let p ∈ Σ2 rΣ3 so that f has a nonzero higher order partial derivative

at p, but all first order partials vanish at p. Let r be the smallest

52

integer such that

∂rfi∂xj1 · · · ∂xjr

(p) 6= 0

and there is a k ∈ 1, . . . , r − 1 such that

∂r−1fi

∂xj1 · · · ∂xjk · · · ∂xjr(p) = 0,

where the hat over the partial with respect to xjk omits that derivative.

Denote the set of all such p ∈ Σ2 r Σ3 by Xirk. Since there are only

countably many such Xirk’s, we only need to show that any given Xirk

has µ(f(Xirk)) = 0.

Consider the C∞ map θ : Rm → R defined by

θ =∂r−1fi

∂xj1 · · · ∂xjk · · · ∂xjr.

Then 0 ∈ R is a regular value of θ, and so θ−1(0) is a submanifold of Rm

with dim θ−1(0) = dim Rm−dim R = m−1. By the induction hypothe-

sis, µ(f(Σf |θ−1(0))) = 0 for all n ∈ 1, . . . , dim θ−1(0) = 1, . . . ,m−1.

Further, µ(f(Σf |θ−1(0))) = 0 for n = m since µ(f(θ−1(0))) = 0 in

Rn = Rm. But Xirk ⊂ θ−1(0) and so clearly Xirk ⊂ Σf |θ−1(0). Thus,

µ(f(Xirk)) = 0 and, therefore, µ(f(Σ2 r Σ3)) = 0.

(b) µ(f(Σ3)) = 0:

There exists an open neighborhood U of Σ3 on which, for some i and

j,∂f i

∂xj6= 0. The Implicit Function Theorem (Theorem 3) provides,

with a possible restriction of U , an open set A×B ⊂ Rm−1×R and a

53

diffeomorphism h : A × B → U such that (fi h)(x1, . . . , xm−1, t) = t

for (x, t) ∈ A × B. If necessary, reorder coordinates so that fi = fn.

Now,

(f |U h) : A×B ⊂ Rm−1 × R→ Rn−1 × R, f(x, t) = (ut(x), t),

where ut : A → Rn−1 is smooth for each t ∈ B. Now, (x, t) ∈ Σf iff

x ∈ Σut . Thus,

Σf ∩ h(A×B) =⋃t∈B

h(Σut × t).

Since dimA = n− 1, the induction hypothesis gives

µn−1(ut(Σut)) = 0,

where µn−1 denotes Lebesgue measure in Rn−1. Now, by Fubini’s The-

orem,

µn

(⋃t∈B

(f h)(Σut × t)

)=

∫B

µn−1(ut(Σut)) dt =

∫B

0 dt = 0.

Thus, µ(f(Σ3 ∩ U)) = 0. Since µ(f(Σ3)) ≤ µ(f(Σ3 ∩ U)), this shows

that µ(f(Σ3)) = 0.

This completes the induction step. So µ(f(Σ2 ∪ Σ3)) = 0 in Rj for all

f : Rk → Rj, where k ∈ 1, . . . ,m − 1 and j ∈ 1, . . . , k implies

µ(f(Σ2 ∪ Σ3)) = 0 in Rn for f : Rm → Rn, where n ∈ 1, . . . ,m.

This proves that µ(f(Σ2 ∪ Σ3)) = 0.

54

Now, µ(f(Σf )) ≤ µ(f(Σ1)) + µ(f(Σ2 ∪ Σ3)) = 0.

4.4 Approximation Lemmas in Rn

Lemma 10 (Morse). If U ⊆ Rn is open and f : U → R is C2, then the set of linear

mappings L : Rn → R for which the function f +L has degenerate critical points has

measure zero in (Rn)∗ ∼= Rn, where (Rn)∗ is the dual space to Rn.

For “almost all” linear mappings L : Rn → R, the function f + L has only

nondegenerate critical points.

Proof. Consider the manifold U × (Rn)∗. Then

M = (x, L) : d(f(x) + L(x)) = 0

is a submanifold of U × (Rn)∗. To see this, consider the map

ϕ : U × (Rn)∗ → (R2n)∗ given by ϕ(x, L) = d(f(x) + L(x)).

Now, for any x ∈ U and L ∈ (Rn)∗, we have that ϕ(x, L) is a linear map. If the

matrix of L in the standard basis is

M(L) =

[L1 L2 · · · Ln

],

then the matrix of ϕ(x, L) is

M(ϕ(x, L)) =

[∂(f + L)

∂x1

∣∣∣∣x

· · · ∂(f + L)

∂xn

∣∣∣∣x

∂(f + L)

∂L1

∣∣∣∣x

· · · ∂(f + L)

∂Ln

∣∣∣∣x

]=

[∂f

∂x1

∣∣∣∣x

+ L1 · · ·∂f

∂xn

∣∣∣∣x

+ Ln x1 · · · xn

]

55

in the standard basis. Thus, the matrix for dϕ(x, L) ∈ HomR(Rn × (Rn)∗, (R2n)∗) ∼=

HomR(R2n,R2n) in the standard basis (i.e., the Jacobian matrix) is

J(ϕ(x, L)) =

(∂

∂xj

∣∣∣∣(x,L)

(∂f

∂xi

∣∣∣∣x

+ Li

)) (∂

∂Lj

∣∣∣∣(x,L)

(∂f

∂xi

∣∣∣∣x

+ Li

))(

∂

∂xj

∣∣∣∣(x,L)

(xi)

) (∂

∂Lj

∣∣∣∣(x,L)

(xi)

)

=

∂2f

∂x21

∣∣∣∣x

· · · ∂2f

∂xn∂x1

∣∣∣∣x

1 · · · 0

.... . .

......

. . ....

∂2f

∂x1∂xn

∣∣∣∣x

· · · ∂2f

∂x2n

∣∣∣∣x

0 · · · 1

1 · · · 0 0 · · · 0

.... . .

......

. . ....

0 · · · 1 0 · · · 0

.

=

f∗∗ 1

1 0

,where each block is n×n. Clearly, dϕ 6= 0. So every value ϕ(x, L) is a regular value of

ϕ. In particular the operator 0 ∈ (R2n)∗ is a regular value and therefore its preimage

is a submanifold of U × (Rn)∗. This means that M = (x, L) : d(f(x) + L(x)) = 0

is a submanifold of U × (Rn)∗.

Since d(f(x)+L(x)) = df(x)+dL(x) = df(x)+L, we have d(f(x)+L(x)) = 0

iff L = −df(x). Thus, the map

ψ : U →M given by ψ : x 7→ (x,−df(x))

56

is a diffeomorphism of U onto M . Bijectivity of ψ is clear. Smoothness of ψ can

be seen from the fact that ψ is the diagonal product of the identity map with the

differential of f , both of which are smooth. The inverse of ψ is just the projection

onto the first factor, which is smooth.

Further, each (x, L) ∈M corresponds to a critical point x ∈ U of f +L. This

critical point is degenerate precisely when the Hessian matrix

(∂2f

∂xi∂xj

)is singular

since

(∂2(f + L)

∂xi∂xj(x)

)=

(∂2f

∂xi∂xj(x)

)+

(∂2L

∂xi∂xj(x)

)=

(∂2f

∂xi∂xj(x)

).

