Homology Theory, Morse Theory, and the Morse Homology …€¦ · Homology Theory, Morse Theory, and the Morse Homology Theorem Alex Stephanus June 8, 2016 Abstract This thesis develops

Homology Theory, Morse Theory, and the Morse

Homology Theorem

Alex Stephanus

June 8, 2016

Abstract

This thesis develops the Morse Homology Theorem, first starting bymotivation for and developing of the notion of homology groups of a mani-fold. Following this, we introduce some introductory Morse Theory, specif-ically the process of building a CW complex which is homotopy equivalentto a given manifold. Finally, we introduce the machinery required for theproof of the Morse Homology Theorem, which relates the homology groupsgenerated by the n-cells of a manifold to the homology groups generatedby critical points of certain functions on the manifold.

Contents

1 Introduction 1

2 Motivation for Homology Groups 2

3 Homology 5

4 Morse Theory: Constructing CW Complexes 13

5 Gradient Flows, Stable and Unstable Manifolds, and Morse-Smale functions 21

6 Corollaries to the λ-lemma 24

7 The Morse Homology Theorem 26

8 Acknowledgements 36

1 Introduction

Classification of manifolds has been a central goal of topology since the field hasexisted, and even today it remains a key focus of the field. In the last hundred

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years or so algebraic topology has sought to ascribe groups to manifolds and,in doing so, apply insights from algebra to the topological problem of manifioldclassification. Within the past 50 years even more disciplines have been pulledin to help understand this problem. In this thesis we build up the theory ofMorse Homology, which links the algebraic structures of homology with themore geometric flow structure that particular real-valued functions induce on amanifold.

We begin by developing some of the machinery of homology theory, whichseeks to understand a manifold’s different-dimensional ”holes” in an algebraicsense. We show some important properties of homology groups, for instancethat they are a homotopy invariant of a space, and explore the relative homol-ogy groups of a space, which provide a way to ignore certain subspaces of atopological space. The other important result that we prove in this section isthat, for a CW complex, the homology which arises from the cell structure isidentical to the more traditional singular homology.

Following the equivalence of cellular and singular homology we turn to MorseTheory to construct a way to put a cell structure on a manifold. As it turnsout, a smooth real-valued function with some nice properties is all that we needto determine a cell structure on a manifold, and in this section we flesh outthe process for determining a cell structure given an appropriate function on amanifold.

Finally, we bring these two theories together to prove the Morse HomologyTheorem. This theorem, which is essentially the capstone result of this thesis,states that that singular homology groups of a manifold coincide exactly withthe homology groups generated by the manifold’s critical points.

This thesis is meant to be largely self-contained, if the reader has a fairlygood footing in basic abstract algebra, differential topology, and Riemanniangeometry then everything should roll out smoothly. However, due to the needto build up multiple disciplines some proofs are omitted for the sake of brevityand readability. In general, if a result is stated without proof it is likely becausethe proof is both long and not terribly insightful for what we’re building to.

2 Motivation for Homology Groups

Rather than just dive into homology theory, we begin with the motivating ex-ample of homotopy groups of a space. These groups are very intuitively definedand can give insight into how a space is structured. However, homotopy groupsare also problematic in some ways. Some of these problems provide motiva-tion for the development of homology theory which, althouth not as intuitivelydefined, is generally much better-behaved.

2.1 The Fundamental Group

One of the simplest tools for studying manifolds is the fundamental group, π1.The fundamental group of a manifold X is the group of based loops in X, i.e.

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maps α : [0, 1] → X such that α(0) = α(1). subject to the relation that twoloops are considered equivalent if they are homotopically identical. The groupoperation is concatenation of loops, specifically concatenating two paths α andβ to get a path α ◦ β, defined as follows:

(α ◦ β)(l) =

{α(2l) if 0 ≤ l ≤ .5β(2l − 1) if .5 < l ≤ 1

The set of maps combined with this concatenation operation satisfies the groupaxioms as follows.

1. Closure: If one concatenates two loops α and β which both begin and endat some point x0 in X then their concatenation is also clearly a loop in Xbased at x0.

2. Identity: The identity is given by the path which simply stays at x0

3. Associativity: Given 3 paths α, β, and γ the order of composition doesnot matter.

4. Inversion: Given a loop α based at x0, let α−1 be defined as α−1(l) =α(1− l). The homotopy given by

(ft)(l) =

α(2l) if 0 ≤ l ≤ t

2

α(t) if t2 ≤ l ≤ 1− t

2

α−1(2l − 1) if 1− t2 ≤ l ≤ 1

is smooth with f0(l) = x0 and f1(l) = (α ◦α−1)(l), so α ◦α−1 ' β, whereβ is the identity loop.

The fundamental group is useful for a variety of reasons, but one of the mostimportant is its ease of computation. Van Kampen’s Theorem gives a relativelysimple way to calculate the fundamental group of a space by splitting it up intosmaller spaces. A simplified statement of the theorem is as follows:

Theorem 2.1 (van Kampen’s Theorem) Let X be a topological space, and letU1 and U2 be two open, path-connected subspaces of X. If U1 ∩ U2 is path con-nected and nonempty then X is path connected and the inclusion morphismsdraw the commutative pushout below:

π1(U1)

π1(U1 ∩ U2)

i1 -

π1(U1) ∗π1(U1∩U2) π1(U2) k--

π1(X)-

π1(U2)

--

i2-

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The natural morphism k is an isomorphism, that is to say that the funda-mental group X is the free product of the fundamental groups of 1 and 2 withamalgamation of the fundamental group of their intersection.

The proof of van Kampen’s Theorem is omitted here, as it is fairly involvedand not particularly relevant to us. For a full proof see pg. 53 of [1].

Van Kampen’s Theorem, as it describes spaces which are built out of smallerspaces, leads naturally to the question of whether there is a convenient way toconstruct spaces out of smaller spaces. As it turns out there is and these spaces,known as CW complexes, are central to the study of algebraic topology.

2.2 CW Complexes

In algebraic topology, it is often helpful to look at topological spaces whichare endowed with a useful structure. One such set of spaces considered to beparticularly nice are called CW complexes. A CW complex is constructed byinductively gluing together disks of increasing dimension along their boundaries.The definition of a CW complex is below, note that en is taken to be the opendisk of dimension n. When necessary, we will refer to the boundary of this diskas en.

1. Start with a discrete set X0, regarding the points of this set to be 0-cells.

2. Inductively build the n-skeleton, Xn by attaching n-cells enα to Xn−1 viamaps φα : Sn−1 → Xn−1.

3. This process can either terminate at some finite n, in which case X = Xn,or it can go on infinitely, in which case X =

⋃iX

i

CW Complexes are particularly well-behaved spaces, with this inherent in-ductive structure supplying many useful properties. For instance, in a CWcomplex X, the fundamental group is the free group with generators comingfrom the 1-skeleton of X and relations coming from the 2-skeleton of X.

2.3 Homotopy Groups

The fundamental group allows us to characterize manifolds in a low-dimensionalmanner. Anywhere a loop cannot be contracted to a point it means that thereis some sort of hole in the manifold. For instance, the circle S1 has funda-mental group isomorphic to Z, which is indicative of the area in the center ofthe circle which a loop cannot be contracted through. However, Sn is simplyconnected for n > 1, that is to say it has a trivial fundamental group. Thesehigher-dimensional spheres still have holes in a similar manner to S1, but thatinformation is not captured by the fundamental group. This makes sense, as thefundamental group considers low-dimensional maps and homotopies. In fact, ina CW complex the fundamental group can be calculated knowing only the 2-skeleton of the complex. This lack of higher-dimensional information provided

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by the fundamental group leads naturally to the question of whether higher-dimensional analogs exist. The fundamental group considers maps of S1 intospaces and homotopies of those maps, and similarly the nth homotopy groupis constructed in a similar manner, only one considers maps and homotopies ofmaps of Sn. The main issue with homotopy groups is that they cannot gener-ally be computed as easily as the fundamental group. Van Kampen’s Theoremgives a nice way to calculate the fundamental group of a space, but it does notgeneralize into higher dimensions.

3 Homology

Homotopy groups, while useful for examining a space’s structure, have the draw-back that they are not terribly easy to use and they are not always algebraicallysimple. For instance, the fundamental group of a space is not necessarily abelian,a property that we would like to have. So, in an effort to avoid some of the com-putational difficulties associated with homotopy groups, we develop homologytheory. The definition is much less intuitive at first, but it has a similar goal ofputting an algebraic structure on the n-dimensional holes of a topological space.

3.1 Simplices

The theory of homology is developed primarily by considering simplices, sobefore we develop homology theory, it is appropriate to first define a simplex.An n-dimensional simplex, denoted as ∆n, is a generalization of a 2-d triangleinto higher dimensions. n-simplices are defined by sets of points [v0, . . . , vn]subject to the condition that no set of k points lies in a hyperplane of dimensionless than k− 1, for 2 < k ≤ n+ 1. Simplices are meant to generalize the notionof a triangle into higher dimensions, so this condition is a generalization of theidea that one cannot have a triangle with three collinear points.

∆n = {(v0t0, . . . , vntn) ∈ Rn |∑i

ti = 1 and ti ≥ 0 for every i}

The order of a simplex’s vertices is relevant when considering boundary mapslater, so we keep the order given by the subscripts of the vertices.

Additionally, we consider the boundary of ∆n to be the union of the simplicesof dimension n− 1 formed when deleting a vertex from the original n-simplex,i.e. the simplices [v0, . . . , vi, . . . , vn], with vi indicating that the ith vertex hasbeen removed. These (n−1)-simplices are called the faces of ∆n. One importantconvention to take is that in the faces of ∆n, or rather any subsimplex ∆k of∆n, the vertices of ∆k are ordered as they are in ∆n. Additionally, we orientthe edges of ∆n, [vi, vj ] towards the higher-ordered vertex. This orientation ofedges carries over into all subsimplices of ∆n.

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3.2 Singular Homology

Singular homology stems from so-called ’singular’ mappings of simplices into aspace X. These maps need only be continuous, so the term singular captures theidea that a map need not necessarily resemble a simplex in its image. So, wedefine a singular n-simplex to be a map σ : ∆n → X. To start with, we definethe group Cn(X) to be the free abelian group whose basis is the set of singularn-simplices in X. The elements of Cn(X), n-chains, are finite sums of the form∑i niσi, with ni in Z and σi singular n-simplices of X.Homology concerns itself with how the n-simplices of X attach to the n− 1-

simplices of X, and in order to examine that, we define the following boundarymap ∂n : Cn(X)→ Cn−1(X), below.

