Classification of Two-dimensional ManifoldsUsing Morse Theory

Akihiko Nishimura

August 10, 2010

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Contents

0 Introduction 3

1 Morse Theory 51.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Morse Function and its Relation to the Topology of a Manifold 7

2 Classification of Two-dimensional Manifolds 232.0 Zero and One dimensional manifolds . . . . . . . . . . . . . . 232.1 Useful Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2 Proof of the Classification Theorem . . . . . . . . . . . . . . . 37

3 Further Directions 413.1 Handle Decomposition of Three-manifold and Heegard Splitting 423.2 Poincare Conjecture and Classification of Three-manifolds . . 443.3 Exotic R4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Acknowledgment 47

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0 Introduction

The classification of manifolds has been an active area of interest in topologyand geometry. While a number of results have been proven about the subject,many questions still remain open. The subject may also be of interest totheoretical physicists as the Superstring theory conjectures that the universeis actually ten-dimensional and that the additional six dimensions (on topof the four space-time dimensions) is made of a six-dimensional compactmanifold of the type called Calabi-Yau space.

The classification of two-dimensional manifolds by homeomorphism typeand by diffeomorphism type has been known for a long time. However, clas-sifying the manifolds in dimension three or higher turn out to be much morecomplex problem. Even for three-dimensional manifolds, the classification byhomeomorphism type has not been established, though recent work of Perel-man showed that a three-dimensional manifold can be at least decomposedinto some “elementary” pieces each of which we understand well. In four orhigher dimensions, no results come close to a complete classification.

In this paper, we introduce the readers to Morse theory and, using theMorse theoretic tools, prove the complete classification by diffeomorphismtype of compact orientable two-dimensional manifolds. Our proof of theclassification is more elementary than a classical proof of the result, relyingmostly on analysis techniques and avoiding machinery from algebraic topol-ogy.

Morse theory has proven to be a powerful tool for studying a topolog-ical structure of a smooth manifold. Roughly speaking, the idea of Morsetheory is to define a certain type of real-valued function on a manifold anddecompose the manifolds into pieces by “slicing” it at the level sets of thefunction. For the two-dimensional manifolds, this decomposition will provideus with enough information to classify them completely by their diffeomor-phism type. The decomposition also provides us a lot of information aboutthe higher-dimensional manifolds, though not quite enough to establish acomplete classification.

Incidentally, Morse theory is closely related to the Superstring theory.To illustrate this, consider the event in which two particles collides to forma single particle then splitting again into two particles. The premise of theSuperstring theory is that the particles are actually one-dimensional man-ifolds diffeomorphic to S1. Hence, from the Superstring theory viewpoint,the trajectory of the particles in the previously described event forms a two-

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dimensional manifold M (see Figure 0.1). If we let t : M → [0, 1] be the

Figure 0.1: Two particles colliding and then splitting

projection onto a local time coordinate, then the real-valued function t turnsout to be a Morse function. Thus we could study the trajectory M of theparticles using Morse theory.

In the last section of this paper, we indicate further directions for the read-ers by giving a short survey of the results that have been established for threeand four-dimensional manifolds. We discuss Heegard splitting and explainwhy the classification becomes much more difficult in three-dimension. Wethen give a brief account of the recent work by Perelman on three-dimensionalmanifolds, which led to a proof of the famous Poincare’s conjecture. Finally,we look at an “exotic R4” to introduce the readers to some surprising behav-ior of four-dimensional manifolds.

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1 Morse Theory

1.1 Preliminaries

We here introduce notations and definitions necessary for the later discus-sion. Some of the facts here will be stated with only a sketch of a proof or noproof at all since they are fairly elementary results from differential topology.

A subset M of a Euclidean space is called an n-dimensional smooth mani-fold (with boundary) if every point of M possesses a neighborhood diffeomor-phic to an open set in the upper half space Hn; more precisely, for all p ∈Mthere is a neighborhood U of p and a diffeomorphism x = (x1, . . . , xn) fromU onto an open subset of Hn. The diffeomorphism x is called a coordinatefunction or coordinate system on U . When we make a choice of a coordinatefunction x on a neighborhood of p ∈M , we will always without loss of gener-ality assume that x(p) = 0. We define the tangent space of a manifold M atp ∈M as the image of the derivative map d(x−1)0 where x : U ⊂M → Rn isa coordinate function defined on a neighborhood of p.1 When two coordinatefunctions x and y have an overlapping domain, the function g = y ◦ x−1 is adiffeomorphism on an open subset of Hn and is called a change of coordinate(function).

Let M and N be two smooth manifolds. Let h : N → M be a smoothmap, and let x and y be coordinate functions on neighborhoods of q ∈ Nand h(q) ∈M respectively. Note that a map g = y ◦x−1 is a diffeomorphismon an open set of an Euclidean space. We define a derivative or differentialof h at q ∈ N by dhq = dy−1 ◦ dg ◦ dx : TqN → Th(q)M . This definitionis independent of a choice of particular coordinate functions. A map h iscalled an immersion if the differential map dhq is injective for each q ∈ N . Amap h is called proper if the preimage of a compact set in M is compact inN . An immersion that is injective and proper is called an embedding. It canbe shown that an embedding h : N → M maps N diffeomorphically onto asubmanifold of M . In particular, if N is compact, the image h(N) under animmersion h is always a submanifold of M .

Let M ⊂ Rm be a n-dimensional smooth manifold, and f be a smoothreal-valued function on M . A point p ∈ M is called a critical point of fif the differential dfp is zero. The value f(p) at a critical point p is called

1More generally, the tangent space at p ∈ M can be defined as a space of directionalderivatives at p. All of the following discussions and proofs will hold with this more generaldefinition.

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the critical value. A critical point p is called non-degenerate if the Hessianmatrix

Hess(f ◦ x−1)0 =

(∂(f ◦ x−1)

∂xi∂xj(0)

)is invertible for some coordinate function x : M → Rn defined on a neigh-borhood of p. One can check that this is well-defined as follows. If g is adiffeomorphism defined on an open set in Rn such that g(0) = 0, then

Hess(f ◦ x−1 ◦ g)0 = dgt0 Hess(f ◦ x−1)0 dg0 (1.1)

where dgt0 denotes the transpose of dg0. (Note that the above formula holdsonly if p is a critical point.) This shows that, if Hess(f ◦ x−1)0 is invertiblefor one coordinate function x, then in fact Hess(f ◦ y−1)0 is invertible forany coordinate function y.