Consider the projection

π : M → (Rn)∗ given by π : (x, L) 7→ L.

Since L = −df(x) for L = π(x, L), we have

π ψ : U → (Rn)∗ given by π ψ : x 7→ −df(x)

and

J((π ψ)(x)) =

∂

∂x1

∣∣∣∣x

(− ∂f

∂x1

)· · · ∂

∂xn

∣∣∣∣x

(− ∂f

∂x1

)...

. . ....

∂

∂x1

∣∣∣∣x

(− ∂f

∂xn

)· · · ∂

∂xn

∣∣∣∣x

(− ∂f

∂xn

)

so that d(πψ)(x) = −f∗∗. Thus, πψ is critical at x ∈ U precisely when the Hessian

of f is singular. Thus, (x, L) ∈M corresponds to a degenerate critical point of f +L

exactly when d(π ψ)(x) = 0. In particular, f + L has a degenerate critical point iff

57

L is a critical value of π ψ : U → (Rn)∗ ∼= Rn. Since ψ is C1 and π is C∞, we have

that π ψ is C1. By Sard’s Theorem, the image of critical points of π ψ has measure

zero in (Rn)∗. This proves the lemma.

Lemma 11. Let K ⊆ U be a compact subset of an open set U ⊆ Rn. If f : U → Rn

is C2 and has only nondegenerate critical points in K, then there exists a δ > 0 such

that if

1. g : U → R is C2

2.

∣∣∣∣ ∂f∂xi (x)− ∂g

∂xi(x)

∣∣∣∣ < δ for all x ∈ K, all i, j = 1, . . . , n

3.

∣∣∣∣ ∂2f

∂xi∂xj(x)− ∂2g

∂xi∂xj(x)

∣∣∣∣ < δ for all x ∈ K, all i, j = 1, . . . , n

then g likewise has only nondegenerate critical points in K.

Lemma 12. Suppose h : U → U ′ is a diffeomorphism of one open subset of Rn onto

another and carries the compact set K ⊂ U onto K ′ ⊂ U ′. Then for any ε > 0, there

exists δ > 0 such that if f : U ′ → R satisfies

1. is smooth on U ′ 2. |f(x)| < δ 3.

∣∣∣∣ ∂f∂xi (x)

∣∣∣∣ < δ 4.

∣∣∣∣ ∂2f

∂xi∂xj(x)

∣∣∣∣ < δ

for all x ∈ K ′ and all i, j = 1, . . . , n, then f h satisfies

1. |(f h)(x)| < ε 2.

∣∣∣∣∂(f h)

∂xi(x)

∣∣∣∣ < ε 3.

∣∣∣∣∂2(f h)

∂xi∂xj(x)

∣∣∣∣ < ε

for all x ∈ K and all i, j = 1, . . . , n.

CHAPTER 5

COBORDISMS AND ANALYSIS

5.1 Basic Definitions

Definition 67. (W ;V0, V1) is a smooth manifold triad if

1. W is a compact smooth n-manifold

2. V0 and V1 are two closed submanifolds of W which are open in ∂W

3. ∂W = V0 q V1.

Definition 68. Let M0 and M1 be two closed smooth n-manifolds (i.e., M0, M1

compact, ∂M0 = ∂M1 = ∅). A cobordism c : M0 →M1 is a 5-tuple

c = (W ;V0, V1;h0, h1),

where

1. (W ;V0, V1) is a smooth manifold triad

2. h0 : V0 →M0 is a diffeomorphism

3. h1 : V1 →M1 is a diffeomorphism.

Example 4. Let L1 ⊂ R3 be the set

L1 =

x ∈ R3 : −1 ≤ −x2 + y2 + z2 ≤ 1 and |x|√y2 + z2 < sinh(1) cosh(1)

.

Let

∂LL1 =x ∈ L1 : −x2 + y2 + z2 = −1

58

59

Figure 5.1. The Manifold L1

be the “left boundary” and

∂RL1 =x ∈ L1 : −x2 + y2 + z2 = +1

be the “right boundary.” Then L1 can be given a smooth manifold structure and

∂L1 = ∂LL1 q ∂RL1. See Figure 5.1.

Definition 69. Two cobordisms c : M0 →M1 and c′ : M0 →M1, where

c = (W ;V0, V1;h0, h1) and c′ = (W ′;V ′0 , V′1 ;h′0, h

′1)

are equivalent if there exists a diffeomorphism g : W → W ′ carrying V0 to V ′0 and

V1 to V ′1 such that the following diagrams commute:

V0

M0

h0

???

????

????

??V0 V ′0

g|V0 // V ′0

M0

h′0

and

V1

M1

h1

???

????

????

??V1 V ′1

g|V1 // V ′1

M1

h′1

60

Figure 5.2. Level Surfaces of L1

For completeness, it should be mentioned here that cobordisms form a category

whose objects are closed manifolds and whose morphisms are equivalence classes of

cobordisms. Given a triad (W ;V, V ′), one writes W : V → V ′.

5.2 Morse Theory

Definition 70. A Morse function on a smooth manifold triad (W ;V0, V1) is a

smooth function f : W → [a, b] such that

1. f−1(a) = V0 and f−1(b) = V1

2. All the critical points of f are interior (lie in W r ∂W ) and are non-

degenerate.

Morse’s Lemma implies that the critical points of a Morse function are isolated.

Compactness ofW implies that a Morse function has only finitely many critical points.

Example 5. The manifold L1 (see Example 4) has the Morse function f : L1 →

[−1, 1] given by f(x, y, z) = −x2 + y2 + z2. Some level surfaces are depicted in Figure

5.2. Notice that f−1(−1) = ∂LL1 and f−1(+1) = ∂RL1, and that 0 ∈ R3 is a non-

degenerate critical point of f , of index λ = 1.

Definition 71. The Morse number µ of a smooth manifold triad (W ;V0, V1) is the

minimum over all Morse functions f of the number of critical points of f .

61

The following pages are devoted to showing that every smooth manifold triad

possesses a Morse function.

Lemma 13. There exists a smooth function f : W → [0, 1] with f−1(0) = V0,

f−1(1) = V1, such that f has no critical point in a neighborhood of ∂W .

Proof. Let (h1, U1), . . . , (hk, Uk) be an atlas for W such that no Ui meets both V0

and V1, and that if Ui ∩ ∂W 6= ∅ the coordinate map hi : Ui → Rn+ carries Ui onto

B1(0) ∩ Rn+.

On each set Ui define a map

fi : Ui → [0, 1]

as follows. Denote hi(p) = (x1(p), . . . , xn(p)). If Ui ∩ V0 6= ∅, let fi be the map

fi(p) = (πn hi)(p) = xn(p).

If Ui ∩ V1 6= ∅, let fi be the map

fi(p) = 1− (πn hi)(p) = 1− xn(p).

If Ui ∩ (V0 ∪ V1) = ∅, set fi(p) = 12

for all p ∈ W . Choose a partition of unity ϕi

subordinate to the cover Ui and define a map

f : W → [0, 1] by f(p) =k∑i=1

ϕi(p)fi(p),

where fi(p) is understood to have the value 0 outside Ui. Then f is clearly a well-

defined smooth map to [0, 1] with f−1(0) = V0 and f−1(1) = V1.