∂n(σ) =∑i

(−1)iσ | [v0, . . . , vi, . . . , vn]

Restricting σ to the set [v0, . . . , vi, . . . , vn] gives us a new (n − 1)-chain, withthe (−1)i precipitating out of the orientation of the edges which are removed.An important property of this boundary map follows from the definition:

Lemma 3.1 For any n-simplex, ∂n−1∂n = 0

Proof The proof follows from simply composing ∂n with ∂n− 1

∂n(σ) =∑i

(−1)iσ | [v0, . . . , vi, . . . , vn] so, we get that

∂n−1∂n(σ) =∑j<i

(−1)i(−1)jσ | [v0, . . . , vj , . . . , vi, . . . , vn] +

∑j>i

(−1)i(−1)j−1σ | [v0, . . . , vi, . . . , vj , . . . , vn]

As j ranges through all values other than i, switching j and i in the second sumcancels out the first sum, so the sum is zero when taken over all possible valuesof i and j.

If we look at all of these maps for a given space X

. . .∂n+1−−−→ Cn(X)

∂n−→ Cn−1(X)∂n−1−−−→ . . .

∂1−→ C0(X)∂0−→ 0

we get what is called a chain complex on the space, a series of maps betweenabelian groups with the property ∂n∂n+1 = 0.

The composition ∂n∂n+1 being identically zero is equivalent to the statementthat the image of ∂n+1 is contained in the kernel of ∂n. As Im(∂n+1) is a (notnecessarily proper) subgroup of Ker(∂n), the quotient Ker(∂n)/Im(∂n+1)exists. We define the nth singular homology group via this quotient.

Hn(X) = Ker(∂n)/Im(∂n−1)

We can think of Ker(∂n) as the group of cycles of n−simplices, and Im(∂n+1)to be the group of boundaries (this group contains precisely the chains of n-simplices which form the boundary of some (n + 1)-chain). In this sense, ho-mology groups are obtained by taking cycles mod boundaries.

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3.3 Homotopy Invariance of Homology Groups

One particularly important property of singular homology groups is that theyare invariant under homotopy, that is to say that if two spaces X and Y arehomotopy equivalent then they have isomorphic homology groups. This sec-tion contains the proof of that fact, which stems from the homomorphism ofhomology groups which is induced by a map between two spaces.

This homomorphism comes from exactly where one would expect it to, takinga map f : X → Y and composing each σ : ∆n → X with f . This composition ofmaps induces a homomorphism f∗ : Hn(X)→ Hn(Y ) in the following manner.If we denote the composition fσ as f#(σ), then we get that each f#(σ) is amap from ∆n into Y . We can then extend this map linearly to chains in Y bydefining f#(Σiniσi) = Σinif#(σi) The crucial detail of this composition is thefollowing chain of equalities.

f#∂(σ) = f#(Σi(−1)iσ|[v0, . . . , vi, . . . , vn])

= Σi(−1)ifσ|[v0, . . . , vi, . . . , vn] = ∂f#(σ)

Where ∂ is the same boundary operator defined previously. Because ∂ and fcommute, we can construct the following commutative diagram.

. . . - Cn+1(X)∂- Cn(X)

∂- Cn−1(X) - . . .

. . . - Cn+1(Y )

f#?

∂- Cn(Y )

f#?

∂- Cn−1(Y )

f#?

- . . .

These f ′s define a chain map from the chain complex of X into that of Y . Wecan see that f# takes cycles to cycles and boundaries to boundaries by observingthat ∂α = 0 gives ∂(f#α) = f#(∂α) = 0, and also that f#(∂β) = ∂(f#β). So,f# induces a homomorphism f∗ : Hn(X) → Hn(Y ). The following theorem isone of the more important results we will be using from homology theory, andit will show up throughout this thesis.

Theorem 3.2 If two maps f, g : X → Y are homotopic, then they induce thesame homomorphism f∗ = g∗ : Hn(X)→ Hn(Y )

Proof They key to this proof is the following way to divide ∆n × I into sim-plices. Let ∆n × 0 be given by [v0, . . . , vnt] and ∆n × 1 by [w0, . . . , wn] subjectto the condition that vi and wi have the same image under the natural pro-jection ∆n × I → ∆n. We can then go from [v0, . . . , vn] to [w0, . . . , wn] bystitching them together with a sequence of n-simplices. These simplices are ofthe form [v0, . . . , vi, wi+1, . . . , wn] for every i with −1 ≤ i ≤ n. Between eachsuccessive pair of these simplices, i.e. between [v0, . . . , vi, wi+1, . . . , wn] and[v0, . . . , vi−1, wi, wi+1, . . . , wn], is an (n + 1)-simplex. The union of all of these(n+ 1)-simplices spans the region ∆n × I.

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Following this we define a new operator, called a prism operator, using ahomotopy F : X × I → Y and a simplex ∆n → X. Composing these two maps,we get a map F ◦ (σ × I) : ∆n × I → X × I → Y. Then, we define the prismoperator P : Cn(X)→ Cn+1(Y ) as follows

P (σ) =∑I

(−1)iF ◦ (σ × 1)|[v0, . . . , vi, wi, . . . , wn]

These operators have the property that ∂P = g# − f# − P∂ (for verification ofthis property see the proof of 2.10 in [1]). Using this information, we know thatif α ∈ Cn(X) is a cycle then we have g#(α)−f#(α) = ∂P (α)+P∂(α) = ∂P (α).The last equality comes from the fact that ∂α = 0 (as it is a cycle). Thus, g#−f#

is a boundary, which we know from before means that g# and f# determine thesame homology class. As such, g# and f# are equal on the homology class ofα, and we are done.

This statement implies immediately that for a homotopy equivalence f :X → Y the induced homomorphisms f∗ : Hn(X)→ Hn(Y ) are in fact isomor-phisms.

This is one of the most important results we will take from homology theory,and we will combine with soon-to-come results about the homology of a CWcomplex to do a lot of heavy lifting later.

3.4 Exact Sequences, Relative Homology, and the Mayer-Vietoris Sequence

A question that comes up naturally when studying spaces is how subspaces fitinto and define the larger space. Spaces can often be constructed inductivelyfrom smaller spaces, so understanding the behavior of a larger space relativeto its subspaces is typically very useful information to have. Homology is noexception to this rule, and the concept of a relative homology group attemptsto understand these relationships. Relative homology groups essentially conveyhow the homology of a space changes when we ignore one of its subspaces.However, before we rigorize this idea we will define something called an exactsequence, which is a structure that is quite important for the understanding ofrelative homology groups. We have already defined a chain complex, that is aseries of homomorphisms from the chain groups of a space into each other:

. . .∂n+2−−−→ Cn+1(X)

∂n+1−−−→ Cn(X)∂n−→ Cn−1(X)

∂n−1−−−→ . . .

with the additional information that ∂n∂n+1 = 0 for every n, and we alsoknow that this condition is equivalent to the inclusion Im(∂n+1) ⊆ Ker(∂n).The notion of an exact sequence strengthens this condition from an inclusionto an equivalence. That is to say that an exact sequence is a sequence ofhomomorphisms between abelian groups

. . .αn+2−−−→ An+1

αn+1−−−→ Anαn−−→ An−1

αn−1−−−→ . . .

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with the additional constraint that Im(αn+1) = Ker(αn). The inclusionsnecessary for this sequence to be a chain complex are of course satisfied, andbecause the two groups are identical it means that Ker(∂n)/Im(∂n+1) = Hn(X)is trivial.

Here we will just list a few properties of exact sequences that follow imme-diately from the definition of an exact sequence. They come up often enoughthat it is worth listing them out.

1. 0→ Aα−→ B is exact if and only if Ker(α) = 0, so if α is injective.

2. Aα−→ B → 0 is exact iff Im(α) = B, so if α is surjective. It follows

naturally then that

3. 0→ Aα−→ B → 0 is exact iff α is an isomorphism.

4. 0→ Aα−→ B

β−→ C → 0 is exact if and only if α is injective, β is surjective,and Im(α) = Ker(β). In this case we get that C is isomorphic to B/Im(α).This property is not as obvious as the other three, but it will be relevantlater.

One particularly important result concerning exact sequences is the followingtheorem.

Theorem 3.3 If A is a nonempty closed subspace of a space X, and is also adeformation retract of a neighborhood in X, then there exists an exact sequence

. . . Hn(A)i∗−→ Hn(X)

j∗−→ Hn(X/A)∂−→

Hn−1(A)i∗−→ Hn−1(X)

j∗−→ . . .j∗−→ H0(X/A)→ 0

where i∗ is the homomorphism induced by the inclusion i : A ↪→ X and j∗ isinduced by the quotient map j : X → X/A.

It should be noted that the groups H are the reduced homology groups of anonempty space X. These groups are the homology groups of the chain complex

. . .→ C2(X)∂−→ C1(X)

∂−→ C0(X)ε−→ Z→ 0

with the map ε(∑i niσi) =

∑i ni. Hn(x) ≈ Hn(X) for n > 0, and H0(X) ≈

H0(X)⊕ Z.A proof of this theorem and a construction of the boundary map ∂ can

be found in Chapter 2 of [1] (Theorem 2.13 and proof). An important bit ofterminology to remember is that if a space X and subspace A ⊆ X satisfy thetheorem’s conditions then we call (X,A) a good pair.

We have already posed the question of how to understand how the homologyof X changes when you ”ignore” one of its subspaces, and one natural idea isto examine the homology of the space when you quotient out the subspace.Relative homology groups do just that, and are defined in the following way soas to effectively ignore the homology of X generated by the subspace.

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Given a space X and a subspace A ⊆ X, we define Cn(X,A) to be thequotient Cn(X)/Cn(A) so any chain in A is quotiented out. Chains in A haveboundaries in A, so the boundary map ∂ takes Cn(A) into Cn−1(A). This meansthat the boundary map induces a quotient map ∂ : Cn(X,A) → Cn−1(X,A).So, we get a similar chain complex

. . .∂n+1−−−→ Cn(X,A)

∂n−→ Cn(X,A)∂nn−1−−−−→ . . .