When p ∈M is a non-degenerate critical point, there is a quadratic formQx associated with the Hessian matrix of f ◦ x−1 defined by

Qx(v) = vtHess(f ◦ x−1)0 v for all v ∈ Rn

Denote the maximum dimension of a subspace of Rn on which Qx is negativedefinite by λ(Qx). For a non-degenerate critical point p, we define the indexof f at p to be λ(Qx) for some coordinate function x. To see this is well-defined, let g be a diffeomorphism defined on an open set in Rn such thatg(0) = 0 and suppose {v1, . . . , vk} is a set of the maximum number oflinearly independent vectors such that Qx(vi) < 0 for each i. Then using theequation (1.1), we have

Qg◦x(dg−10 vi) = (dg−1

0 vi)tHess(f ◦ x−1 ◦ g)0 (dg−1

0 vi)

= (dg−10 vi)

t dgt0 Hess(f ◦ x−1)0 dg0 (dg−10 vi)

= vti Hess(f ◦ x−1)0 vi

= Qx(vi)

< 0

This shows that λ(Qx) ≤ λ(Qg◦x). A similar argument shows that the reverseinequality also holds. Hence the index of f is well-defined. We sometimesdenote the index of f at p by λf (p) or simply λ(p) when the function f isclear from the context. When we state the Morse Lemma, we will see thatthe index of f at a non-degenerate critical point can be intuitively consideredas the number of linearly independent directions along which f is decreasing.

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Remark. If one defines the tangent space at p ∈M intrinsically as a space ofdirectional derivatives, then one can also define the Hessian of f at a criticalpoint p intrinsically as a symmetric bilinear functional on TpM . Then onecan define the index of f at p to be the maximum dimension of a subspace ofTpM on which the Hessian is positive definite. With this intrinsic definitionit is clear that the index of f at p does not depend on a choice of a coordinatefunction. For more details about the intrinsic definitions, see [DC].

1.2 Morse Function and its Relation to the Topologyof a Manifold

We now define a Morse function and explain its relation to the topologyof a manifold. Let M ⊂ Rm be a n-dimensional smooth manifold withoutboundary.

Definition 1.1. A smooth function f : M → R is called a Morse functionif all of its critical points are non-degenerate.

A Morse function can provide a great deal of information about the topol-ogy of a manifold as we will see soon. However, the theory may not be ofmuch use if a Morse function is a very special kind of function that appearsrather infrequently. The following theorem shows that a Morse function onM not only exists but also is “generic” in some sense.

Theorem 1.2. Let f : M → R be any smooth function. For a ∈ Rm, definefa : M → Rn by

fa(p) = f(p) + a1p1 + . . .+ ampm for all p ∈M ⊂ Rm

where a = (a1, . . . , am) and p = (p1, . . . , pm). Then the function fa is Morsefor almost every a ∈ Rm.

Proof. See [GP] page 43-45.

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We now reveal the relationship between a Morse function and the topologyof a manifold. The following theorem tells us that local behavior of a Morsefunction in a neighborhood of a non-degenerate critical point is completelydetermined by its index.

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Theorem 1.3 (Morse Lemma). Let p ∈M be a non-degenerate critical pointof a smooth real-valued function f and λ be the index of f at p. Then thereexists a local coordinate function x = (x1, . . . ,xn) : U ⊂ M → Rn definedon a neighborhood U of p such that, for all q ∈ U ,

f(q) = f(p)− (x1(q))2 − . . .− (xλ(q))2 + (xλ+1(q))2 + . . .+ (xn(q))2

Or equivalently, for all x ∈ x(U),

f ◦ x−1(x) = f(p)− (xi)2 − . . .− (xλ)

2 + (xλ+1)2 + . . .+ (xn)2

Proof. See [M1] page 6-8.

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From the Morse lemma, we immediately get the following corollary.

Corollary 1.4. Non-generate critical points are isolated.

Here is another theorem relating the topology of a manifold and criticalpoints of a function on the manifold. We first introduce a notation; fix asmooth real-valued function f on a manifold M and, for a ∈ R, let

Ma = f−1(−∞, a] = {p ∈M : f(p) ≤ a}

Theorem 1.5. Let f be a smooth real-valued function on M . Suppose thatthe set f−1[a, b] ⊂M is compact and that f−1(a−ε, b+ε) contains no criticalpoints of f for some ε > 0. Then Ma and M b are diffeomorphic.

To prove the preceding theorem, we first establish a lemma. Recall thata vector field V on M is a map from M to the ambient space Rm such thatV (p) ∈ TpM for all p ∈ M . A smooth vector field on M is closely relatedto a collection of diffeomorphisms from M onto itself called a 1-parametergroup of diffeomorphisms.

Definition 1.6. A 1-parameter group of diffeomorphisms of a manifold Mis a smooth map φ : R ×M →M such that

1. for each t ∈ R the map φt : M → M defined by φt(p) = φ(t, p) is adiffeomorphism of M onto itself

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

Note that the second property implies that φ0 = idM . Also note that, ifwe fix p ∈ M , we can consider φp : t → φ(t, p) as a curve in M . Given a1-paremeter group of diffeomorphisms, we can define a smooth vector fieldV on M by

V (p) = φ′p(0) for all p ∈M

By the second property of a 1-parameter group of diffeomorphisms, we haveφp(t+ s) = φφ(t,p)(s). By differentiating with respect to s, we see that

φ′p(t) = φ′φ(t,p)(0)

= V (φ(t, p))

This vector field V is said to generate the group φ.

Lemma 1.7. A compactly supported smooth vector field V on M generatesa unique 1-parameter group of diffeomorphisms i.e. there exists a uniquesmooth map φ : R ×M →M such that

1. for each t ∈ R, the map φt : M → M defined by φt(p) = φ(t, p) is adiffeomorphism of M onto itself

2. φt+s = φt ◦ φs

3. φ0 = idM

4. ∂φ∂t

(t, p) = V (φ(t, p)) for all t ∈ R, p ∈M .

Proof. By considering the vector field dx(V ) on an open set of Rn, we canuse the existence and the uniqueness of the solution to a differential equa-tion in Rn to find a local solution to the differential equation ∂φ

∂t= V ◦ φ.