62

Finally, we verify that df 6= 0 on ∂W . Suppose q ∈ V0. Then, for some i,

q ∈ Ui and ϕi(q) > 0 (since∑ϕi(q) = 1). We have

∂f

∂xn=

i∑j=1

fj∂ϕj∂xn

+

ϕ1∂f1

∂xn+ · · ·+ ϕi

∂fi∂xn

+ · · ·+ ϕn∂fn∂xn

. (5.1)

Now, fj(q) = 0 for all j = 1, . . . , k. So at q the first summand is zero. All the

derivatives∂fj∂xn

(q) = 1 for j = 1, . . . , k. Therefore, ϕi(q)∂fi∂xn

(q) > 0 and each term

in the sum is non-negative. Thus,∂f

∂xn(q) 6= 0.

Suppose now that q ∈ V1. Then, for some i, q ∈ Ui and ϕi(q) > 0. We have

again Eq. (5.1). Now, fj(q) = 1 for all j = 1, . . . , k. So the first sum becomes simply

i∑j=1

fj∂ϕj∂xn

=i∑

j=1

∂ϕj∂xn

=∂

∂xn

k∑j=1

ϕj =∂

∂xn(1) = 0.

All the derivatives∂fj∂xn

(q) = −1 for j = 1, . . . , k. Therefore, ϕi(q)∂fi∂xn

(q) < 0 and

each term in the sum is non-positive. Thus,∂f

∂xn(q) 6= 0.

It follows that df 6= 0 on ∂W , and hence df 6= 0 in a neighborhood of ∂W .

Let F (M,R) denote the set of smooth real-valued functions on a compact

manifold-with-boundary M . In order to construct a topology on F (M,R), first, let

(hα, Uα) be a finite atlas for M . Take Cα to be a compact refinement of Uα.

Let δ > 0 and define N(δ) ⊆ F (M,R) to be the set of all g : M → R such that

1. ∀α

∀x∈hα(Cα)

|(g h−1α )(x)| < δ

2. ∀α

∀x∈hα(Cα)

∀i=1,...,n

∣∣∣∣∂(g h−1α )

∂xi(x)

∣∣∣∣ < δ

3. ∀α

∀x∈hα(Cα)

∀i,j=1,...,n

∣∣∣∣∂2(g h−1α )

∂xi∂xj(x)

∣∣∣∣ < δ.

63

Take N(δ) as a base of neighborhoods of the zero function. Since F (M,R) can be

given the obvious additive group structure, any other function f ∈ F (M,R) can be

given a neighborhood basis by taking sets N(f, δ) = f +N(δ) as the basis elements.

Therefore, g ∈ N(f, δ) iff

1. ∀α

∀x∈hα(Cα)

|(f h−1α )(x)− (g h−1

α )(x)| < δ

2. ∀α

∀x∈hα(Cα)

∀i=1,...,n

∣∣∣∣∂(f h−1α )

∂xi(x)− ∂(g h−1

α )

∂xi(x)

∣∣∣∣ < δ

3. ∀α

∀x∈hα(Cα)

∀i,j=1,...,n

∣∣∣∣∂2(f h−1α )

∂xi∂xj(x)− ∂2(g h−1

α )

∂xi∂xj(x)

∣∣∣∣ < δ.

Lemma 12 in Chapter 4 guarantees that the three properties of N(f, δ) do not depend

on the choice of the atlas or the compact refinement. Thus, the sets N(f, δ) form a

suitable topological basis for F (M,R). The topology generated by this basis is called

the C2 topology on F (M,R).

Theorem 14. If M is a compact manifold without boundary, the Morse functions

form an open dense subset of F (M,R) in the C2 topology.

Theorem 15. On any triad (W ;V0, V1), there exists a Morse function.

Lemma 14. Let f : W → [0, 1] be a Morse function for the triad (W ;V0, V1) with

critical points p1, . . . , pk. Then f can be approximated by a Morse function g with the

same critical points such that g(pi) 6= g(pj) for i 6= j.

Lemma 15. Let f : (W ;V0, V1)→ ([0, 1], 0, 1) be a Morse function, and suppose that

0 < c < 1 where c is not a critical value of f . The both f−1[0, c] and f−1[c, 1] are

smooth manifolds with boundary.

Corollary 4. Any cobordism can be expressed as a composition of cobordisms with

Morse number 1.

64

5.3 Elementary Cobordisms

Definition 72. Let f : W → [a, b] be a Morse function for the triad (W n;V, V ′). A

vector field X on W n is a gradient-like vector field for the Morse function f if

1. X(f) > 0 on W r Σf , and

2. given p ∈ Σf of index λ, there is a neighborhood U 3 p and a coordinate

system x : U → Rn given by x = (x,y) = (x1, . . . , xλ, xλ+1, . . . , xn) such

that, for q ∈ U ,

f(q) = f(p)− ‖x(q)‖2 + ‖y(q)‖2 ,

Xq = −x1(q)∂

∂x1

∣∣∣∣q

− · · · − xλ(q) ∂

∂xλ

∣∣∣∣q

+ xλ+1(q)∂

∂xλ+1

∣∣∣∣q

+ · · ·+ xn(q)∂

∂xn

∣∣∣∣q

.

Example 6. Consider the triad (W ;V, V ′) given in Figure 5.3 as a subset of R3

with p = 0. Near p, W is the quadric surface defined implicitly by the equation

z = −x2 + y2. A Morse function for (W ;V, V ′) is f(x, y) = −x2 + y2, the projec-

tion onto the z-axis. (The coordinates (x, y) near p are formally given by the chart

(x, y, z) 7→ (x, y), the projection onto the (x, y)-plane.) Notice that p is a nondegen-

erate critical point for f . A gradient-like vector field for f is

Xq = −x(q)∂

∂x

∣∣∣∣q

+ y(q)∂

∂y

∣∣∣∣q

=1

2∇f.

Lemma 16. For every Morse function f on a triad (W n;V, V ′) there exists a gradient-

like vector field X.

65

Figure 5.3. A gradient-like vector field on a triad (W ;V, V ′).

Proof. Let f be a Morse function on (W n;V, V ′). Such an f exists by Theorem 15.

First, suppose f has only one critical point p. By the Morse Lemma (Lemma 9 in

Chapter 4) there are coordinates (x,y) = (x1, . . . , xλ, xλ+1, . . . , xn) in a neighborhood

U0 of p such that f = f(p)− ‖x‖2 + ‖y‖2 throughout U0.

Each point p′ ∈ W r U0 is not a critical point of f . By the Implicit Function

Theorem (Theorem 3 in Chapter 1) there exist coordinates x′1, . . . , x′n in a neighbor-

hood U ′ of p′ such that f = constant + x′1 in U ′.

Let U be a neighborhood of p such that U ⊂ U0. By the previous paragraph

and the fact that W r U0 is compact, there are neighborhoods U1, . . . , Uk such that

1. W r U0 ⊂k⋃i=1

Ui,

2. U ∪ Ui = ∅, i = 1, . . . , k, and

3. Ui has coordinates xi1, . . . , xin and f = constant + xi1 on Ui, i = 1, . . . , k.

On U0 there is a vector field whose coordinates are (−x1, . . . ,−xλ, xλ+1, . . . , xn),

and on Ui there is the vector field∂

∂xi1with coordinates (1, 0, . . . , 0), i = 1, . . . , k.