We know the relation ∂n∂n+1 = 0 holds in Cn(X), so it holds for the quotient-induced operator as well. The homology groups of this chain complex are definedto be the relative homology groups Hn(X,A). Elements of the group Hn(X,A)are represented by relative cycles, which are n-chains whose boundaries lie com-pletely in A.

Relative homology groups share a similar property to Theorem 3.3, in thatthey fit into the following exact sequence:

. . . Hn(A)i∗−→ Hn(X)

j∗−→ Hn(X,A)∂−→

Hn−1(A)i∗−→ Hn−1(X)

j∗−→ . . .j∗−→ H0(X,A)→ 0

where i∗ and j∗ are the homomorphisms induced by the inclusions A ↪→ Xand (X, ∅) ↪→ (X,A). The boundary operator ∂ : Cn(X,A) → Cn−1(X,A) isdefined by considering the boundaries of relative cycles which represent elementsof Hn(X,A). If a relative cycle α represents an element [α] then the boundarymap sends [α] to the class of the boundary of α in Hn(A). We refer to theoperator ∂ in this sequence as a connecting homomorphism, a definition whichwill be important later. One important property of this map ∂ is that it is anatural transformation, which is to say that for a map f : (X,A)→ (Y,B), thediagram below is commutative.

. . . - Hn(A)i∗- Hn(X)

j∗- Hn(X,A)∂- Hn−1(A) - . . .

. . . - Hn(B)

f∗?

i∗- Hn(Y )

f∗?

j∗- Hn(Y,B)

f∗?

∂- Hn−1(B)

f∗?

- . . .

A more thorough discussion of naturality can be found in Chapter 2 of [1].One additional result that we will use once later is the Excision Theorem.

This theorem lays out conditions under which a a subset Z ⊂ A ⊂ X can bedeleted without affecting the relative homology groups Hn(X,A). The state-ment of the theorem is as follows.

Theorem 3.4 Given subspaces Z ⊂ A ⊂ X such that Z is contained in theinterior of A, then the inclusion (X − Z,A − Z) ↪→ (X,A) induces isomor-phisms HN (X − Z,A − Z) → Hn(X,A) for all n. Equivalently, for subspacesA,B ⊂ X whose interiors cover X, the inclusion (B,A ∩B) ↪→ (X,A) inducesisomorphisms Hn(B,A ∩B)→ Hn(X,A) for all n.

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We only use this theorem once, and the proof is quite laborous, so it is omittedhere. For a fully fleshed-out proof of the excision theorem the reader is directedto Theorem 2.20 in [1].

As we mentioned earlier, one thing we would very much like to have is a wayto build up the homology of a space X by considering it as the union of smallerspaces. The fundamental group has this property in van Kampen’s Theorem,but this does not generalize effectively to higher dimensions, one problem withhomotopy groups. As it turns out, homology groups have a largely identicalproperty, the Mayer-Vietoris sequence. This is a long exact sequence that existsfor a space X and two subspaces A,B ⊆ X such that X is the union of theinteriors of A and B. The sequence is as follows:

. . .→ Hn(A ∩B)(i∗,j∗)−−−−→ Hn(A)⊕Hn(B)

k∗−l∗−−−−→ Hn(X)∂∗−→ Hn−1(A ∩B)→

. . .→ H0(X)→ 0

Here the homomorphisms are induced by the maps i : A ∩ B ↪→ A, j :A ∩ B ↪→ B, k : A ↪→ X, and l : B ↪→ X. We will not use Mayer-Vietorissequences at all, but as an analogue for van Kampen’s theorem they they area property of homology groups worth mentioning. One can find a derivation ofthe Mayer-Vietoris sequence in Chapter 2 of [1].

3.5 Cellular Homology

We already know that a cell structure on a space is a very nice thing to have,and we get many useful properties from that. That niceness extends to thehomology groups of a space–it turns out that we can compute cellular homologygroups directly from the cell structure of a space, and these homology groupsare identical to the singular homology groups introduced before. As such, theycarry all of the properties from earlier as well.

To begin, there are a few relevant facts which are worth establishing aboutthe homology groups of a CW complex. They are as follows:

If a space X is a CW complex then the following three properties are true.

1. 1. Hk(Xn, Xn−1) is zero for k 6= n and for k = n is free abelian with abasis generated by the n-cells of X.

2. 2. Hk(Xn) = 0 for k > n Particularly, if X is of finite dimension thenHk(X) = 0 for k >dim(X).

3. 3. The map Hk(Xn)→ Hk(X) arising naturally from the inclusion Xn ↪→X is an isomorphism for k < n and is surjective for k = n.

The proof of these statements can be found in [1], proposition 2.34. We willnot reproduce it here as they are not central results, but rather useful pieces ofinformation.

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Theorem 3.5 HCWn (X) ≈ Hn(X), where Hn(X) is the nth singular homology

group, and HCWn (X) is the nth cellular homology group, which we will now

define.

Let X be a CW complex, then by stitching together pieces of the exactsequences for the relative homology groups we arrive at the following diagram:

0

0 Hn(Xn+1) ≈ Hn(X)

-

Hn(Xn)

-

-

. . . - Hn+1(Xn+1, Xn)dn+1 -

∂n+1 -

Hn(Xn, Xn−1)dn -

jn

-

Hn−1(Xn−1, Xn−2) - . . .

Hn−1(Xn−1)jn−1

-

∂n-

0

-

Note that the isomorphism Hn(Xn+1) ≈ Hn(X) comes from fact 3 from before.In this diagram the maps dn and dn+1 are defined simply by the compositions

jn−1∂n and jn∂n−1 in the diagram. It we look at dndn+1, we can see that itis the composition jn−1∂njn∂n+1. which contains the composition ∂njn, whichis contained in the exact sequence for (Xn, Xn−1). As such, the compositiondndn+1 is zero, so the sequence traversed by the maps dk is a chain complex(Hn(Xn, Xn−1) is a free abelian group generated by the n-cells of X, so the restof the requirements of something to be a chain complex are satisfied). We callthe homology groups of this complex the cellular homology groups of X, andwrite them as HCW

n (X). The proof that these are isomorphic to the singularhomology groupsHn(X) comes from considering the relationships between mapsin the above diagram.

Proof We know from property 4 of exact sequences listed earlier that Hn(X)is isomorphic to Hn(Xn)/Im(∂n+1). Because jn is injective, it gives rise to anisomorphism between Im(∂n+1) ⊆ Hn(Xn) and Im(jn∂n+1) ⊆ Hn(Xn, Xn−1).Note that Im(jn∂n+1) = Im(dn+1), so this means that Im(∂n+1) ≈ Im(dn). Ad-ditionally, because jn−1 is also injective we know by the same argument thatKer(dn) ≈ Ker(∂n).Because Im(∂n+1) ≈ Im(dn+1) and Ker(dn) ≈ Ker(∂n), we know that thequotients are also isomorphic, so that we get Hn(X) ≈ Hn(Xn)/Im(∂n+1) ≈Ker(dn)/Im(dn+1) = HCW

n (X).

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This equivalence is quite important to us, as the cellular homology groupsof a space X can be computed directly from a cell structure on that space. Sin-gular homology groups, on the other hand, are rooted in monstrous free abeliangroups, often with uncountably many generators. Computation becomes muchmore straightforward then as the groups required to compute cellular homologiesare so much more manageable than those required for singular homology.

4 Morse Theory: Constructing CW Complexes

In light of the equivalence of cellular homology and singular homology, it isnatural that we would like to know how reliably we can apply the techniquesof cellular homology to manifolds in general. In order to define the cellularhomology groups of a manifold the manifold has to have a cell structure onit, so which manifolds can be given a cell structure is therefore an importantquestion for us to try and answer. Morse theory gives us with an answer to thatquestion by providing a way to obtain a cell structure on a manifold M using afunction on the manifold with certain important properties.

Before we begin proving any results in this section, it is worth taking a littlebit of time to state some definitions which we will use here and throughoutthe rest of this thesis. For starters, any time we consider a manifold M it willbe a smooth manifold, ”smooth” here meaning C∞. If we consider a smoothfunction f : M → R, a point p in M is called a critical point of f if the mapf∗ : TMp → TRf(p) is zero. The image of p is called a critical value of f. If apoint q ∈ R is not a critical value we call it a regular value.

A bit of notation we will use quite often is Ma, defined as

Ma = {x ∈M |f(x) ≤ a}.

If a is a regular value, then Ma is a smooth manifold with boundary.We call a critical point p non-degenerate if the matrix

(∂2f

∂xi∂xj(p))

is nonsingular, and we call the functional associated to this matrix the Hessianof f at p. Obviously, if p is a non-degenerate critical point of f then the abovematrix must be of full rank at p. An additional piece of terminology we will useis calling a smooth function f : M → R a Morse function if it has no degeneratecritical points.

One last important definition is that of the index of a bilinear functional.For a functional H, the index of the functional is the maximum dimension of asubspace V on which H is negative definite. Specifically, we will call the indexof the Hessian on TMp the index of f at p. The index of a critical point is acrucial concept throughout this section, as the construction of the CW complexassociated to a manifold M stems precisely from the indices of the critical pointsof a function on M. The following lemma, which we will prove half of, lays the

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groundwork for this construction as well as gives a more tangible meaning tothe index of a critical point.

Lemma 4.1 If p is a non-degenerate critical point of f(from the previous lemma),then in a neighborhood U of p there exists a local coordinate system (y1, . . . , yn)with yi(p) = 0 for all i and such that the identity

f = f(p)− (y1)2 − . . .− (yλ)2 + (yλ+1)2 + . . .+ (yn)2

holds through U. In this case, λ is the index of f at p.

Proof The part of this lemma that we will prove is that if such an f exists,then λ must be the index of f at p. If we take the coordinate system (z1, . . . , zn)at p such that

f(q) = f(p)− z1(q)2 − . . .− zλ(q)2 + zλ+1(q)2 + . . .+ zn(q)2

then we know that

∂2f

∂zi∂zj(p) =

−2 if i = j ≤ λ2 if i = j > λ

0 if i 6= j

This is equivalent to the matrix of the Hessian of f with respect to the basisgenerated by the coordinate functions at p being diagonal, with λ (-2)’s followedby (n − λ) 2’s along the diagonal of the matrix. So, there is a λ-dimensionalsubspace W of TMp where the Hessian is negative definite, and an (n − λ)-dimensional subspace V where the Hessian is positive definite. Each of thesesubspaces is the largest of their respective types, as if there were a negativedefinite subspace of dimension greater than λ then it would intersect with V ,which cannot be the case as V is positive definite. A similar argument showsthat V is also the positive definite subspace of maximal dimension. So, λ is theindex of the Hessian at p by the definition of index.