One only needs to check that the local solutions “patch together” smoothlyto give a global solution satisfying the properties of 1-parameter group ofdiffeomorphisms. For the details, see [M1] page 10-11.

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Recall that a Riemannian metric <,> on M is a collection of inner prod-ucts <,>p: TpM × TpM → R defined by

〈v, w〉p =m∑i=1

viwi for all v, w ∈ TpM ⊂ Rm

for each p ∈ M .2 For convenience, we will be less formal and denote each<,>p simply by <,>. Also recall that the gradient vector field of f on M ,denoted ∇f , is a unique vector field such that 〈∇fp, v〉 = dfp(v) for allp ∈ M and v ∈ TpM . One can check that ∇f is smooth if f is smooth andthat ∇f is always orthogonal to the level sets f−1{c} for c ∈ R. The ideaof the proof of the theorem 1.5 is to let the level set f−1{b} flow down tof−1{a} along the tragectries orthogonal to the level sets of f .

Proof of Theorem 1.5. Since f is smooth, the preimage f−1(a − ε, b + ε) isopen and f−1[a, b] is closed. Hence we can construct a smooth functionρ : M → R such that

1. ρ = 1‖∇f‖2 on f−1[a, b]

2. support(ρ) ⊂ f−1(a− ε, b+ ε)

3. ρ ≥ 0 on M

Now define a smooth vector field V on M by

V (p) = −ρ(p) ∇fp for all p ∈M

Then by our construction the vector field V satisfies the assumptions ofLemma 1.7 and hence generates a 1-parameter group of diffeomorphism φ.Using the properties of φ, we obtain

∂(f ◦ φ)

∂t(t, p) = dfφ(t,p)

(∂φ

∂t(t, p)

)=

⟨∇fφ(t,p),

∂φ

∂t(t, p)

⟩=⟨∇fφ(t,p), −ρ(φ(t, p)) ∇fφ(t,p)

⟩= −ρ(φ(t, p))‖∇fφ(t,p)‖2

2More generally, a Riemannian metric <,> on M is any smooth collection of innerproducts on TpM for each p. The smoothness here means that, for any two smooth vectorfield V1 and V2, a map:p → 〈V1(p), V2(p)〉 defines a smooth real-valued function on M .The particular Riemannian metric we chose is often called the “standard metric”.

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for all t ∈ R and p ∈ M . Since ρ(q) = 1‖∇fq‖2 for q ∈ f−1[a, b], the previous

equation implies

∂(f ◦ φ)

∂t(t, p) = −1 if φ(t, p) ∈ f−1[a, b] (1.2)

Also, since ρ ≥ 0 by our construction,

∂(f ◦ φ)

∂t(t, p) ≤ 0 for all t ∈ R and p ∈M (1.3)

Now we show that φb−a is a diffeomorphism from M b to Ma. We claimthe followings:

Claim 1. For all p ∈M , if f(p) ≤ b, then f ◦ φb−a(p) ≤ a.

Claim 2. For all p ∈M , if f(p) > b, then f ◦ φb−a(p) > a.

Before we prove the claims, note that Claim 1 shows that φb−a maps M b

into Ma and Claim 2 shows that φb−a maps M b onto Ma. This proves thatthe restriction φb−a|Mb defines a diffeomorphism from M b onto Ma. Hence itremains only to prove the claims to complete the proof of the theorem.

Let p ∈ f−1(a, b) and define a set Ip by

Ip = {s ∈ [0, f(p)− a) : a ≤ f ◦ φ(s, p) ≤ b}

It is not difficult to show that Ip is a non-empty, closed, and open subsetof [0, f(p) − a). Hence Ip = [0, f(p) − a). Together with Equation 1.2, thisproves

f ◦ φ(s, p) = f(p)− s for all s ∈ [0, f(p)− a] (1.4)

Now, let p ∈ M b. To prove Claim 1, we need show that f ◦ φb−a(p) ≤ a. Iff(p) ≤ a, then by Equation 1.3 we have

f ◦ φb−a(p) ≤ f(p) ≤ a ≤ b

and we are done. If a ≤ f(p) ≤ b, then by Equation 1.4 we have

f ◦ φf(p)−a(p) = a

Since b− a ≥ f(p)− a in this case, Equation 1.3 implies

f ◦ φb−a(p) ≤ f ◦ φf(p)−a(p) = a

This proves Claim 1. Claim 2 can be proved in a similar way.

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By modifying the proof of Theorem 1.5, we can obtain another usefulresult.

Theorem 1.8. Under the same conditions as in Theorem 1.5, the set f−1[a, b]is diffeomorphic to a “cylinder” f−1(a)× [a, b].

Proof. Let V be a smooth vector field as in the proof of Theorem 1.5 and φbe a 1-parameter group of diffeomorphisms generated by V . Define a maph : f−1(a)× [0, (b− a)]→M by

h(q, s) = φ(−s, q)

for each (q, s) ∈ f−1(a) × [0, (b − a)]. From Lemma 1.10 (see below), it

Figure 1.2

follows that h is a diffeomorphism from f−1(a)× [a, b] onto its image. Usingarguments similar to those in the proof of Theorem 1.5, one can check thatthe image of h is in fact f−1[a, b].

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To state Lemma 1.10, we first define the notion of transversality.

Definition 1.9 (Transversality). Let N be a submanifold of M and V be asmooth vector field on M . We say that N and V are transversal or intersecttransversally if

TpN + {V (p)} = TpM for each p ∈ N

Figure 1.3: transversalFigure 1.4: not transversal

Lemma 1.10. Let M be an n-dimensional manifold and g be an embed-ding from an (n − 1)-dimensional manifold N into M . Also let φ be a1-parameter group of diffeomorphisms generated by a smooth vector field Von M . If the submanifold g(N) intersects transversally with V , then a maph : N × [−r, r]→M defined by

h(q, s) = φ(s, g(q)) for each (q, s) ∈ N × [−r, r]

is a diffeomorphism onto its image.

Proof. Note that the transversality condition in particular implies V (q) 6= 0for all q ∈ g(N). Using this fact and the properties of a 1-parameter groupof diffeomorphisms, it is not difficult to check that h is a bijection ontoits image. To prove that h is actually a diffeomorphism, by virtue of theInverse Function Theorem it suffices to show that dh(q,s) has rank n for each(q, s) ∈ N × [−r, r].