Piece together these vector fields using a partition of unity subordinate to the cover

U0, U1, . . . , Uk, obtaining a vector field X on W . Then X is the required gradient-like

vector field for f .

66

Figure 5.4. The neighborhoods U,U0, U1, U2 in the existence construction aredrawn schematically for the triad in Example 6.

If f has more than one critical point, repeat this procedure for each critical

point, taking care to make sure the Ui’s still lie outside of some open neighborhood

of each critical point.

From now on, the triad (W ;V0, V1) will be identified with the cobordism

(W ;V0, V1; i0, i1), where i0 : V0 → V0 and i1 : V1 → V1 are the identity maps.

Definition 73. A triad (W ;V0, V1) is a said to be a product cobordism if it is

diffeomorphic to the triad (V0 × [0, 1];V0 × 0 , V0 × 1).

Theorem 16. If the Morse number µ of the triad (W ;V0, V1) is zero, then (W ;V0, V1)

is a product cobordism.

Proof. Let f : W → [0, 1] be a Morse function with no critical points. By Lemma

16, there exists a gradient-like vector field X for f . Since there are no critical points

for f , the map X(f) : W → R is strictly positive. One obtains a new gradient-like

67

Figure 5.5. Integral curves for X. See [11] p. 147.

vector field X by setting

X(p) =1

X(f)(p)X(p)

for all p ∈ W . Now X(f)(p) = 1 for all p ∈ W .

If p ∈ ∂W , then f expressed in some coordinate system x1, . . . , xn, xn ≥ 0,

about p extends to a smooth function f : U → R, where U ⊂ Rn is open. Similarly, X

expressed in this coordinate system also extends to U ⊂ Rn. Thus, the fundamental

existence and uniqueness theorem for ordinary differential equations applies locally

to W (see [11] Ch. 5).

Let ϕ : [a, b]→ W be any integral curve for X. Then

d

dt(f ϕ) = X(f) = 1.

Hence, f(ϕ(t)) = t+ t0, where t0 ∈ R is a constant. Making the change of parameter,

ψ(s) = ϕ(s− t0), we obtain an integral curve which satisfies f(ψ(s)) = s.

Each integral curve can be extended uniquely over a maximal interval. This

interval must be [0, 1] since W is compact. Indeed, integral curves must originate on

V0 and terminate on V1 since ξ(f) > 0, f(V0) = 0, and f(V1) = 1. See Figure 5.5.

68

Thus, for each y ∈ W there is a unique maximal integral curve ψy : [0, 1]→ W

which passes through y and satisfies f(ψy(s)) = s. Furthermore, ψy(s) is a smooth

function of both variables. (See p. 147 in [11]).

The required diffeomorphism h : V0× [0, 1]→ W is now given by the formulas

h(y0, s) = ψy0(s) and h−1(y) = (ψy(0), f(y)).

Corollary 5 (Collar Neighborhood Theorem). Let W be a compact smooth manifold

with boundary. There exists a neighborhood of ∂W (called a collar neighborhood)

diffeomorphic to ∂W × [0, 1).

Definition 74. A connected, closed submanifold Mn−1 ⊂ W n r ∂W n is said to be

two-sided if some neighborhood of Mn−1 on W n is cut into two components when

Mn−1 is deleted.

Corollary 6 (The Bicollaring Theorem). Suppose that every component of a smooth

manifold M of W is compact and two-sided. Then there exists a “bicollar” neigh-

borhood of M in W diffeomorphic to M × (−1, 1) in such a way that M corresponds

to M × 0.

The collaring and bicollar theorems remain valid without the compactness

conditions.

Theorem 17. Let (W ;V0, V1) and (W ′;V ′0 , V′1) be two smooth manifold triads and

h : V1 → V ′1 a diffeomorphism. Then there exists a smoothness structure S for

W ∪h W ′ compatible with the given structures on W and W ′. S is unique up to

diffeomorphism leaving V0, h(V1) = V ′1 , and V ′2 fixed.

69

Lemma 17. Let (W ;V0, V1) and (W ′;V ′1 , V′2) be two smooth manifold triads with

Morse functions f , f ′ to [0, 1], [1, 2], respectively. Construct gradient-like vector

fields X and X ′ on W and W ′, respectively, normalized to unity so that X(f) = 1

and X ′(f ′) = 1 except in a small neighborhood of each critical point. Given a diffeo-

morphism h : V1 → V ′1 there is a unique smoothness structure on W ∪hW ′, compatible

with the given structures on W and W ′, so that f and f ′ piece together to give a smooth

function on W ∪hW ′ and X and X ′ piece together to give a smooth vector field.

Corollary 7. Given the situation in Lemma 17, we have

µ(W ∪hW ′;V0, V′2) ≤ µ(W ;V0, V1) + µ(W ′;V ′1 , V

′2),

where µ is the Morse number of the triad.

The following focuses on cobordisms of Morse number 1. In particular, Defini-

tion 75 below sets up the analytical structure around a critical point to be used for the

rest of the thesis. The structure is developed by applying Morse’s Lemma (Lemma

9) to the Morse function, achieving a particularly nice coordinate system around the

critical point. Close enough to the critical point, the preimage of a regular value ε2

satisfies the equation

−‖x‖2 + ‖y‖2 = ε2,

which has the solution

(x,y) = ε(u sinh(t),v cosh(t)), u ∈ Sλ, v ∈ Sn−λ, t ∈ R,

70

provided λ ≥ 1 and n− λ ≥ 1. Indeed,

−‖x‖2 + ‖y‖2 = −λ∑i=1

(εui sinh(t))2 +n∑

i=λ+1

(εvi cosh(t))2

= ε2[(− sinh2(t))(u21 + · · ·+ u2

λ) + cosh2(t)(v2λ+1 + · · ·+ v2

n)]

= ε2[− sinh2(t) + cosh2(t)]

= ε2.

Definition 75. Let (W ;V, V ′) be a smooth manifold triad with a Morse function

f : W → R and a gradient-like vector field X for f . Suppose p ∈ W is a critical

point, and V0 = f−1(c0) and V1 = f−1(c1) are levels such that c0 < f(p) < c1 and that

c = f(p) is the only critical value in the interval [c0, c1].

Since X is a gradient-like vector field for f , there is a neighborhood U of p in

W , and a coordinate diffeomorphism g : Bn2ε → U so that

(f g)(x,y) = c− ‖x‖2 + ‖y‖2

and so that X has coordinates (−x1, . . . ,−xλ, xλ+1, . . . , xn) throughout U , for some

1 ≤ λ ≤ n and some ε > 0. Here x = (x1, . . . , xλ) ∈ Rλ and y = (xλ+1, . . . , xn) ∈ Rn−λ.

Set

V−ε = f−1(c− ε2) and Vε = f−1(c+ ε2).

Assume that

4ε2 < min(|c− c0|, |c− c1|),

71

Figure 5.6. V0, V1, Vε, and V−ε are drawn schematically for a 2-manifoldwith µ = λ = 1.

so that V−ε lies between V0 and f−1(c) and Vε lies between f−1(c) and V1. See Figure

5.6.