We will not prove the second half of this lemma, that such a coordinatesystem exists, as the proof revolves around manipulating and transforming dif-ferent coordinate systems and is not helpful to understanding the rest of thissection. The proof of the first half is helpful as it helps ground the idea of theindex of a critical point as well as demonstrates its importance.

The process for building a CW complex for a manifold M boils down totraversing the manifold by considering Ma for increasing values of a and at-taching cells at critical points of f. The following theorems detail the process ofattaching these cells, culminating eventually with a theorem which describes thecell structure associated to a manifold. To begin, we show that this structuredepends entirely on the critical points of f.

Theorem 4.2 Suppose f is a smooth real-valued function on a manifold M. As-sume also that a and b are numbers in R with a < b, and that the set f−1[a, b]

14

is compact and contains no critical points of f . It follows that Ma is diffeomor-phic to M b. In fact, Ma is a deformation retract of M b, so the inclusion mapMa ↪→M b is a homotopy equivalence.

Before we proceed with the proof of this theorem, we must introduce anothercouple of important definitions.

To start, we define the gradient vector field of a function on a Riemannianmanifold. Consider a Riemannian manifold (M, g), and let < X,Y > denotethe inner product of two tangent vectors provided by the metric. For a smoothfunction f : M → R, the gradient vector field of f with respect to the metric gis the smooth vector field ∇f such that for every smooth vector field V on M,< ∇f, V >= df(V ) = V · f . It should be noted that ∇f is unique.

Additionally, we must define a 1-parameter group of diffeomorphisms on amanifold M. A 1-parameter group of diffeomorphisms on a manifold M is acontinuous action of the real numbers on M given by the map φ : R×M →Mwith the following two properties.

1. For every t ∈ R the map φt : M → M defined as φtp = φ(t, p) is adiffeomorphism of M onto M.

2. φt ◦ φs = φt+s for all pairs t, s ∈ R

An important fact about 1-parameter groups which will be used in the fol-lowing proof is as follows:

Lemma 4.3 A smooth vector field on M which vanishes outside of a compactset K ⊂M generates a unique 1-parameter group of diffeomorphisms for M.

For a proof of this result, the reader is directed to Section I.2 of [2] (Lemma 2.4and proof). With these two definitions in mind, we proceed with the proof ofthe theorem.

Proof Essentially, what we want to do is push M b down to Ma orthogonallythrough the level sets f−1(c) for c in [a, b]. This should deform M b smoothlyinto Ma and provide us with a suitable deformation retraction.In order to do this, we take ρ to be smooth function ρ : M → R such that

ρ =1

< ∇f,∇f >

within the set f−1[a, b], and vanishes outside of a compact neighborhood of thisset. We then let X be the vector field defined as

Xq = ρ(q)(∇f)q

So, this vector field points in the same direction as ∇f , but with inverse mag-nitude. This set satisfies the conditions of Lemma 4.3, so it generates a unique1-parameter group of diffeomorphisms φt : M → M. Now, we fix some q in Mand, if it is contained in the set f−1[a, b] then we know that

df(φt(q))

dt= <

dφt(q)

dt,∇f > = < X,∇f > = 1.

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So, we know that the map t→ f(φt(q)) is then linear with derivative equal to 1when f(φt(q)) is between a and b. Because of this, φb−a maps Ma diffeomorphi-cally onto M b. We then define the 1-parameter family of maps rt : M b → M b

as

rt(q) =

{q if f(q) ≤ a (i.e. if q is in Ma

φt(a−f(q))(q) if a ≤ f(q) ≤ b

For this family of maps, r0 is the identity map on M b, and the homotopy-equivalent r1 retracts M b onto Ma. This is the definition of a deformationretraction, so we are done.

Theorem 4.4 Let f : M → R be a smooth function, and suppose p in M isa non-degenerate critical point of index λ. Say f(p) = c, and assume that forsome ε > 0 f−1[c− ε, c+ ε] is compact and contains no other critical points off . Then, for all ε small enough, M c+ε is homotopy equivalent to M c−ε with aλ-cell attached to it.

Proof The soul of this proof is contained in two separate parts. The first step isto select a neighborhood H of p in M and show that M c−ε∪eλ is a deformationretract of M c ∪H. Following this, we use the preceding theorem to show thatM c−ε∪H is a deformation retract of M c+ε. Combining these 2 retractions thengives the result we’re looking for. To begin, we look for the retraction fromM c−ε ∪H to M c−ε ∪ eλ. We want to choose a coordinate system u1, . . . , un ina neighborhood U of p where the following property holds

f = f(p)− (u1)2 − . . .− (uλ)2 + (uλ + 1)2 + . . .+ (un)2

By the Morse Lemma we know that such a coordinate system exists, as p isnon-degenerate. We also know that uk(p) = 0 for all 1 ≤ k ≤ n. Following this,we choose a small enough ε > 0 such that the following two properties hold.

1. f−1[c− ε, c+ ε] is compact and only contains p as a critical point.

2. U ′s image in Rn coming from this coordinate system contains the closedball

{(u1, . . . , un)|∑i

(ui)2 ≤ 2ε}

We then define eλ to be the set of points in U where

(u1)2 + . . .+ (uλ)2 ≤ ε and uλ+1, . . . , un = 0

It is worth noting that the intersection of eλ ∩M c−ε is the boundary eλ. Nowthat we have established exactly the λ-cell we are using, we proceed to establishthe two deformation retractions we are looking for. To do so, we construct anew function F : M → R as follows. First we take a function µ : R → R withthe following properties.

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1. µ(0) > ε

2. µ(r) = 0 for r ≥ 2ε

3. −1 ≤ dµdr ≤ 0 for all r. We refer to

dµdr

as µ′.

We now use µ to define F by slightly perturbing f in the region U . That is, wedefine F as

F =

{f for x outside of the neighborhood U

f − µ((u1)2 + . . .+ (uλ)2 + 2(uλ+1)2 + . . .+ 2(un)2) for x contained in U

For the following steps, it simplifies computations if we define two functionsξ, η : U → [0,∞) as follows:

ξ(u1, . . . , un) = (u1)2 + . . .+ (uλ)2

η(u1, . . . , un) = (uλ+1)2 + . . .+ (un)2

Intuitively, they represent the magnitude of a point (u1, . . . , un)’s presence in thenegative definite and positive definite subspaces of TMp. Following immediatelyfrom these definitions, we know that f = c + ξ − η and that F = c − ξ + η −µ(ξ + 2η).To start, we know that the region F−1(−∞, c+ ε] is identical to f−1(∞, c+ ε] =M c+ε. This is because when ξ + 2η ≤ 2ε we know that µ(ξ + 2η) = 0 Whenξ + 2η ≤ 2ε we know the following:

F ≤ f = c− ξ + η ≤ c ≤ ξ

2≤ c+ ε

We also know that F has the same critical point set as f on M . The functionsare equal outside of U, so we look in the region U . Examining the partials

∂F

∂ξ= −1− µ′(ξ + 2η)

∂F

∂η= 1− 2µ′(ξ + 2η)

and observing that

dF =∂F

∂ξdξ +

∂F

∂ηdη

we can see that dF = 0 only when dξ and dη are zero. However, this happensonly at the origin, so the origin can be the only critical point of F in U. So, thecritical points of f and F are identical.

Now, we look at the region F−1[c − ε, c + ε]. We know from previouslythat F−1(−∞, c + ε] is identical to M c+ε, and because F ≤ f in all of M, weknow that M c−ε ⊆ F−1(−∞, c− ε]. From these two observations, we know thatF−1[c− ε, c+ ε] ⊆ f−1[c− ε, c+ ε]. As F−1[c− ε, c+ ε] is closed and a subset ofcompact f−1[c− ε, c+ ε], it is compact as well. As a subset of f−1[c− ε, c+ ε],

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it also contains no critical points of F other than p. However, we know thatF (p) = c−µ(0) < c− ε, so F−1[c− ε, c+ ε] does not contain p as a critical pointeither. As F−1[c−ε, c+ε] contains no critical points, we know from the previoustheorem that F−1(−∞, c−ε] is a deformation retract of F−1(−∞, c+ε] = M c+ε.For convenience’s sake, we will refer to F−1(−∞, c− ε] as M c−ε ∪H, where wedefine H to be the closure of (F−1(−∞, c− ε]−M c−ε).

We now consider the cell eλ which consists of all points q with ξ(q) ≤ ε, andη(q) = 0. We know that ∂F

∂ξ < 0 so for all q ∈ eλ we have that F (q) ≤ F (p) <

c− ε. However, f(q) ≥ c− ε for q ∈ eλ, so eλ is in the region H.Now, we seek to prove that M c−ε ∪ eλ is a deformation retract of M c−ε ∪H.

We proceed simply by defining the deformation retraction that we want. Wedenote this deformation retraction rt : M c−ε ∪H → M c−ε ∪H, and it will bedefined differently for different parts of this space. For starters, we can definert to be the identity outside of U. Within U, the definition is a bit more com-plicated, taking on the form of three cases depending on the value of ξ.

The first case is when ξ ≤ ε, in which case we let rt be the transformation

(u1, . . . , un)→ (u1, . . . , uλ, tuλ+1, . . . , tun).

So, r1 is simply the identity map on this region and r0 maps (u1, . . . , un) to(u1, . . . , uλ, 0, 0, . . . , 0), with the last (n− λ) coordinates equal to zero. This isprecisely the region eλ.

The second case is for the region ε ≤ ξ ≤ η + ε. In this region, we define rtas mapping

(u1, . . . , un)→ (u1, . . . , uλ, stuλ+1, . . . , stu

n).

We define the number st as

st = t+ (1− t)(ξ − εη

)12 .