Note that∂h

∂q(q, s) = dφs|g(q)dgq

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for each (q, s) ∈ N × [−r, r]. The above equation in particular implies that(∂h∂q

(q, s))

has rank (n− 1). It also implies

Image

(∂h

∂q(q, s)

)= dφs|g(q)

(Tg(q)g(N)

)Thus if we could further show that ∂h

∂s(q, s) /∈ dφs|g(q)

(Tg(q)g(N)

), then that

would imply dh(q,s) has rank n.Observe that

φ(s+ r, g(q)) = φs ◦ φ(r, g(q))

By differentiating with respect to r, we obtain

∂φ

∂s(s, g(q)) = dφs|g(q)

∂φ

∂s(0, g(q))

= dφs|g(q)(V (g(q))

)Thus we have shown

∂h

∂s(q, s) =

∂φ

∂s(s, g(q))

= dφs|g(q)(V (g(q))

)By the transversality assumption, we know V (g(q)) /∈ Tg(q)g(N). Hence theabove equation shows that ∂h

∂s(q, s) /∈ dφs|g(q)

(Tg(q)g(N)

)as dφs|g(q) is an

isomorphism between the tangent spaces of M . This proves that dh(q,s) hasrank n for each (q, s) ∈ N × [−r, r]. From the Inverse Function Theorem, itfollows that the bijection h is in fact a diffeomorphism, completing the proofof Lemma 1.10.

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Recall that non-degenerate critical points are isolated (Corollary 1.4).Hence a Morse function f onM has the isolated critical points. If additionallyM is compact, then the critical values of f are also isolated. Consequently,if the set f−1[a, b] contains no critical points, then there is ε > 0 for whichf−1(a− ε, b+ ε) contains no critical points. Together with Theorem 1.5 andTheorem 1.8, the preceding argument proves:

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Corollary 1.11. Suppose that f is a Morse function on a compact mani-fold M and that f−1[a, b] contains no critical points. Then Ma and M b arediffeomorphic, and f−1[a, b] is diffeomorphic to f−1(a)× [0, 1].

We now discuss the change in the topology of a manifold when thereis a critical point in f−1[a, b]. We first introduce a notion of handle at-tachment. A k-handle is a hyper-cylinder Bk × Bl for some l ∈ N whereBk = {x ∈ Rk : ‖x‖ ≤ 1}, and we denote a k-handle by Hk. We call theproduct Bk × Bl as “k-handle” regardless of l because we are mainly inter-ested the boundary component Sk−1×Bl of a handle and often the dimensionl is just a “complementary” dimension so that k and l add up to some spec-ified dimension. Now suppose M is n-dimensional and g is an embeddingfrom the boundary component Sk−1 × Bn−k of a k-handle onto a subset ofthe boundary of M . By identifying the submanifold h(Sk−1 × Bn−k) of ∂Mand the boundary component Sk−1 × Bn−k of a k-handle, we can define anew manifold M ∪g Hk; this process is called a k-handle attachment to themanifold M . Of course this is not a rigorous definition, but defining a han-

Figure 1.5: handle attachment

dle attachment rigorously involves technical details beyond the scope of thispaper. Instead, we will rigorously define what it means for a manifold to beobtained by a handle attachment, which is all we need for the purpose of thispaper. For the precise definition of a handle attachment, see Section 2.1 of[N].

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Figure 1.6: A torus M where the height corresponds to the value of thefunction f

Definition 1.12. Let M be a n-dimensional manifold with boundary andN ⊂ M be a submanifold with boundary of the same dimension. The man-ifold M is said to be obtained by attaching a k-handle to N if there is ahomeomorphism h−1 from the closure of M − N to a k-handle such thatits restriction to ∂(M − N) is a diffeomorphism onto the boundary compo-nent Sk−1×Bn−k of k-handle. When such a homeomorphism h−1 exists, themanifold M is called N with k-handle attached.

Now we are ready to state a theorem which describes the change in thetopology a manifold near a critical point of a Morse function.

Theorem 1.13. Let f be a smooth real-valued function on M . Suppose thatp ∈M is a non-degenerate critical point of f with index λ. Let c = f(p) andsuppose that f−1[c− ε, c+ ε] is compact and contains no crtical points otherthan p for some ε > 0. Then M c+ε is diffeomorphic to M c−ε with a λ-handleattached.

The proof will follow the approach of [M1] and [N]. Figure 1.6 indicatesthe idea of the proof in the special case when f is a hight function on atorus. We first modify the function f in a small neighborhood of p to obtaina new function F such that F−1(−∞, c+ ε] = M c+ε and that F−1[c− ε, c+ ε]is compact and contains no critical point. Then using Corollary 1.11, weargue that M c+ε is diffeomorphic to F−1(−∞, c − ε]. Finally, we use our

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understanding, from Morse Lemma, of the local behavior of f near p to showthat F−1(−∞, c− ε] is M c−ε with a λ-handle attached.

Proof. By Morse Lemma, there is a local coordinate system x in a neighbor-hood U of p such that

f = c− (x1)2 − . . .− (xλ)2 + (xλ+1)2 + . . .+ (xn)2

throughout U . Choose ε > 0 so that

1. the region f−1[c − ε, c + ε] is compact and contains no critical pointsother than p.

2. the image under x contains the closed ball B2ε = {q ∈ Rn : ‖q‖ ≤ 2ε}.

(see Figure 1.7). We now construct a function F : M → R as follows. First

Figure 1.7: The image of x. The heavily shaded region corresponds to theregion {f ◦ x−1 < c− ε}. The lightly shaded region corresponds to the re-gion {c− ε < f ◦ x−1 < c}. The dotted region corresponds to the region{c < f ◦ x−1 < c+ ε}

construct a smooth function ρ : [0,+∞)→ [0,+∞) such that

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1. ρ(0) > ε

2. support(ρ) ⊂ B2ε

3. −1 < ρ′ ≤ 0 on H

where ρ′ = dρdr

. Define a smooth function F : M → R by

F = f outside U

F = f − ρ(

(x1)2 + . . .+ (xλ)2 + 2(xλ+1)2 + . . .+ 2(xn)2

)on U

Note that, since ρ is supported by B2ε, F differs from f only on the (preimageof) ellipsoid

{q ∈ U : (x1)2 + . . .+ (xλ)2 + 2(xλ+1)2 + . . .+ 2(xn)2 ≤ 2ε}

Also note that F ≤ f on M because ρ is a nonnegative-valued function. Forconvenience, we define functions ξ, η : U → R by

ξ = (x1)2 + . . .+ (xλ)2, η = (xλ+1)2 + . . .+ (xn)2

With this notation, we have f = c− ξ + η and

F = c− ξ + η − ρ(ξ + 2η) on U

We now claim the followings.