The characteristic embedding

ϕL : Sλ−1 × Bn−λ → V0

is obtained as follows. First define an embedding

ϕ : Sλ−1 × Bn−λ → V−ε

by

ϕ(u, θv) = g(εu cosh(θ), εv sinh(θ)),

where u ∈ Sλ−1, v ∈ Sn−λ−1, and θ ∈ [0, 1). Starting at the point ϕ(u, θv) in V−ε

the integral curve of X is a non-singular curve which leads from ϕ(u, θv) back to

some well-defined point ϕL(u, θv) in V0. Define the left-hand sphere SL of p in V0

to be the image ϕL(Sλ−1 × 0). Notice that SL is just the intersection of V0 with

72

Figure 5.7. The left-hand sphere SL and the left-hand disk DL are shownschematically for a 2-manifold with µ = λ = 1. Here ϕL : −1, 1 × (−1, 1)→ V0.

all integral curves of X leading to the critical point p. The left-hand disk DL is a

smoothly embedded disk with boundary SL, defined to be the union of the segments of

these integral curves beginning in SL and ending at p. See 5.7.

Similarly the characteristic embedding

ϕR : Bλ × Sn−λ−1 → V1

is obtained by embedding Bλ × Sn−λ−1 → Vε by

(θu,v) 7→ g(εu sinh(θ), εv cosh(θ))

and then translating the image to V1. The right-hand sphere SR of p in V1 is defined

to be ϕR(0 × Sn−λ−1). It is the boundary of the right-hand disk DR, defined as

the union of segments of integral curves of X beginning at p and ending in SR. See

Figure 5.8.

73

Figure 5.8. The right-hand sphere SR and the right-hand disk DR are shownschematically for a 2-manifold with µ = λ = 1. Here ϕR : (−1, 1)× −1, 1 → V1.

Definition 76. An elementary cobordism is a smooth manifold triad (W ;V, V ′)

possessing a Morse function f with exactly one critical point p. The index of

(W ;V, V ′) is the index of p with respect to f .

An elementary cobordism (W ;V, V ′) is not a product cobordism and so by

Theorem 16 has a Morse number µ(W ;V, V ′) = 1. Also, the index of an elementary

cobordism is well-defined. That is, it is independent of the choice of the Morse

function f , and hence p.

The next two results show that performing surgery corresponds to passing a

critical point of index λ of a Morse function on an n-manifold.

Theorem 18. If V ′ = χ(V, ϕ) can be obtained from V by surgery of type (λ, n− λ),

then there exists an elementary cobordism (W ;V, V ′) and a Morse function f : W →

R with exactly one critical point, of index λ.

74

Figure 5.9. BoundingCurves of L1 ⊂ R2

Figure 5.10. Bounding Surfaces of L2 ⊂ R3

Proof. Let

Lλ =

(x,y) ∈ Rλ × Rn−λ : −1 ≤ −‖x‖2 + ‖y‖2 ≤ 1 and ‖x‖ ‖y‖ < (sinh 1)(cosh 1).

(See Figures 5.9 and 5.10.)

Then Lλ is a differentiable manifold with two boundaries, given by

−‖x‖2 + ‖y‖2 = −1 and − ‖x‖2 + ‖y‖2 = 1.

The “left” boundary, −‖x‖2 + ‖y‖2 = −1, is diffeomorphic to Sλ−1×Bn−λ under the

correspondence

(u, θv)↔ (u cosh θ,v sinh θ), θ ∈ [0, 1),u ∈ Sλ−1,v ∈ Sn−λ−1.

75

The “right” boundary, −‖x‖2 + ‖y‖2 = 1, is diffeomorphic to Bλ× Sn−λ−1 under the

correspondence

(θu,v)↔ (u sinh θ,v cosh θ), θ ∈ [0, 1),u ∈ Sλ−1,v ∈ Sn−λ−1.

Consider the orthogonal trajectories of the surfaces given by the equations

−‖x‖2 + ‖y‖2 = k, k ∈ R.

Given a point (x,y), the trajectory which passes through it can be parameterized in

the form

t 7→ (t−1x, ty).

If x = 0, this trajectory is a straight line segment shooting away from the origin;

if y = 0, this trajectory is a straight line directed toward the origin. For both x

and y nonzero, this trajectory is a hyperbola which starts at some well-defined point

(u cosh θ,v sinh θ) on the left boundary of Lλ and is directed toward the corresponding

point (u sinh θ,v cosh θ) on the right boundary.

Now, construct an n-manifold W = ω(V, ϕ) as follows. Start with the disjoint

sum

((V r ϕ

(Sλ−1 × 0

))× D1

)q Lλ.

For each u ∈ Sλ−1, v ∈ Sn−λ−1, θ ∈ (0, 1), and c ∈ D1, identify the point (ϕ (u, θv) , c)

in the first summand with the unique point (x,y) ∈ Lλ such that

76

1. −‖x‖2 + ‖y‖2 = c

2. (x,y) lies on the orthogonal trajectory which passes through the point

(u cosh θ,v sinh θ).

This correspondence defines a diffeomorphism

ϕ(Sλ−1 ×

(Bn−λ r 0

))× D1 ←→ Lλ ∩

((Rλ r 0

)×(Rn−λ r 0

)).

So ω(V, ϕ) is a well-defined smooth manifold.

This manifold ω(V, ϕ) has two boundaries, each corresponding to the values

c = −‖x‖2 + ‖y‖2 = ±1.

The left boundary, c = −1, can be identified with V , letting z ∈ V correspond to

(z,−1) ∈(V r ϕ

(Sλ−1 × 0

))× D1 for z /∈ ϕ

(Sλ−1 × 0

)(u cosh θ,v sinh θ) ∈ Lλ for z = ϕ(u, θv).

The right boundary can be identified with χ(V, ϕ) via the correspondences

z←→ (z,+1), z ∈ V r ϕ(Sλ−1 × 0

)(θu,v)←→ (u sinh θ,v cosh θ), (θu,v) ∈ Bλ × Sn−λ−1.

A function f : ω(V, ϕ)→ R is defined by:

f(z, c) = c for (z, c) ∈(V r ϕ

(Sλ−1 × 0

))× D1

f(x,y) = −‖x‖2 + ‖y‖2 for (x,y) ∈ Lλ.

77

Then f is a well-defined Morse function with one critical point, of index λ.

Theorem 19. Let (W ;V, V ′) be an elementary cobordism with characteristic em-

bedding ϕL : Sλ−1 × Bn−λ → V . Then (W ;V, V ′) is diffeomorphic to the triad

(ω (V, ϕL) ;V, χ (V, ϕL)).

Theorem 20. Let (W ;V, V ′) be an elementary cobordism possessing a Morse function

with one critical point, of index λ. Let DL be the left-hand disk associated to a fixed

gradient-like vector field. Then V ∪ DL is a deformation retract of W .

Corollary 8. With (W ;V, V ′) an elementary cobordism possessing a Morse function

with one critical point, of index λ, and DL the left-hand disk associated to a fixed

gradient-like vector field, the relative homology groups

Hn(W,V ) ∼=

Z, n = λ

0 , otherwise.