We have s1 is simply 1, and s0 is ( ξ−εη )12 . So, r1 is the identity map, and r0

takes (u1, . . . , un) to (u1, . . . , uλ, ( ξ−εη )12uλ+1, . . . , ( ξ−εη )

12un). Taking f(r0(u1, . . . , un)),

we get f = c− ξ + η( ξ−εη ) = c− ε. So, r0 takes all of region 2 into f−1(c− ε).In the third case, we consider the region η+ε ≤ ξ, we just define rt to be the

identity. It should be noted that when ξ = η+ε in case 2 we get that st is equal to1 so rt is the identity map. Thus, we have defined a deformation retraction fromF−1(−∞, c+ ε] to M c−ε ∪ eλ. Composing this with the deformation retractionfrom M c+ε to F−1(−∞, c+ ε] we have a deformation retraction from M c+ε toM c−ε ∪ eλ and are done.

Now that we know how to attach cells to critical points of f, we can buildup a cell structure on a manifold M using the following theorem.

Theorem 4.5 If f is a Morse function on a manifold M and Ma is compactfor each a, then M is homotopy-equivalent to a CW complex, with each criticalpoint of index λ corresponding to a different λ-cell.

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Before we begin the proof of this theorem, we first prove two additionallemmas which the proof hinges on. The lemmas and their proofs are as follows.

Lemma 4.6 Let φ0 and φ1 be homotopic maps from the sphere eλ to a spaceX. Then, the identity map of X extends to a homotopy equivalence

k : X ∪φ0 eλ → X ∪φ1 e

λ

Proof We prove this simply by defining the homotopy equivalence k as follows.k(x) = x for x in X

k(tu) = 2tu for 0 ≤ t ≤ 12 and u in eλ

k(tu) = φ2−2tφ(u) for 12 ≤ t ≤ 1 and u in eλ

φt indicates a homotopy map between φ0 and φ1 (we know this exists from theconditions of the lemma). tu also indicates multiplying a unit vector in eλ witht ∈ [0, 1].

Lemma 4.7 If we let φ : eλ → X be an attaching map then any homotopyequivalence f : X → Y extends to a homotopy equivalence

F : X ∪φ eλ → Y ∪fφ eλ

Proof Let us start off by defining a function F as follows:

F =

{f for x in X

identity for x in eλ

Let g : Y → X be a homotopy inverse to f and define then G : Y ∪fφ eλ →X ∪gfφ eλ in a similar manner to F . That is, set G|Y = g andG|eλ is the identity map.

We know then that gfφ is homotopic to φ, so it follows from the previouslemma that there is a homotopy equivalence

k : X ∪gfφ eλ → X ∪φ eλ

Firstly, we prove that the triple-composition

kGF : X ∪φ eλ → X ∪φ eλ

is homotopic to the identity map on X ∪φ eλ. Let ht be a homotopy betweengf and the identity function. Then, from the definitions of k, F, and G we getthat

kGF (x) = gf(x) for x in X

kGF (tu) = 2tu for 0 ≤ t ≤ 12 and u in eλ

kGF (tu) = h2−2tφ(u) for 12 ≤ t ≤ 1 and u in eλ

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From this set of equations we can define an appropriate homotopy

qτ : X ∪φ eλ → X ∪φ eλ

This homotopy is as follows:qτ (x) = hτ (x) for x in X

qτ (tu) = 21+τ for 0 ≤ t ≤ 1+τ

2 and u in eλ

qτ (tu) = h2−2t+τφ(u) for 1+τ2 ≤ t ≤ 1 and u in eλ

In order to complete the proof of this lemma, it is important to prove thefollowing statement about left and right homotopy inverses.

Aside: If a map F has a left homotopy inverse L and a right homotopy inverseR, then F is a homotopy equivalence and R and L are two-sided inverses.

The proof is relatively simple. LF ' identity and FR ' identity give thatL ' L(FR) = (LF )R ' R, so RF ' LF ' identity. This shows that R is a2-sided inverse, and a similar proof shows that L is as well, so we are done.

With this in mind, we examine the functions we have defined for this proof.We know that kGF ' identity, so we know that F has a left homotopy inverse.However, were we to construct G via a function g such that f were a homotopyinverse of g, we would have that kFG ' identity, so G has a left homotopyinverse.

From Lemma 4.6, we know that k is a homotopy equivalence, so that meansthat k(GF ) ' identity implies that (GF )k ' identity. This means that G hasa right homotopy inverse, which from the previous aside indicates that G is a2-sided homotopy inverse of (Fk). So, we know that (Fk)G ' identity. Fromthis and the knowledge that F has a left homotopy inverse, we get that F is ahomotopy equivalence and we are done.

With these two lemmas proved, we continue on to the proof of the maintheorem.

Proof Let c1 < c2 < c3 < . . . be the critical values of a smooth functionf : M → R. The sequence has no accumulation point since we know that eachMa is compact. Additionally, if R 3 a < c1 then we know that Ma is empty.So, suppose that a 6= cn for any n, and that Ma is homotopy equivalent toa CW complex. Let us take c to be the smallest ci > a. By the precedingtheorems, we know that M c+ε is homotopic to M c−ε ∪φ1

eλ1 ∪φ2. . .∪φj(c) eλj(c)

for attachment maps φ1, . . . , φj(c). We also know that there exists a homotopyequivalence h : M c−ε →Ma, and we have assumed that a homotopy equivalenceh′ : Ma → K exists, where K is a CW complex.

In this case, we have that for each i, h′ ◦ h ◦ φi is homotopic to a mapψi, which attaches the cell eλi onto the (i− 1)-skeleton of K. So, we have thatK∪ψ1

eλ1∪ψ2. . .∪ψj(c) eλj(c) is again a CW complex, and is homotopy equivalent

to M c+ε by Lemmas 4.6 and 4.7.

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Inducting, we know that Ma′ is homotopy equivalent to a CW complex forevery a′. If M has all its critical points contained in a compact set Ma then weare done, as even if M is not compact, if all its critical points are compact thenour result follows from Theorem 4.2. If M has its critical points contained insome noncompact set then we get an infinite chain

Ma1 ⊂ Ma2 ⊂ Ma3 ⊂ . . .

K1

?⊂ K2

?⊂ K3

?⊂ . . .

where each downward arrow is a homotopy equivalence coming from earlier inthe proof. Define K to be the union of the Ki in the limit topology, and letg : M → K be the limiting map. As g is the limit of homotopy equivalences, ginduces isomorphisms of homotopy groups of every dimension. This completesthe proof for the case of a noncompact interval being needed to encompass allthe critical points of M , and thus completes the proof of the Theorem.

So, given a Morse function on a manifold M, we can use that function tobuild up a CW complex which is homotopy-equivalent to M. In light of thehomotopy-invariance of the homology groups of a space, this means that wecan apply the techniques of cellular homology to any smooth manifold that wecan find a Morse function on. However, the need for a Morse function on Mraises the question of how often we can find such a function. We get a verywell-behaved structure on a manifold if there is a Morse function on it, but itwould be a shame if Morse functions were too uncommon to be really useful.

Luckily, Morse functions are very easy to find for any smooth manifold M.We will not detail the process here, but in Section I.6 of [1], Milnor shows that forany smooth manifold M there exists a Morse function for which Ma is compactfor every a in R. So, for any smooth manifold M we can find a Morse functionon M, which means that every smooth manifold is homotopy-equivalent to aCW complex.

5 Gradient Flows, Stable and Unstable Mani-folds, and Morse-Smale functions

In this section we will lay much of the groundwork for the Morse HomologyTheorem. We begin by defining gradient flows on a manifold and stating someof their properties, and following that we explore stable and unstable manifolds,which are an important way of characterizing a flow on a manifold. Specifically,for a flow on a manifold, the stable and unstable manifolds of a critical pointrigorize the idea of spaces flowing into and out of that point. With stable andunstable manifolds developed we will define Morse-Smale functions, which areMorse functions that satisfy an additional constraint on the intersections oftheir stable and unstable manifolds. We then detail some important results

21

concerning flow lines between critical points which we will use extensively inthe proof of the Morse Homology Theorem.

5.1 Gradient Flows

Before we proved Theorem 4.2 earlier, we defined the gradient vector field ofa function on a manifold, and we also defined a 1-parameter group of diffeo-morphisms on a manifold. In this section, we will combine the two conceptsto create a 1-parameter group of diffeomorphisms which is generated by thegradient vector field of a function f. Specifically, we define a gradient flow ona manifold as follows: given a smooth function f on a manifold M, we take φtto be the 1-parameter group of diffeomorphisms generated by the −∇f. This isthe group with

φ0(x) = x andddtφt(x) = −(∇f)(φt(x)).

Given this group φt, we can define the curve γx : (a, b)→M by

λx(t) = φt(x)

For every x ∈ M . This curve is called a gradient flow line. This is due to thefollowing 2 results, Propositions 3.18 and 3.19 from [3], presented here withoutproof.

Lemma 5.1 Every smooth function f : M → R on a finite-dimensional smoothRiemannian manifold (M, g) decreases along its gradient flow lines.

Lemma 5.2 Let f : M → R be a Morse function on a finite dimensionalcompact smooth Riemannian manifold (M, g). Then every gradient flow line fbegins and ends at a critical point of M.

5.2 Stable and unstable manifolds

Another important definition which we will use is that of stable and unstablemanifolds. These rigorize the ideas of flows into and out of a point. Thesemanifolds are defined for any non-degenerate critical point p ∈ M of a smoothfunction f .

The stable manifold of p is the set W s(p) = {x ∈M | limt→∞

φt(x) = p}

The unstable manifold of p is the set Wu(p) = {x ∈M | limt→−∞

φt(x) = p}

Essentially, the stable manifold of the point p is the set of points which flowinto p under the 1-parameter group φt, and the unstable manifold is the set ofpoints which flow out of p under φt. There is an extremely important theoremto do with the stable and unstable manifolds of critical points, whose proof iscovered through most of Chapter 4 of [3].

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Theorem 5.3 Let f : M → R be a Morse function on a compact smoothRiemannian manifold (M, g) of dimension m <∞. If p ∈M is a critical pointof f, then the tangent space at p splits as

TpM = T spM⊕

TupM

where the Hessian is positive definite on T spM and negative definite on TupM.Additionally, the stable and stable manifolds are surjective images of smoothembeddings

Es : T spM →W s(p) ⊆M

Eu : TupM →Wu(p) ⊆M

So, W s(p) is a smoothly embedded open disk of dimension m− λp, and Wu(p)is a smoothly embedded open disk of dimension λp, where λp is the index of p.

