Differential Topology and the h-Cobordism Theorem
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Transcript of Differential Topology and the h-Cobordism Theorem
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 .
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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|),
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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
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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.
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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.
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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.
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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 ).
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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
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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
87
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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