A generator for Hλ(W,V ) is represented by DL.

Proof. By Theorem 20, V ∪ DL is a deformation retract of W . So

H∗(W,V ) ∼= H∗(V ∪ DL, V ).

78

Let U = V r SL. Then U ⊂ int(V ). By excision,

H∗(V ∪ DL, V ) ∼= H∗((V ∪ DL) r U, V r U)

= H∗(DL ∪ SL,SL)

= H∗(DL, SL)

∼=

Z, n = λ

0 , otherwise.

5.4 Rearrangement of Cobordisms

From now on c will denote a cobordism rather than an equivalence class of

cobordisms.

Definition 77. If a composition cc′ of two elementary cobordisms is equivalent to a

composition dd′ of two elementary cobordisms such that

index(c) = index(d′) and index(c′) = index(d)

then we say that the composition cc′ can be rearranged.

We state a theorem which guarantees that, given a Morse function and a

gradient-like vector field on cc′, the composition can be rearranged if SR ∩ SL = ∅.

Theorem 21. Let (W ;V0, V1) be a triad with Morse function f having two critical

points, p and p′. Suppose that for some choice of gradient-like vector field X, the

compact set Kp of points on trajectories going to or from p is disjoint from the compact

79

set Kp′ of points on trajectories going to or from p′. If f(W ) = [0, 1] and a, a′ ∈ (0, 1),

then there exists a new Morse function g such that

1. X is a gradient-like vector field for g,

2. the critical points of g are still p and p′, with g(p) = a and g(p′) = a′,

3. g agrees with f near V0 ∪ V1 and equals f plus a constant in some neigh-

borhood of p and in some neighborhood of p′.

Theorem 22 (Rearrangement Theorem). Any cobordism c may be expressed as a

composition

c = c0c1c2 · · · cn, n = dim c,

where each cobordism ci admits a Morse function with only critical points of index i

all located on the same level.

Corollary 9. Given any Morse function f on a triad (W ;V0, V1), there exists a new

Morse function f , called self-indexing, which has the same set of critical points as

f , each with the same index, and which has the properties

1. f(V0) = −12

2. f(V1) = n+ 12

3. f(p) = index(p) for each p ∈ Σf .

CHAPTER 6

THE H-COBORDISM THEOREM

6.1 Cancellation Theorems

Definition 78. Two submanifolds Mm, Nn ⊂ V v are said to have transverse in-

tersection (or to intersect transversely) if at each point q ∈M ∩N the tangent

space to V at q is spanned by the vectors tangent to M and the vectors tangent to N .

(If m+n < v this is impossible, so transverse intersection simply means M∩N = ∅.)

Theorem 23 (First Cancellation Theorem). If the intersection of SR with S′L 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 X on an arbitrarily small

neighborhood of the single trajectory T from p to p′ producing a nowhere zero vector

field X ′ whose trajectories all proceed from V0 to V1. Further, X ′ is a gradient-like

vector field for a Morse function f ′ without critical points that agrees with f near

V0 ∪ V1.

Let M1 and M2 be smooth submanifolds of dimensions m1 and m2 in a smooth

manifold M of dimension m = m1+m2 that intersect in points p1, . . . , pk transversely.

Suppose that M1 is oriented and that the normal bundle ν(M2) of M2 in M is oriented.

At pi choose a positively oriented m1-frame X1, . . . , Xm1 of linearly independent vec-

tors spanning TMpi . Since the intersection at pi is transverse, the vectors X1, . . . , Xm1

represent a basis for the fibre at pi of the normal bundle ν(M2).

Definition 79. The intersection number of M1 and M2 at pi is defined to

be +1 or −1 according to whether the vectors X1, . . . , Xm1 represent a positively or

80

81

negatively oriented basis for the fiber at pi of ν(M2). The intersection number

M2 ·M1 of M1 and M2 is the sum of the intersection numbers at the points pi.

Theorem 24 (Second Cancellation Theorem). Let (W n;V0, V1) be a triad with a

Morse function f having a gradient-like vector field X. Suppose p and p′ are critical

points of f with indices λ and λ + 1, respectively. Assume that f(p) < 12< f(p′)

and that an orientation has been given to the left-hand sphere S′L in V = f−1(12) and

also to the normal bundle in V of the right-hand sphere SR. Suppose W , V0, and V1

are simply connected, and λ ≥ 2 and λ + 1 ≤ n − 3. If SR · S′L = ±1, then W n is

diffeomorphic to V0 × [0, 1]. In fact, if SR · S′L = ±1, then X can be altered near V

so that the right- and left-hand spheres in V intersect in a single point, transversely;

and the conclusions of the First Cancellation Theorem (Theorem 23) then apply.

It is the case that V = f−1(12) is also simply connected. This is a result of

applying Van Kampen’s theorem to get

π1(V ) ∼= π1(Dn−λR (p) ∪ V ∪ Dλ+1

L (q)),

where the restrictions λ ≥ 2 and n − λ ≥ 3 are employed. By Theorem 20, the

inclusion

DR(p) ∪ V ∪ DL(q) → W

is a homotopy equivalence. So π1(V ) ∼= π1(W ) = 1.

Because of Theorem 25 below, the Second Cancellation Theorem is true even

with the single dimension restriction n ≥ 6.

Theorem 25 (H. Whitney (1944) [13]). Let M and N be closed smooth transversely

intersecting submanifolds of dimensions m and n in the smooth (m+ n)-manifold V

82

(without boundary). Suppose that M is oriented and that the normal bundle ν(N) of

N in V is oriented. Further, suppose that m + n ≥ 5 and n ≥ 3. In case n = 1

or n = 2, suppose that the inclusion induced map π1(V rN)→ π1(V ) is one-to-one

into.

Let p, q ∈M ∩N be points with opposite intersection numbers such that there

exists a loop L contractible in V that consists of a smoothly embedded path from p to

q in M followed by a smoothly embedded path from q to p in N , where both paths miss

(M ∩N) r p, q.

With these assumptions, there exists an isotopy ht, t ∈ I, of the identity

idV : V → V such that

1. The isotopy fixes idV near (M ∩N) r p, q

2. h1(M) ∩N = (M ∩N) r p, q.

If M and N are connected, n ≥ 2 and V is simply connected, no explicit

assumption about the loop L is required.

By turning the triad around (reversing f), one arrives at the following corollary

of the Second Cancellation Theorem (Theorem 24).

Corollary 10. Theorem 24 is also true when the dimension conditions are replaced

by λ ≥ 3 and λ+ 1 ≤ n− 2.

Definition 80. Suppose W is a compact oriented smooth n-dimensional manifold.

Then ∂W is given a well-defined orientation, called the induced orientation, by

saying that an (n − 1)-frame τ1, . . . , τn−1 of vectors tangent to ∂W at some point

p ∈ ∂W is positively oriented if the n-frame ν, τ1, . . . , τn−1 is positively oriented in

TWp, where ν is any vector at p tangent to W but not to ∂W and pointing out of W

(i.e., ν is outward normal to ∂W ).

83

Alternatively, one specifies J∂W K ∈ Hn−1(∂W ) as the induced orientation

generator for ∂W , where J∂W K is the image of the orientation generator JW K ∈

Hn(W,∂W ) for W under the boundary homomorphism Hn(W,∂W )→ Hn−1(∂W ) of

the exact sequence for the pair (W,∂W ).