With stable and unstable manifolds defined, we can use them to define aMorse-Smale function on M, which we will be basing the proof of the MorseHomology Theorem on.

5.3 Morse-Smale Functions

We have already dealt with Morse functions on a manifold M. That is, smoothfunctions f : M → R which have no degenerate critical points. However, astronger condition on the function, called the Morse-Smale transversality con-dition, places additional constraints on the stable and unstable manifolds of thecritical points of f. This condition is as follows:

A Morse function on a finite-dimensional smooth Riemannian Manifold (M,g)is said to satisfy the Morse-Smale transversality condition if the stable and un-stable manifolds of f intersect transversally, that is to say that

Wu(q) tW s(p)

for all critical points p, q of f. If a Morse function satisfies this condition thenwe call if a Morse-Smale function.

Stemming directly from this transversality condition is a categorization ofintersections of the stable and unstable manifolds of different critical points.

Lemma 5.4 Let f : M → R be Morse-Smale function on a finite-dimensionalcompact smooth manifold (M, g). If p and q are critical points of f with non-trivial intersection, then Wu(q) ∩W s(p) is an embedded (λq − λp)-dimensionalsubmanifold of M .

Proof The proof comes simply from the fact that Wu(q) is a λq-dimensionalsubmanifold of M, and W s(p) is a (m− λ)-dimensional submanifold. So, theirintersection is then a submanifold whose dimension dim(Wu(q)∩W s(p)) is equalto

dimWu(q)+ dimW s(p)−m = λq + (m− λp)−m = λq − λp

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For the future, we will refer to this intersection Wu(q)∩W s(p) as W (q, p). Thus,W (q, p) denotes the flow out of a critical point q into a different critical pointp.

An immediate consequence of this is the following corollary:

Corollary 5.5 IF F : M → R is a Morse-Smale function on a finite-dimensionalcompact smooth Riemannian manifold (M,g), then the index of critical pointsdecreases strictly along gradient flow lines. Specifically, if p and q are criticalpoints of f with W (q, p) 6= then λq > λp.

The proof of this comes simply from observing that if W (q, p) 6= , then thermust be at least one flow line from q into p. Flow lines have dimension 1, sodimW (q, p) = λq − λp ≥ 1.

One more result about Morse-Smale functions that is worth noting is thefollowing Kupka-Smale Theorem, concerning how many functions on a manifoldsatisfy the Morse-Smale transversality condition.

Theorem 5.6 If (M,g) is a finite-dimensional compact smooth Riemannianmanifold, then the set of Morse-Smale gradient vector fields of class Cr is ageneric subset of the set of all Cr gradient vector fields on M for 1 ≤ r ≤ ∞.

For a proof of the Kupka-Smale Theorem the reader is directed to Section 6.1of [3]. The reason we mention it here is that almost all of the remaining resultsof this thesis will be rooted in Morse-Smale functions. As such, it is comfortingto know that they are so universal.

6 Corollaries to the λ-lemma

In this section we list some important results for the proof of the Morse Homol-ogy Theorem. For proofs of these results, we direct the reader to section 6.3 of[3]. The reason we do not go through the proofs here is that they all dependon the λ−lemma, a result whose statement and proof are very intricate and notparticularly illuminatory for us. Rather, the meat of the λ-lemma for us are thecorollaries which follow it. The results are as follows:

Lemma 6.1 Let M be a smooth manifold, and suppose that p and q are hy-perbolic fixed points of φ, a diffeomorphism of M. If Wu(q) and W s(p) have apoint of transverse intersection, then

Wu(q) ⊇Wu

As Wu(q) and Wu(p) can be thought of as the flows out of q and p respectively,this inclusion effectively means that all of the flow out of p is contained in theclosure of q′s outflow. The next result has to do with the flows from one pointinto a second and then into a third.

24

Lemma 6.2 Let M be a smooth manifold, and let p1, p2, and p3 be three hy-perbolic fixed points of φ, a diffeomorphism from M to itself. If Wu(p3 andW s(p2) have non-null transverse intersection, and the same is true for Wu(p2)and W s(p1) then Wu(p3) and W s(p1) have a point of transverse intersection,and the inclusion

W (p3, p1) ⊆W (p3, p2)⋃W (p2, p1)

⋃{p1, p2, p3}

This is effectively a transitive property for the outflow from a fixed point p3,through p2, and into p1. In fact, when the diffeomorphism in this precedinglemma is generated by a gradient vector field of a Morse-Smale function f onM , then this result holds for any trio of critical points of the function f . Withthis in mind, we can define a partial ordering of the critical points of f . For anytwo critical points p and q for which W (q, p) 6= ∅ holds, we say that q � p.

From this partial ordering we get the following two inclusions:

Corollary 6.3 For any critical point q of f : M → R the inclusion below holds.

Wu(q) ⊇⋃q�p

Wu(p)

Additionally, if p and q are critical points of f such that q � p, then

W (q, p) ⊇⋃

q�q�p�p

W (q, p)

The following lemmas show that the opposite inclusions also hold, that is tosay that the inclusions in the above lemmas and corollary are actually equalities.

Lemma 6.4 If Wu(q) ∩Wu(p) 6= , then p ∈Wu(q)

Lemma 6.5 If p 6= q is a critical point of f : M → R such that p ∈ Wu(q),then Wu(q) intersects W s(p)− p.

Corollary 6.6 Suppose that q and p are critical points of f : M → R, andWu(q) ∩W s(p) 6= . Then

Wu(q) ⊆⋃q�p

Wu(p).

That is to say that q � p.

Corollary 6.7 For any critical point q of f : M → R we have that

Wu(q) =⋃q�p

Wu(p)

Additionally,

W s(q) =⋃r�q

W s(r)

25

Corollary 6.8 If p and q are critical points of f : M → R and q � p then

W9q, p) = Wu(q) ∩W s(p) =⋃

q�q�p�p

W (q, p)

where this union is taken over all critical points between p and q under thepartial order taken earlier.

Corollary 6.9 If p and q are critical points of relative index one (λq = λp = 1),then

W (q, p) = W (q, p) ∪ p, q

Additionally, W (q, p) has finitely many components, that is to say that there area finite number of gradient flow lines from q to p.

With these results under our belt we can proceed to the culminating resultof this thesis, the Morse Homology Theorem.

7 The Morse Homology Theorem

The result we have been building to is the Morse Homology Theorem, whichidentifies the singular homology groups of a manifold M with the homologygroups generated by the critical points of a Morse-Smale function f : M → R.However, in order to do this we must define the homology groups generated bythe critical points of f. To do this, we introduce the Morse-Smale-Witten chaincomplex of a function f on a manifold .

Definition Let f : M → R be a Morse-Smale function on a compact smoothoriented Riemannian manifold M of dimension M < ∞, and assume that ori-entations for the unstable manifolds of f have been chosen. Let Ck(f) be thefree abelian group generated by Crk(f), and define

C∗(f) =

m⊕k=0

Ck(f).

We now define a homomorphism ∂k : Ck(f)→ Ck−1(f) by

∂j(q) =∑

p∈Crk−1(f)

n(q, p)p.

where n(q, p) is defined below. We call this the Morse-Smale-Witten boundaryoperator, and the pair (C∗(f), ∂∗) is the Morse-Smale-Witten chain complex off.

The Morse Homology Theorem states that the Morse-Smale-Witten chaincomplex is in fact a chain complex and, furthermore, that the homology groupsit generates are isomorphic to the singular homology groups Hn(M)

26

7.1 Orientations and n(q, p)

We have defined the Morse-Smale-Witten boundary operator already and in-volved in the summation is the number n(q, p). However, we have not explicitlydefined what this number is. Effectively, this number n(q, p) is a way of count-ing the gradient flow lines from q to p while accounting for the orientations ofeverything[ELABORATE ON TH IS]

Before we can do this, though, we must first establish some preliminaryconventions for how we are orient

So, we take q and p to be critical points of relative index one. Let’s saythat λq = k and λp = k − 1. Let us assume that q � p (If q does not precedep then there are no gradient flows between q and p, so n(q, p) = 0), and letγ : R→M be a gradient flow line from q to p. At any point x in γ(R), we cancomplete (−∇f)(x) to a positive basis of TxW

u(q), ((−∇f)(x), Bux ). If we take apositive basis Bsx of TxW

s(p), then (Bxs , Bux ) is a basis for TxM (as Bux has k−1

elements, and Bsx has (m−k+ 1) elements). If (Bsx, Bux ) is a positively-oriented

basis for TxM then we assign +1 to this flow γ. If not, then we assign -1 to γ.

7.2 Index Pairs

To start, we let φt be a 1-parameter group of diffeomorphisms on a locallycompact metric space M. A subset S of M is called an invariant subset ifφt(S) = S for every t ∈ R. Following this, we can define the maximal invariantsubset I(N) of a subset N of M as follows:

I(N) = {x ∈ N |φt(x) ∈ N for every t ∈ R}

Furthermore, a compact invariant subset S is called isolated if there existsa compact neighborhood N of S such that I(N) = S. Following this, we candefine index pairs as follows. Given an isolated compact invariant subset S, anindex pair (N,L) for S is a pair of compact sets L ⊂ N such that the followingproperties hold.

1. S = I((N − L)) ⊆ int(N − L)

2. x ∈ L and x · [0, t] ⊆ N implies that x · [0, t] ⊆ L. We denote this propertyby saying that L is positively invariant in N.

3. If an orbit of a point leaves N , it has to go through L first. More specif-ically, if for a point x in N we have that x · t /∈ N for some t > 0, thenthere must be some t′ ∈ [0, t] with the property that x · [0, t] ⊆ N andx · t′ ∈ L.

A couple of key results regarding index groups are the following theorems(Theorems 7.14 and 7.15 in [3]).

Theorem 7.1 Every isolated compact invariant set admits an index pair

27

Theorem 7.2 If S is an isolated compact invariant set and (N,L) and (N , L)are two index pairs for S, then N/L and N/L are homotopy equivalent as pointedspaces via maps induced by the flow φ.

We will not prove these theorems here, but a proof of the first theorem canbe found in Section 4.1 of [4] and Section 7.5 of [3] is dedicated to proving thesecond theorem.

An important property that an index pair (N,L) can have is for the inclusionmap L ↪→ N to be a cofibration. If that is the case, then we say that the indexpair is regular.