Claim 1. F−1(−∞, c+ ε] = M c+ε

Claim 2. The region F−1[c − ε, c + ε] is compact and contains no criticalpoints.

Proof of Claim 1. Within the ellipsoid {ξ + 2η ≤ 2ε}, we have

f = c− ξ + η

≤ c+1

2ξ + η

≤ c+ ε

Hence F ≤ f ≤ c+ ε on the ellipsoid. Outside the ellipsoid, the functions fand F coincide. Thus we have shown f ≤ c+ ε if and only if F ≤ c+ ε, andthis proves Claim 1.

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Proof of Claim 2. We first show that F−1[c − ε, c + ε] is compact. SinceF ≤ f , we have F−1(−∞, c − ε) ⊃ f−1(−∞, c − ε). Also, by Claim 1 weknow F−1(−∞, c+ ε] = f−1(−∞, c+ ε]. Thus

F−1[c− ε, c+ ε] ⊂ f−1[c− ε, c+ ε] (1.5)

Since f−1[c − ε, c + ε] is compact by assumption, the above inclusion showsF−1[c− ε, c+ ε] is compact.

Next, we show that the functions f and F have the same critical points.The functions f and F coincide outside U and hence have the same criticalpoints in that region. Within U , we have

d(F ◦ x−1) =∂F

∂ξd(ξ ◦ x−1) +

∂F

∂ηd(η ◦ x−1)

where∂F

∂ξ= −1− ρ′(ξ + 2η) < 0

∂F

∂η= 1− 2ρ′(ξ + 2η) ≥ 1

d(ξ ◦ x−1)(x) = (2x1, . . . , 2xλ, 0, . . . , 0)

d(η ◦ x−1)(x) = (0, . . . , 0, 2xλ+1, . . . , 2xn)

Hence d(F ◦ x−1)(x) = 0 if and only if x = 0, showing that p is the onlycritical point of F within U . This proves that f and F have the same criticalpoints within U as well as outside U .

Now consider the region F−1[c − ε, c + ε]. We have shown thatF−1[c− ε, c+ ε] ⊂ f−1[c− ε, c+ ε] and that f and F have the same criti-cal points. Furthermore, by assumption f−1[c− ε, c + ε] contains no criticalpoint other than p. Hence F−1[c− ε, c + ε] contains no critical point exceptpossibly at p. However we know p /∈ F−1[c− ε, c+ ε] because

F (p) = f(p)− ρ(ξ(p) + 2η(p))

= c− ρ(0)

< c− ε

Thus F−1[c− ε, c+ ε] contains no critical point. This completes the proof ofClaim 2.

19

From Claim 2 and Corollary 1.11, it follows that F−1(−∞, c− ε] andF−1(−∞, c + ε] are diffeomorphic. Moreover, by Claim 1, we know thatF−1(−∞, c+ ε] = M c+ε. Thus, F−1(−∞, c− ε] is diffeomorphic to M c+ε.

It remains to show that F−1(−∞, c − ε] is given as M c−ε with λ-handleattached. LetH denotes the closure of F−1(−∞, c−ε]−M c−ε (see Figure 1.8).We need to show that H is homeomorphic to Bλ×Bn−λ. Note that H is an

Figure 1.8: H is the white region in the middle. The shaded region corre-sponds to M c−ε, and the dotted region corresponds to F−1[c− ε, c+ ε]

intersection of F−1(−∞, c − ε] and f−1[c − ε,+∞). Hence the region H isdescribed by the system of inequalities

ξ + η ≤ 2ε−ξ + η ≥ −ε−ξ + η − ρ(ξ + 2η) ≤ −ε

Define a subset R of R2 by

{(x, y) ∈ R2 : x, y ≥ 0, x+ y ≤ 2ε,−x+ y − ρ(x+ 2y) ≤ −ε,−x+ y ≥ −ε}

20

Figure 1.9

We will show that R is homeomorphic to a rectangle. Figure 1.9 should helpyou to follow the argument below.

First consider the polygonal region

{x, y ≥ 0, y − x ≥ 0, x+ y ≤ 2ε}

Define a function ux(y) by

ux(y) = −x+ y − ρ(x+ 2y) + ε

Since −1 < ρ′ ≤ 0 by our construction, we have ρ(x) > ρ(0)− x. Hence

ux(0) = −x− ρ(x) + ε

< −ρ(0) + ε

< 0

On the other hand, ux(y) → +∞ as y → +∞. Since ux(y) is a strictlyincreasing function of y, our argument shows that there exists a unique solu-tion y = s(x) of the equation ux(y) = 0. Using the implicit function theoremit follows that s is smooth function of x and

ds

dx(x) =

1− ρ′(x+ 2s(x))

1 + ρ′(x+ 2s(x))∈ [0, 1]

Note that the point Q, the intersection of two lines x+y = 2ε and y−x = −ε,lies on the graph of s. Also, since s′ ∈ [0, 1], the slope-1 segment AQ lies

21

below the graph of s (see figure). We now see that the region R is describedby the system of inequalities

{x, y ≥ 0, y ≤ s(x), y − x ≥ −ε}

From this, it is not difficult to see that there is a homeomorphism g from Rto a cube [0, 1] × [0, 1] such that the vertices O,A,Q, P are mapped to thevertices

(0, 0), (1, 0), (1, 1), (0, 1)

(See Figure 1.9)We finally construct a homeomorphism from H to Bλ × Bn−λ. Define a

projection (rλ, rn−λ) from H onto [0, 1]× [0, 1] by

(rλ(p), rn−λ(p)) = g (ξ(p), η(p))

for each p ∈ H. Write the coordinate function x as (xλ,xn−λ) where xλ(p) ∈Rλ and xn−λ(p) ∈ Rn−λ. Define a map h : H → Bλ ×Bn−λ by

h =

(rλ

xλ‖xλ‖

, rn−λxn−λ‖xn−λ‖

)Note that xλ

‖xλ‖= xλ

ξ1/2and xn−λ

η1/2are the angular coordinates of Sλ and Sn−λ.