Theorem 26. Suppose a cobordism c is represented by the triad (W ;V, V ′), f is a

Morse function for c, and c = c1c2 · · · cn, where each cλ is such that f has critical

points all on the same level with index λ. Set Wλ = c1 · · · cλ for λ = 0, 1, . . . , n and

W−1 = V so that

V = W−1 ⊆ W0 ⊆ W1 ⊆ · · · ⊆ Wn = W.

Define

Cλ = Hλ(Wλ,Wλ−1) ∼= H∗(Wλ,Wλ−1)

and let ∂ : Cλ → Cλ−1 be the boundary homomorphism for the exact sequence of the

triple Wλ−2 ⊆ Wλ−1 ⊆ Wλ (see Theorem 9). Then C∗ = Cλ, ∂ is a chain complex.

That is, ∂2 = 0. Moreoever, Hλ(C∗) ∼= Hλ(W,V ) for all λ = 0, . . . , n.

Theorem 27 (Poincare Duality). If (W ;V, V ′) is a smooth manifold triad of dimen-

sion n and W is oriented, then for λ = 0, . . . , n

Hλ(W,V ) ∼= Hn−λ(W,V ′).

Theorem 28 (Basis Theorem). Suppose (W ;V, V ′) is a triad of dimension n pos-

sessing a Morse function f with all critical points having index λ and lying on the

same level. Let X be a gradient-like vector field for f . Assume that 2 ≤ λ ≤ n − 2

84

and that W is connected. Then, given any basis for Hλ(W,V ), there exists a Morse

function f ′ and a gradient-like vector field X ′ for f ′ which agree with f and X in a

neighborhood of V ∪ V ′ and are such that f ′ has the same critical points as f , all on

the same level, and the left-hand disks for X ′, when suitably oriented, determine the

given basis.

Theorem 29 (Product Cobordism Theorem). Suppose (W ;V, V ′) is a triad of di-

mension n ≥ 6 possessing a Morse function with no critical points of indices 0, 1

or n − 1, n. Furthermore, assume that W,V and V ′ are all simply connected (hence

orientable) and that H∗(W,V ) = 0. Then (W ;V, V ′) is a product cobordism.

Proof. Set c = (W ;V, V ′). By the Rearrangement Theorem (Thm. 22), there is a

Morse function f such that c = c2c3 · · · cn−2, where cλ has critical points all on one

level and each with index λ. From Theorem 26, there is a sequence of free abelian

groups

Cn−2∂n−2−→ Cn−3

∂n−3−→ · · ·Cλ+2∂λ+2−→ Cλ

∂λ−→ · · · ∂3−→ C2.

Pick a basis zλ1 , . . . , zλkλ

for ker ∂λ ⊆ Cλ. H∗(W,V ) = 0 implies im ∂λ+1 = ker ∂λ.

Thus, choose bλ1 , . . . , bλkλ−1

∈ Cλ so that ∂λ(bλi ) = zλ−1

i for i = 1, · · · , kλ−1. Nowzλikλi=1∪bλikλ−1

i=1is a basis for Cλ.

Since 2 ≤ λ < λ + 1 ≤ n − 2, the Basis Theorem (Theorem 28) applies. So

one can find a Morse function f ′ and a gradient-like vector field X ′ on c such that

the left hand disks DL of cλ and cλ+1 represent the chosen bases for Cλ and Cλ+1.

Let p and q be critical points of cλ and cλ+1 corresponding to zλ1 and bλ+11 .

By increasing f ′ in a neighborhood of p and decreasing f ′ in a neighborhood of q we

85

obtain cλcλ+1 = c′λcpcqc′λ+1, where cp has only the critical point p and cq has only the

critical point q. Let V0 denote the level manifold between cp and cq.

Now, cpcq and its two end manifolds are simply connected (c.f. the remarks

following Theorem 24). Since ∂(bλ+11 ) = zλ1 , the spheres SR(p) and SL(q) in V0 have

intersection number ±1. By the Second Cancellation Theorem (Theorem 24), cpcq is

a product cobordism and f ′ and X ′ can be altered on the interior of cpcq so that f ′

has no critical points there. Repeating this process as many times as possible removes

all critical points. Therefore, by Theorem 16, (W ;V, V ′) is a product cobordism.

Theorem 30.

Index 0. If H0(W,V ) = 0, the critical points of index 0 can 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 “trades” the

critical points of index 1 for an equal number of critical points of index 3.)

6.2 Proof of the h-Cobordism Theorem

Theorem 31 (The h-Cobordism Theorem). Suppose the triad (W n;V, V ′) has the

properties

1. W , V , and V ′ are simply connected,

2. H∗(W,V ) = 0, and

3. dimW = n ≥ 6.

Then W is diffeomorphic to V × [0, 1].

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First, note that H∗(W,V ) = 0 is equivalent to the condition H∗(W,V′) = 0 by

Poincare Duality (Thm. 27).

Proof. By Corollary 9, there is a self-indexing Morse function f for (W ;V, V ′). The-

orem 30 kills all critical points of index 0 or 1. Now replace f by −f so that a critical

point of index λ for f becomes a critical point of index n−λ for −f . Using Theorem

30 with −f eliminates critical points of indices n and n− 1 for f . Now Theorem 29

gives the result.

Theorem 31 gets its name from the definition below and the remark which

follows.

Definition 81. A triad (W ;V, V ′) = 0 is an h-cobordism and V is said to be

h-cobordant to V ′ if both V and V ′ are deformation retracts of W .

Conditions 1 and 2 in Theorem 31 imply that (W ;V, V ′) is an h-cobordism.

The key result of the thesis is the following corollary of Theorem 31.

Corollary 11. Two simply connected closed smooth manifolds of dimensions ≥ 5

that are h-cobordant are diffeomorphic.

6.3 Applications of the h-Cobordism Theorem

Proposition 1 (The Generalized Poincare Conjecture in Dimensions ≥ 5). If M is

a closed simply-connected smooth n-manifold, n ≥ 5, with the (integral) homology of

the n-sphere Sn, then M is homeomorphic to Sn. If n = 5 or 6, M is diffeomorphic

to Sn.

See [9].

Corollary 12. If M is a closed smooth n-manifold, n ≥ 5, which is a homotopy

n-sphere (i.e., is of the homotopy type of Sn), then M is homeomorphic to Sn.

REFERENCES

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88

[1] G. Birkhoff and S. MacLane, A Survey of Modern Algebra, Macmillan, NewYork, 3rd Edition, 1965.

[2] G. Bredon, Geometry and Topology, Springer-Verlag, New York, 1993.

[3] W. Fleming, Functions of Several Variables, Springer-Verlag, New York, 2ndEdition, 1977.

[4] M. Hirsch, Differential Topology, Springer-Verlag, New York, 1976.

[5] A. Kosinski, Differential Manifolds, Dover Publications, Mineola, 1993.

[6] J. Lambek, Lectures on Rings and Modules, Blaisdell Publishing, Waltham,1966.

[7] J. Milnor, Morse Theory, Princeton University Press, Princeton, 1963.