We will begin our examination of index pairs by constructing them for thecritical points of a Morse function f. We know for any q ∈ Crk(f) the tan-gent space TqM splits into T sqM ⊕ Tuq M, with Tuq M = TqW

u(q) and T sqM =TqW

s(q). We also know that W s(q) and Wu(q) intersect transversally at q. So,we get a coordinate chart φ : U → TqM around q that maps W s(q) ∩ U intoT sqM and Wu(q) ∩ U into Tuq M. Now, we take ε > 0 and define

Dsε = {v ∈ T sqM |‖v‖ ≤ ε} and

Duε = {v ∈ Tuq M |‖v‖ ≤ ε}

Then, let Nq = φ−1q (Ds

ε × Duε ) and Lq = φ−1

q (Dsε × ∂Du

ε ). Nq is an isolatingneighborhood for {q}, a compact invariant set, and (Nq, Lq) is an index pair forthis set. Now, we can see that the index pair (Nq, Lq) is regular, and we havethat

(Nq, Lq) ≈ (Dsε ×Du

ε , Dsε × ∂Du

ε ) ' (Duε , ∂Dε),

So Nq/Lq is a pointed sphere of dimension λq. So, Hj(Nq, Lq) ≈ Hj(Duε , ∂D

uε ) ≈

Hj(Duε /∂D

uε , ∗) ≈ Hj(Nq/Lq, ∗) for every j. Additionally, because an orienta-

tion of Tuq M determines a generator of Hk(Nq, Lq) ≈ Hk(Duε , ∂D

εu) ≈ Z. There-

fore, we get an identification

Ck(f) ≈⊕

q∈Crk(f)

Hk(Nq, Lq).

This identification is quite important to us for proving the Morse Homologytheorem.

One particular set of index pairs we will be relying quite heavily on is thefollowing sequence, which provides index pairs for the flows on M determinedby −∇f. To start, we define

W (k, j) =⋃

j≤λp≤λq≤k

W (q, p)

where 0 ≤ j ≤ k ≤ m. We know that these spaces are compact as they arethe images of closed disks in a compact manifold. So, let us take a compactneighborhood N of W (k, j) such that Cr(f) ∩ N = Cr(f) ∩W (k, j), then weknow that

I(N) = {x ∈ N |φt(x) ∈ N for all t ∈ R} = W (k, j).

28

So, we know that W (k, j) is an isolated compact invariant set for 0 ≤ j ≤ k ≤ m.We know from Corollary 6.7 that the sets

W sj =

⋃j≤λp

W s(p) and

Wuj =

⋃λp≤j

Wu(p)

are compact for all j = 0, . . . ,m. Set Nm = M , and then choose a cofiberedcompact neighborhood Nm−1 of Wu

m−1 in M which is positively invariant in Nmand fulfills the property Nm−1 ∩W s

m = ∅. One such neighborhood is

Nm −⋃

q∈Crm(f)

intNq.

With Nm−1 fulfilling these conditions (Nm, Nm−1) is a regular index pairfor Crm(f). Similarly, we then choose a cofibered compact neighborhood Nm−2

of Wum−2 in Nm−1 which is positively invariant in Nm−1 and satisfies Nm−2 ∩

W sm−1 = ∅. We then know that (Nm−1, Nm−2) is an index pair for Crm−1(f)

and (Nm, Nm−2) is an index pair for W (m,m− 1). Repeating this choosing ofcofibered compact neighborhoods establishes a sequence

∅ = N−1 ⊆ N0 ⊆ N1 ⊆ . . . ⊆ Nm = M

with the property that (Nk, Nj−1) is a regular index pair for W (k, j) for allvalues 0 ≤ j ≤ k ≤ m.

Before we proceed, we would first like to define an important tool in provingthe Morse Homology Theorem. Given the homology exact sequence of the tripleN0 ⊆ N1 ⊆ N2

0 - Ck(N1, N0)i- Ck(N2, N0)

j- Ck(N2, N1) - 0

0 - Ck−1(N1, N0)?

i- Ck−1(N2, N0)

∂k?

j- Ck−1(N2, N1)?

- 0

we define the connecting homomorphism

δ∗ : Hk(N2, N1)→ Hk−1(N1, N0)

as follows. Let a ∈ Ck(N2, N1) be a k − cycle, and let a ∈ Ck(N2, N0) be thelift j−1(a). In this case, ∂k(a) is a cycle in Ck−1(N2, N0) which comes froma unique cycle a in Ck−1(N1, N0). We define δ∗[a] = [a]. It is worth notingthat δ∗ = j∗ ◦ δk where δk is the connecting homomorphism for the homologyexact sequence of (N2, N1) and j : (N1, ∅) → (N1, N0) is the inclusion map.An important peoperty of this connecting homomorphism is that it is a natural

29

transformation, which is to say that if g : (N2, N1, N0)→ (N2, N1, N0) is a mapof triples then the diagram below is commutative for every k :

Hk(N2, N1)δ∗- Hk−1(N1, N0)

Hk(N2, N1)

g∗?

δ∗- Hk−1(N1, N0)

g∗?

The homotopy equivalences we get from Theorem 7.2 are flow-induced, so thediagram commutes for regular index pairs.

The last step before we start proving the Morse Homology Theorem is con-structing an important group homomorphism ∆k. To start, we consider someMorse-Smale function f : M → R on a finite dimensional smooth compactRiemannian manifold M. Let q ∈ Crk(f) and p ∈ Crk−1. We know thatS = W (q, p) ∪ {p, q} is an isolated compact invariant set, so by Theorem 7.1 itadmits an index pair (N2, N0). Let N1 = N0∪(N2∩M c), where f(p) < c < f(q).Then, (N2, N1) is an index pair for q and (N1, N0) is an index pair for p. Assumethat these index pairs are regular(Which we can do according to Section 5.1 of[4]).

Let (Nq, Lq) be a regular index pair for q and (Np, Lp) be a regular index pairfor p. Let us define the homomorphism ∆k(q, p) : Hk(Nq, Lq) → Hk−1(Np, Lp)via the composition of maps

Hk(Nq, Lq)≈−→ Hk(N2, N1)

δ∗−→ Hk−1(N1, N0)≈−→ Hk−1(Np, Lp).

The first and third isomorphism come from the homotopy equivalence of The-orem 7.1, and δ∗ is the connecting homomorphism in the exact sequence of(N2, N1, N0). Thus, we get the homomorphism

∆k :⊕

q∈Crk(f)

Hk(Nq, Lq)→⊕

p∈Crk−1(f)

Hk−1(Np, Lp)

7.3 Proving the Morse Homology Theorem

With the preceding definitions taken care of, we can proceed with the MorseHomology Theorem. The proof of this theorem depends strongly on the follow-ing lemma, which allows us to construct the commutative diagram on which theMorse Homology Theorem’s proof hinges. The lemma is as follows:

30

Lemma 7.3 The diagram below commutes.

Ck(f)∂k - Ck−1(f)

⊕q∈Crk(f)

Hk(Nq, Lq)

≈?6

∆k-⊕

p∈Crk−1(f)

Hn−1(Np, Lp)

≈?6

Hk(Nk, Nk−1)

≈?

6

δ∗ - Hk−1(Nk−1, Nk−2)

≈?

6

In this diagram, (Nq, Lq) indicates a regular index pair for q ∈ Crk(f), (Np, Lp)is a regular index pair for p ∈ Crk−1(f), ∂k is the Morse-Smale-Witten bound-ary operator, ∆k is the homomorphism defined earlier, and δ∗ is a connectinghomomorphism.

Proof We know that the bottom square commutes because the connecting ho-momorphism δ∗ is a natural transformation and ∆k is constructed using δ∗, sowe proceed with the top square. We will restrict ourselves to the case whereq ∈ Crk(f) and p ∈ Crk−1(f) are the only two critical points in f−1[a, b], withf(p) = a and f(q) = b. If this is not the case then we need only alter f outsidean isolating neighborhood of S = W (q, p)∪{q, p}. This alteration affects neither∆k nor ∂k, so we are fine.

We first start off by defining some sets which will lead to convenient indexpairs. We have already defined the set M c = {x ∈M |f(x) ≤ c}, and we defineMc to be the corresponding set Mc = {x ∈ M |f(x) ≥ c}. Let ε > 0 be verysmall, and let T � 0.

Nq = {x ∈Mc|f(φ−T (x)) ≤ b+ ε}Lq = {x ∈ Nq|f(x) = c}

Np = {x ∈M c|f(φT (x)) ≥ a− ε}Lp = {x ∈ Np|f(φT (x)) = a− ε}

(1)

We also define C = Np ∪Nq, B = Np ∪ Lq, and A = Lp ∪ (Lq −Np).The definitions of Nq, Np, Lq, and Lp are a bit obtuse, so it helps to have an

intuitive way of thinking about them. One can think of Nq as the set of pointsin M which map to a value ≥ c and did not flow from too far above q. Lq isjust the intersection of this set with f−1(c). Np are the the points with value≤ c and which don’t flow too far past p, and Lp is simply the set of these pointswhich start at c.

(Nq, Lq) and (C,B) are index pairs for q, (Np, Lp) and (B,A) are index pairsfor p, and (C,A) are index pairs for S = W (q, p) ∪ {q, p}.

31

We know that, for every x in Wu(q), f(φ−T ) ≤ f(q) = b, so Wu(q) ∩Mc

is contained in Nq. Additionally, with ε > 0 fixed, we get that Nq contracts toWu ∩Mc as T →∞. So, (Nq, Lq) contracts to

(Wu(q) ∩Mc,Wu(q) ∩ f−1(c)) ≈ (Dk, ∂Dk).

As the inclusion ∂Dn ↪→ Dn is a cofibration, we then know that (Nq, Lq) is aregular index pair.

Similarly, Np is a tubular neighborhood of W s(p) ∩M c. This contracts toW s(p) ∩M c as T →∞ in a similar manner, so we know that

(Np, Lp) ≈ (Dk−1 ×Dm−k+1, ∂Dk−1 ×Dm−k+1)

This is also a cofibration, so we get that (Np, Lp) is a regular index pair.Now, we note that Np is a neighborhood of the stable closed disk W s(p)∩M c,

and note that the unstable sphere Wu(q)∩ f−1(c) is contained in Lq. We knowfrom corollary 6.9 that W (q, p) has finitely many components, so know thatNp ∩ Su(q) = Np ∩Wu(q) ∩ f−1(c) has finitely many components, each with asingle point of intersection with W (q, p) for each component. We denote thesecomponents V1, . . . , Vj and the points of intersection Vj ∩W (q, p) as xj .