One can check that h is actually a homeomorphism between H and a λ-handle. This proves that F−1(−∞, c − ε] is in fact given as M c−ε with λ-handle attached, completing the proof of Theorem 1.13.

2

In the proof of Theorem 1.13, we showed that we can modify the functionf locally to obtain a function whose value at a critical point p is smaller thanthat of f . Hence the corollary below follows from the proof of Theorem 1.13.

Corollary 1.14. There is a Morse function f on M such that distinct criticalpoints of f take distinct values.

Theorem 1.13 says that, when p is a critical point of a Morse functionf , the change in the topology of manifold between M f(p)−ε and M f(p)+ε cor-responds to a handle attachment. We occasionally refer to this change inthe topology as the change in the topology of M when passing through the

22

critical point p of f. Repeated applications of Theorem 1.13 to a compact n-dimensional manifold M equipped with a Morse function reveal that M canbe described as n-dimensional balls with a sequence of handle attachments.The readers are encouraged to see [M1] page 1-3 for an example of a handledecomposition using a height function on torus.

Unfortunately, the decomposition of M into handles often do not provideenough information to classify manifolds. There are two issues. The first isthat there are many different way to attach a handle to a manifold. Thisbecomes more problematic in higher dimensions when there are more distinctways of attaching a handle. The second is that the same manifold admitsmany different Morse functions which in turn give rise to different handledecompositions.

Theorem 1.15. For any smooth manifold without boundary M , there is aMorse function f on M such that λ(p) < λ(q) implies f(p) < f(q) for anytwo critical points p and q of f .

Theorem 1.16. For any smooth manifold without boundary M , there is aMorse function f on M which has only one minimum and one maximum.

2 Classification of Two-dimensional Manifolds

As a simple application of the Morse theory, in this section we will exhibitthe classification of compact orientable connected two-dimensional manifoldsby their diffeomorphism type. More precisely, we will show that any compactorientable connected two-dimensional manifold can be expressed as a “unionof two spheres with k-holes.” Recall that a manifold is said to be orientableif there exists a collection of coordinate functions {xα}α∈Γ for some indexingset Γ such that a differential of a change of coordinate function between anytwo maps x,y ∈ {xα}α∈Γ has a positive determinant whenever the domainsof x and y overlap. The Mobius band is a famous example of a non-orientablemanifold.

2.0 Zero and One dimensional manifolds

We first consider the classification of zero and one-dimensional manifolds.Trivially, a zero-dimensional manifold is just a collection of isolated points.

23

Classifying one-dimensional manifolds turns out to be not so difficult either,and the following fact has been known for long time.

Theorem 2.1. Any compact connected one-manifold is diffeomorphic eitherto a closed interval [0, 1] or to a circle S1 = {x ∈ R2 : ‖x‖ = 1}.

Proof. See [GP] page 208-211.

2

Incidentally the above fact implies that a one-manifold is always orientable.It also follows from the above fact that a boundary of a compact two-manifoldconsists of disjoint S1’s (circles). In particular, if a ∈ R is not a critical valueof f , then the set Ma = f−1(−∞, a] is a manifold with boundary and itsboundary f−1(a) consists of disjoint circle components.

When we analyze the structure of a manifold through Morse theoretichandle decomposition, it is important to know the diffeomorphism type ofthe level sets of a Morse function. In fact, our proof of the classification oftwo-manifolds will rely on the classification of one-manifolds.

2.1 Useful Lemmas

Here we establish three important results for our proof of the classificationof two-manifolds. The purpose of the first lemma is to classify the typesof change in the topology of a two-manifold when passing through criticalpoints of a Morse function.

Lemma 2.2. Let M be a compact orientable two-manifold and let p ∈ Mbe a critical point of a Morse function f on M . Suppose that f(p) ∈ [a, b]and that the region f−1[a, b] contains no critical point other than p. Let k bethe number of disjoint circle components of a submanifold f−1(a). Then thenumber of disjoint circle components of a submanifold f−1(b) is either k + 1or k − 1.

In other words, Lemma 2.2 says that, when the function f passes throughits critical point, the number of disjoint circle components either increases ordecreases by one.

Though it is always true that the number of disjoint circle componentschanges by at most one, the orientability of M further guarantees that the

24

Figure 2.10: Possible types of change in the topology of a two-manifold atcritical points

(a) index 0 (b) index 2 (c) index 1 (d) index 1

number cannot stay the same. In fact, if the number of disjoint componentsstays the same when passing through a critical point of f , then that wouldimply that the two-manifold M contains a Mobius band as a submanifold.

In the proof of Lemma 2.2, we assume that the readers are familiar witha definition of an orientation on M as a smooth assignment of ±1 to orderedbases for tangent spaces of M and how an orientation on M induces anorientation on its boundary ∂M . There are two possible ways to induce theboundary orientation depending on the convention one chooses to use, butthe choice makes no difference to the following proof and therefore we willnot specify which convention to use.

Proof of Lemma 2.2. Let λ be the index of the critical point p. We first dealwith the case when the index λ is 0. Let x be a coordinate function as inMorse Lemma on the neighborhood U of p. Choose ε > 0 small enough thatB√ε ⊆ x(U) and f−1[c − ε, c + ε] contains no critical points other than p.Note that the former condition implies that, by our choice of the particularcoordinate function x, f−1(c+ ε) ∩ U is diffeomorphic to a circle. By modi-fying the proof of Theorem 1.5, one can construct, using the gradient flow, adiffeomorphism on M mapping f−1(c+ε)−U to f−1(c−ε) (see Figure 2.11).This proves that the number of disjoint circle components of f−1(c + ε) isone larger than that of f−1(c− ε), completing the proof for the case λ = 0.

Note that an index 2 critical point of f is an index 0 critical point of −f .Hence, by applying the previous result for the case λ = 0 to −f , we see thatthe number of disjoint components decrease by one when passing through anindex 2 critical point.