[8] —–, Lectures on the h-Cobordism Theorem, Princeton University Press, Prince-ton, 1965.

[9] S. Smale, Generalized Poincare’s Conjecture in Dimensions Greater than Four,Annals of Math. vol. 74 (1961), pp. 391-406.

[10] —–, On the Structure of Manifolds, Amer. J. of Math. vol. 84 (1962), pp.387-399.

[11] M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish orPerish, Princeton, Vol. 1, 3rd Edition, 2005.

[12] H. Whitney, Differentiable Manifolds, Annals of Math. vol. 37 (1936), pp. 645-680.

[13] —–, The Self-intersections of a Smooth n-manifold in 2n-space, Annals of Math.vol. 45 (1944), pp. 220-246.

APPENDIX: A THEOREM ON QUADRATIC FORMS

89

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The lemma of Morse (Lemma 9) takes inspiration from the lemma below about

quadratic forms over the real number field. It is included here for reference as a simpler

version of the proof of Morse’s Lemma. Also, the proof given of Morse’s Lemma refers

to the proof below for the process of ensuring that the upper-lefthand entry of the

matrix (hij(x)) is nonzero.

Definition 82. A quadratic function f : Rn → R is a map of the form

f(x1, . . . , xn) =n∑

i,j=1

fijxixj,

where not all fij = 0, and such that fij = fji for all i, j = 1, . . . , n.

Note that any function f : Rn → R of the form

f(x1, . . . , xn) =n∑

i,j=1

fijxixj

can be symmetrized by taking fij = 12

(fij + fji). Then

f(x1, . . . , xn) =n∑

i,j=1

fijxixj

and fij = fji.

Lemma 18. For any quadratic function f : Rn → R, there is a nonsingular linear

transformation T : Rn → Rn of the variables such that

(f T )(x1, . . . , xn) = −(x1)2 − · · · − (xλ)2 + (xλ+1)2 + · · ·+ (xr)2

for some r ≤ n.

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Moreover, the number λ above is uniquely designated by the quadratic function

f . This is Sylvester’s Law of Inertia. See e.g. [1] p.254.

Proof. The proof proceeds in two stages. First, it is shown that f can be diagonalized

by a suitable transformation. Then the variables are scaled to give the desired form

of the theorem.

Claim 1: (Diagonalization) There is a nonsingular linear transformation T

and constants d1, . . . , dn ∈ R such that

(f T )(x1, . . . , xn) =n∑i=1

di(xi)2.

Proof of Claim 1. The proof is by induction. Define the proposition P (k) to be true

iff there exists a nonsingular linear transformation T : Rn → Rn, a quadratic function

f : Rn−k → R, and constants d1, . . . , dk ∈ R such that

(f T )(x1, . . . , xn) =k∑i=1

di(xi)2 + f(xk+1, . . . , xn).

P (1) :

Case 1. f11 6= 0: Define T by

T (x1, . . . , xn) =

(x1 −

n∑j=2

f1j

f11

xj, x2, . . . , xn

).

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Then by symmetry of the fij and completing the square

(f T )(x) =n∑

i,j=1

fij(Tx)i(Tx)j

= f11

(x1 −

n∑j=2

f1j

f11

xj

)2

+ 2n∑i=2

f1i

(x1 −

n∑j=2

f1j

f11

xj

)xi +

n∑i,j=2

fijxixj

= f11

(x1 −n∑j=2

f1j

f11

xj

)2

+ 2n∑i=2

f1i

f11

(x1 −

n∑j=2

f1j

f11

xj

)xi

+n∑

i,j=2

fijxixj

= f11

(x1 −n∑j=2

f1j

f11

xj −n∑i=2

f1i

f11

xi

)2

−

(n∑i=2

f1i

f11

xi

)2+

n∑i,j=2

fijxixj

= f11

(x1)2 −( n∑

i=2

f1i

f11

xi

)2+

n∑i,j=2

fijxixj

= f11

(x1)2 − f11

(n∑i=2

f1i

f11

xi

)2

+n∑

i,j=2

fijxixj

= d1

(x1)2

+ f(x2, . . . , xn),

where

f(x2, . . . , xn) = −f11

(n∑i=2

f1i

f11

xi

)2

+n∑

i,j=2

fijxixj and d1 = f11.

Case 2. f11 = 0: By hypothesis, there is some fk` 6= 0.

Subcase i. fkk 6= 0 for some k > 1. Take S to be the permutation matrix that swaps

x1 and xk. Now, similar to Case 1, define T by

T (x1, . . . , xn) =

(x1 −

n∑j=2

(f S)1j

(f S)11

xj, x2, . . . , xn

),

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where f S is viewed as a quadratic function so that (f S)ij are the coefficients in

the definition. (Note that one still has (f S)ij = (f S)ji.) Then S T is the desired

map.

Subcase ii. fk` 6= 0 where k, ` > 1 and k 6= `. Without loss of generality, assume

k < `. Define S by

S(x1, . . . , xn) = (xk, x`, x3, . . . , x1 − x2︸ ︷︷ ︸kth

, . . . , x1 + x2︸ ︷︷ ︸`th

, . . . , xn).

Then

(f S)(x) = 2fk`(x1 − x2)(x1 + x2) + other terms not (x1)2

= 2fk`(x1)2 − 2fk`(x

2)2 + other terms not (x1)2.

So (f S)11 = 2fk` 6= 0. To make f S into a quadratic function, just symmetrize by

defining

(f S)ij =1

2[(f S)ij + (f S)ji] .

Note that (f S)11 = (f S)11 = 2fk` 6= 0. Now, similar to Case 1, define T by

T (x1, . . . , xn) =

x1 −n∑j=2

(f S)1j

(f S)11

xj, x2, . . . , xn

.

The desired transformation is S T . This finishes to proof of P (1).

To see that P (k) =⇒ P (k + 1), notice that P (k) leaves f with the first k

variables diagonalized and the last n− k variables as arguments of another quadratic

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function f . Applying P (1) to this new quadratic function f gives P (k + 1). This

concludes the diagonalization proof.

Claim 2: (Scaling) Suppose f : Rn → R is a quadratic function of the form

f(x1, . . . , xn) =n∑i=1

di(xi)2.

Then there is a linear transformation T such that

(f T )(x1, . . . , xn) = −(x1)2 − · · · − (xλ)2 + (xλ+1)2 + · · ·+ (xn)2.

Proof of Claim 2. Set

T (x1, . . . , xn) =

(1√|d1|

x1, . . . ,1√|dn|

xn

).

Then

(f T )(x) =n∑i=1

di

(1√|di|

xi

)2

=n∑i=1

di|di|

(xi)2 =n∑i=1

±(xi)2.

Finally, define S to be the permutation matrix so that all the negative signs are on

the first variables and the positive signs on the last variables. Then T S(= S T )

is the desired transformation.

VITA

Quinton Westrich earned his B.S. degrees in physics and mathematics from

Tennessee Technological University in 2008. He has been involved in research in

Lie algebra symmetries with Jurgen Fuchs at Karlstad University in Sweden and

computational astrophysics with Peter Gogelein at Tubingen Universitat in Germany.

He was awarded the Stanley Dolzycki Memorial Scholarship in 2009 and 2010 for this

research.

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