We have already noted that Np is a tubular neighborhood of W s(p) ∩M c,so as W s(p) ∩M c ≈ Dm−k+1 we get a diffeomorphism

Ψp : Np → Dk−1 ×Dm−k+1.

with the following properties:

Ψp(Lp) = ∂Dk−1 ×Dm−k+1

Ψp(Np ∩W s(p)) = {0} ×DM−k+1

Ψp(Vj) = Dk−1 × {θj} for θj ∈ ∂Dm−k+1

So, Vj is a (k− 1)-dimensional manifold which is diffeomorphic to Dk−1 via themaps Ψp,j = π1 ◦Ψp|vj , whose composition takes Vj to Dk−1 as follows:

VjΨp|Vj−−−−→ Dk−1 ×Dm−k+1 π1−→ Dk−1

where the map π1 simply projects a point (x, y) in Dk−1×Dm−k+1 onto its firstcoordinate x in Dk−1. This map takes Lp into ∂Dk−1 and therefore induces theisomorphism

Hk−1(Np, Lp)(Ψp,1)∗≈ Hk−1(Dk−1, ∂Dk−1)

(Ψ−1j )∗≈ Hk−1(Vj , ∂Vj) ≈ Z.

The orientation of TupM determines a generator α of Hk−1(Np, Lp) ≈ Z, and achain representing α also determines a generator for Hk−1(B,A) ≈ Z. Therefore,we can identifyHk−1(Np, Lp) withHk−1(B,A). This generator α is mapped ontoa generator αj of Hk−1(Vj , ∂Vj) by (Ψ−j 1)∗ ◦ (Ψp,1)∗. So, the homology class αj

32

is determined by the orientation which TxjVj inherits from the orientation ofTupM by way of the isomorphism TxjVj → TupM which is determined by theflow φt. We get another orientation of TxjVj coming from Wu(q), as

TxjVj = (−∇f)(xj)⊥ ∩ TxjWu(q) ⊆ TxjWu(q)

where −∇f(xj) is the first vector in a positive basis. Let nj be either +1 or -1depending on whether or not these two orientations for Txj agree or not. So, njis the sign associated to the gradient flow line containing xj .

We know that Su(q) = Wu(q)∩ f−1(c) is the same region as wu(q)∩Lq. So,we consider the diagram below. The vertical maps arise from inclusions and thehorizontal maps are connecting homomorphisms.

Hk(Wu(q) ∩Nq, Su(q))δ∗- Hk−1(Su(q), (Su(q)−

n∐j=1

Vj))

Hk(C,B)

≈?

δ∗ - Hk−1(B,A)

s∗?

Connecting homomorphisms are natural transformations, so this diagram iscommutative. It is important to observe that

Hk−1(Su(q), Su(q)−n∐j=1

Vj) ≈n⊕j=1

Hk−1(Su(q), Su(q)− Vj) ≈ Z.

Additionally, define δ∗,j to be the jth component of the connecting homomor-phism

δ∗ : Hk(Wu(q) ∩Nq, Su(q))→ Hk−1(Su(q), Su(q)−n∐j=1

Vj)

under this identification. Therefore, for any β ∈ Hk(Wu(q)∩Nq, Su(q)) we haveδ∗(β) = δ∗,1(β) + . . .+ δ∗,β and

s∗(δ∗(β)) =

n∑j=1

s∗(δ∗,j(β)) ∈ Hk−1(B,A) ≈ Z

We know that Nq is a thickening of the unstable closed disk Wu(q) ∩Mc =Wu(q) ∩Nq, so we have the following relationship.

(Nq, Lq) ' (Wu(q) ∩Nq,Wu(q) ∩ Lq) = (Wu(q) ∩Nq, Su(q)).

Therefore, Hk(Wu(q) ∩Nq, Su(q)) ≈ Hk(Nq, Lq). We use the following methodto get a generator β for this group. We start with triangulations of the (k− 1)-dimensional closed disks Vj ⊆ Su(q) for all j = 1, . . . , n, and extend these

33

triangulations to a triangulation of the k-dimensional manifold with boundaryWu(q) ∩ Nq ≈ Dk. This triangulation together with the orientation of Wu(q)gives us the generator β. Similarly to α earlier, β also determines a generatorfor Hk(C,B) ≈ Hk(Nq, Lq), so we can identify

Hk(C,B) = Hk(Nq, Lq) = Hk(Wu(q) ∩Nq, Su(q)).

By the excision theorem, we know thatHk−1(Su(q), Su(q)− Vj) ≈ Hk−1(Vj , ∂Vj) ≈Z for all j = 1, . . . , n. In this isomorphism the imgae of the homology classδ∗,j(β) is represented by the original triangulation of Vj together with the orig-inal orientation, so it corresponds to njαj . We know that αj ∈ Hk−1(Vj , ∂Vj)corresponds to α under the isomorphism Hk−1(Np, Lp) ≈ Hk−1(Vj , ∂Vj), andso we get that

s∗(δ∗(β)) =

n∑j=1

njαj = n(q, p)α ∈ Hk−1(B,A) = Hk−1(Np, Lp)

Identifying β with q and α with p we then get the formula

∆k(q) =∑

p∈Crk−1(f)

n(q, p)p = ∂k(q).

Theorem 7.4 (Morse Homology Theorem) The Morse-Smale-Witten chain com-plex (C∗(f), ∂∗) is a chain complex, and its homology is isomorphic to the sin-gular homology Hn(M).

Proof We begin with the commutative diagram below:

Ck+1(f)∂k+1 - Ck(f)

∂k - Ck−1(f)

Hk+1(Nk+1, Nk)

≈?

6

δ∗- Hk(Nk, Nk−1)

≈?

6

δ∗- Hk−1(Nk−1, Nk−2)

≈?

6

where δ∗ is the connecting homomorphism. We know that such a commutativediagram exists from the previous lemma, so we know that the Morse-Smale-Witten boundary operator has the property that ∂k∂k+1 = 0.

Following this, we examine the index pair (Nk, Nk−1).We know thatW (k, k) =Crk(f), and we use the constructions Nq and Lq from the example of gradientflows to produce a new index pair for Crk(f)

(Nk, Lk), setting Nk =⋃

q∈Crk(f)

Nq and Lk =⋃

q∈Crk(f)

Lq

By Theorem 7.2 we know that

Nk/Nk−1 ' Nk/Lk '∨

q∈Crk(f)

Skq

34

AsNk/Nk−1 is homotopic to a wedge sum of k-spheres, we know thatHj(Nk, Nk−1) =0 for j 6= k. We can then plug this into the exact sequence for (Nk, Nk−1)

. . .→ Hj+1(Nk, Nk−1)→ Hj(Nk−1)→ Hj(Nk)→ Hj(Nk, Nk−1),→ . . .

to yield a short exact sequence of the form

0→ Hj(Nk−1)i∗−→ Hj(Nk)→ 0

for all values of j 6= k, k− 1. By the properties listed earlier for exact sequenceswe know then that the map i∗ is an isomorphism for all j 6= k, k − 1. So, weknow that the inclusion map Hj(Nk)→ Hj(M) is an isomorphism when j < k.

We now wish to show that Hj(Nk) is trivial for j > k. To do so we inducton k. Starting at k = 0 we know that Hj(N0) = 0 for j > 0 because (N0, ∅) =(N0, N−1) is an index pair for Cr0(f).

For the inductive step we assume that Hj(Nk−1) = 0 for all j > k − 1 andconsider the exact sequence associated to (Nk, Nk−1) :

. . .→ Hj(Nk−1)→ Hj(Nk)→ Hj(Nk, Nk−1)→ . . .

We know that Hj(Nk, Nk−1) = 0 for j > k from earlier, and we know thatHj(Nk−1) = 0 from the induction hypothesis, so from the properties of exactsequences we know that Hj(Nk−1) maps surjectively onto Hj(Nk). However,Hj(Nk−1) is trivial, so Hj(Nk) must be as well.

Now that we know that Hj(Nk) = 0 for all j > k we have the followingcommitative diagram:

Hk−1(Nk−2 = 0

0 = Hk(Nk−1) - Hk(Nk) - Hk(Nk, Nk−1)δk - Hk−1(Nk−1)

?

Hk−1(Nk−1, Nk−2)

j∗?

δ∗

-

The horizontal sequence is exact as it comes from the exact sequence for (Nk, Nk−1),and the vertical sequence is comes from the exact sequence for the pair (Nk−1, Nk−2).As the vertical sequence is exact, we know from the properties of exact sequencesthat j∗ is injective.

We then know that the diagram below is commutative:

Hk+1(Nk+1, Nk)δk+1- Hk(Nk) - Hk(Nk+1) - 0

Ck+1(f)

≈?

6

∂k+1- Ker(∂k)

≈?

6

- Hk(M)

≈?

6

- 0

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From this diagram, we know that the bottom row is exact as the top rowcomes from the exact sequence for (Nk+1, Nk). So, we know by property (iv) ofexact sequences that Hk(M) ≈ Ker(∂k)/Im(∂(k+1). As that quotient is the nth

homology group of (C∗(f), ∂∗), we get an isomorphism between the homologygroups of (C∗(f), ∂∗) and the singular homology groups H∗(M). This completesthe proof of the Morse Homology Theorem.

8 Acknowledgements

I would like to thank my advisor, Professor Ralph Cohen, for his suggestionof Morse Homology as well as the help and guidance he gave me through thisentire process. I would like to also thank Dan Berwick-Evans, whose classesfirst sparked my interest in topology. And I am of course forever grateful to myparents, whose encouragement and support have been crucial not only throughthis thesis, but throughout my entire collegiate career.

References

[1] Allen Hatcher. Algebraic Topology. Allen Hatcher, 2001.

[2] John Milnor Morse Theory. Princeton University Press, Princeton, N.J.,1963.

[3] Agustin Banyaga and David Hurtubise Lectures on Morse Homology.Springer Science and Business Media, Dordrecht, Netherlands, 2004.

[4] Dietmar Salamon Connected simple systems and the Conley index of iso-lated irwariant sets. Transactions of the American Mathematical Society291 (1985).

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