Now suppose the index λ is 1. In this case, by Theorem 1.13 it followsthat M b is diffeomorphic to Ma with 1-handle attached. Hence there is a

25

Figure 2.11: Index 0 critical point

homeomorphism h : [0, 1]× [0, 1]→M such that its restriction to the bound-ary component [0, 1]× {0, 1} is an embedding onto the boundary of Ma andthat Image(h) ∪Ma is diffeomorphic to M b. So it suffices to show that theboundary of the manifold Image(h) ∪Ma has the number of disjoint com-ponents one larger or smaller than the level set f−1(a) does. Note that theimage of h|[0,1]×{0,1} is contained in at most two disjoint circle components.If two sides of the handles [0, 1] × {0, 1} get attached to two disjoint circlecomponents, then the two circles becomes one circle and hence the numberof disjoint components decrease by one (see Figure 2.12).

Figure 2.12

26

Figure 2.13

Now suppose that the image of h|[0,1]×{0,1} is contained in one circle.Without loss of generality, assume that the induced boundary orientationon ∂(Image(h) ∪ Ma) is given as follows (see Figure 2.13): the orientationon the line segment {0} × [0, 1] is “downward” i.e. sign

(dh(0,t)(−e2)

)= +1

for all t ∈ [0, 1] where e2 = (0, 1). Note that this imposes the inducedboundary orientation on the line segment {1} × [0, 1] to be “upward” i.e.sign

(dh(1,t)(e2)

)= +1 for all t ∈ [0, 1]. Let θ ∈ [0, 2π] be the angular coordi-

nate of S1 and let γ : θ → γ(θ) be an orientation preserving diffeomorphismfrom S1 onto a subset of ∂(Image(h) ∪Ma) such that γ(0) = h((0, 0)). Thiscurve can never come to the point h((1, 1)) because of the given orienta-tion on the line segment {1} × [0, 1]. If the curve reaches the point h((1, 0))before coming back to h((0, 0)), then this would imply that the handle is at-tached over two disjoint circles, contradicting our assumption that the imageof h|[0,1]×{0,1} is contained in one circle. Hence, the curve must come backto h((0, 1)) forming a circle disjoint from the line segment {1} × [0, 1]. Thisalso implies that the circle component containing the line segment {1}× [0, 1]forms another disjoint circle. Thus, when attached to one circle, a 1-handleforms two disjoint circles out of the original one, increasing the number ofdisjoint circle components by one. This completes the proof for the last caseλ = 1.

2

The proof of Lemma 2.2 shows that there are two types of change in

27

the topology of a two-manifold when a Morse function passes through anindex 1 critical point. One corresponds to attaching a handle over a singlecircle which “splits” the circle into two circles. For convenience, we callthis type of change in the topology as a disconnecting modification. Theother corresponds to attaching a handle over two distict circles producinga single circle. We call this type of change in the topology as a connectingmodification.

Figure 2.14: disconnecting modifica-tion

Figure 2.15: connecting modification

Although we classified the changes in the topology as a connecting ordisconnecting modification, one may wonder if the two disconnecting (orconnecting) modifications produce exactly the same manifold; that is, if thetwo manifolds M1 and M2 are both obtained through disconnecting modifica-tions on the same manifold M , are M1 and M2 diffeomorphic to each other?They are obviously homeomorphic to each other by the definition of handleattachment. However, whether they are diffeomorphic turns out to be a sub-tle question (though the answer is “yes”), and we will later use the followinglemma without a proof to complete the classification of two-manifolds.

Lemma 2.3. If a manifold M is obtained by attaching a 1-handle to theboundary component S1×{1} of a cylinder S1×[0, 1], then M is diffeomorphicto B2 −B2 × {0, 1}, a disk with two embedded disks removed.

The next lemma tells you that the values of two critical points of a Morsefunction can be “switched” under certain conditions. The statement of thelemma may sound somewhat complicated, but essentially it says that theindex 1 critical points of a Morse function on a two-manifold can be assumedto be ordered in a way that all the connecting modifications are done beforethe disconnecting modifications. For convenience, we denote the number ofdisjoint circle components of a level set f−1(c) byN (c) when c is not a criticalvalue of f .

28

Figure 2.16: cylinder with 1-handle attached

Lemma 2.4. Let M be a two-manifold and p−, p+ be index 1 critical points ofa Morse function f on M such that f(p−) < f(p+). Suppose thata < f(p−) < c < f(p+) < b and that the region f−1[a, b] contains no criticalpoints of f other than p− and p+. If N (a) = N (b) = k and N (c) = k − 1,then there is a Morse function f on M such that

1. f = f on M − f−1[a, b]

2. f has no critical points in the region f−1[a, b] other than p− and p+

3. f(p−) > f(p+) and N (c) = k + 1 whenever f(p−) > c > f(p+)

In the proof of Lemma 2.4, for simplicity we assume a = −2,f(p−) = −1/3, f(p+) = +1/3, and b = +2. See Figure 2.17, 2.18, and2.19, to help understand the idea of the proof. We first deform the manifoldf−1[0,+∞) using the positive gradient flow of f until the subset of f−1(0)enters the region f−1[1,+∞]. Call this deformed manifold M+. Similarly,we deform the manifold M0 = f−1(+∞, 0] using the negative gradient flowof f until the subset of f−1(0) enters the region M−1. Call this deformedmanifold M−. Then we “glue” the boundary of M− and M+ to obtain a newmanifold diffeomorphic to M . The fact that f < −1 on an open subset of∂M− and that f > 1 on an open subset ∂M+ allows us to do the “glueing”so that the resulting manifold possesses a Morse function with the orders ofthe critical points p− and p+ switched. These “glueing” and “deforming”manifolds are actually done through modifying the Morse functions on themanifolds although we intuitively think of these processes as glueing anddeforming the manifolds themselves.

29

Figure 2.17: A subset of the manifold M containing the critical points p+

and p−

Proof. For simplicity, assume a = −2, f(p−) = −1/3, f(p+) = +1/3, andb = +2. By assumption, the number of disjoint components increase by onewhen passing through the critical point p+. From the proof of Lemma 2.2, weknow that this corresponds to attaching a handle over a single circle. Let Cdenote the circle to which a handle get attached. Without loss of generality,we can assume C ⊂ f−1(0). Now let φ : R×M →M the 1-parameter groupof diffeomorphisms induced by the positive gradient flow ∇f , and considerthe images of C under the diffeomorphisms φt.

Claim 1. There is an open subset I of C and t > 0 such that

f > 1 on φt(I)

Proof of Claim 1. We first show that there is at least one point p ∈ C suchthat φt(p) > 1 for some t > 0. Choose a coordinate function x as in MorseLemma on the neighborhood U of p+, and choose ε > 0 such that Bε ⊂ x(U).Let V = x−1(Bε) and note that the region f−1[−1,+1] − V is compact andcontains no critical point of f .

Hence we can find α > 0 such that ‖∇f‖ ≥ α on f−1[−1,+1] − V forsome α > 0. This implies that for each p ∈ C there is s > 0 such thatf(φ(s, p)) > 1 unless the curve φp(t) eventually enters the region V .

Now suppose f(φ(t, p)) ≤ 1 for all t ∈ R and all p ∈ C. From the previous

30

Figure 2.18: After modifyingf−1(−∞, 0] and f−1[0,+∞) using agradient flow

Figure 2.19: After “glueing” M− andM+

argument, we know this implies that φt(C) ⊂ V for some finite time t. So wecan define a map ψ from C to the boundary of V by ψ(p) = φ(tp, p) wheretp is the unique time for which φ(tp, p) ∈ ∂V . Using the properties of φ, it isnot difficult to show that ψ is a homeomorphism onto its image. Hence x◦ψdefines a homeomorphism from C to the subset of S1 = ∂Bε. This impliesthat ψ is in fact a homeomorphism from C onto ∂V , but this is impossible asp+ is a saddle point. Hence we must have had f(φ(t, p)) > 1 for some t > 0and some p ∈ C.

In fact, since φs is a diffeomorphism for each s, there must be a neigh-borhood I ⊂ C of p such that f > 1 on φt(I). This proves Claim 1.

Now choose t+ > 0 and an open subset I+ of C as in Claim 1 (seeFigure 2.20). Note that, by repeating the arguments of Claim 1 using −fin place of f , we can find t− > 0 and I− so that f < −1 on φ−t−(I−).Without loss of generality assume that f < 2 on φt++ε(C) and that f > −2on φ−t−−ε(C) for some ε > 0. Now let γ be a diffeomorphism from a circleS1 to C and define a map h from a cylinder S1 × [−t− − ε, t+ + ε] to a

31

Figure 2.20

neighborhood of C in M by

h(θ, s) = φ(s, γ(θ))

where θ ∈ [0, 2π] is an angular coordinate of S1 and s ∈ [−t− − ε, t+ + ε].Since the hypothesis of Lemma 1.10 is satisfied, h is a diffeomorphism ontoits image (see Figure 2.21). We now construct a smooth function F+ on the

Figure 2.21: The cylinder and the image of h

upper half S1×[0, t+ +ε] of the cylinder so that we will have f ◦h = F+ wheref is a function satisfying the properties in the statement of this lemma. Firstchoose a smooth function g : [0, 2π]→ R such that

32

1. g = 1 on[

34π, 5

4π]

2. support(g) ⊂[

12π, 3

2π]

3. 0 ≤ g ≤ 1 throughout [0, 2π]

For α > 0, define Gα : [0, 2π]× [−t− − ε, t+ + ε]→ R by

Gα(θ, s) = g(θ) + αs

Claim 1. There is a constant β > 0 and a real-valued function F+ onS1 × [0, t+ + ε] such that

1. F+ has no critical point on S1 × [0, t+ + ε]

2. F+ = Gβ on S1 × [0, 14t+]

3. F+ = f ◦ h on S1 × [t+, t+ + ε]

(See Figure 2.22 and 2.23.)

Figure 2.22: A schematic picture of the functions g and f ◦ h at s = t+

Proof. We exhibit the crucial properties of f ◦h that allow us to construct thedesired function F+ and leave the details of the construction to the readers.

Let h : [0, 2π]× [0, t+ + ε]→ [0, 2π] be an isotopy of S1 such that

33

Figure 2.23: A schematic picture of F+ for a fixed θ

1. hs = idS1 for all s ∈ [0, 12t+]

2. ht+ maps the interval θ ∈[

12π, 3

2π]

into h−1 (φt(I))

(To understand the meaning of the second condition on h, see Figure 2.24)Define a diffeomorphism H on a cylinder by setting H(θ, s) = (h(θ), s) forall (θ, s) ∈ [0, 2π]× [0, t+ + ε]. By our construction we have

f ◦ h ◦ H(θ, t+) > 1 for θ ∈[

1

2π,

3

2π

]f ◦ h ◦ H(θ, t+) > 0 for all θ ∈ [0, 2π]

The previous two properties imply

f ◦ h ◦ H(θ, t+)− g(θ) > 0 for all θ ∈ [0, 2π]

Hence there is β > 0 such that

f ◦ h ◦ H(θ, t+)− g(θ) > 2βt+ for all θ ∈ [0, 2π] (2.6)

This choice of β guarantees the existence of a function on S1 × [0, t+ + ε]satisfying the second and third properties in the statement of this claim. Tosatisfy the first property of F+, we need another property of f ◦h◦H; namely,

∂f ◦ h ◦ H∂s

(θ, s) > 0 for all (θ, s) ∈ [0, 2]× [−t− − ε, t− + ε] (2.7)

34

Figure 2.24: The region h−1 (φt(I)) is where the value of f ◦h(θ, t+) is largerthan 1.

which follows from the following computation:

∂f ◦ h ◦ H∂s

= df

(∂h ◦ H∂s

)

= df

(∂h

∂s

)= df

(∂φ

∂s

)= df (∇f)

= 〈∇f, ∇f〉= ‖∇f‖2

The two properties of f ◦ h ◦ H (see Equation 2.6 and 2.7) guarantees theexistence of the desired function F+ on [0, 2π] × [−t − ε, 0]. This completesthe proof of Claim 1.

Notice that, by imitating the construction of F+, we can construct afunction F− on the lower half S1 × [−t− − ε, 0] of the cylinder such that

1. F− has no critical point on S1 × [−t− − ε, 0]

2. F− = Gβ on S1 × [−14t−, 0]

3. F− = f ◦ h+ 1 on S1 × [−t− − ε,−t−]

35

(To construct F−, we may have to replace β with an appropriate smallerpositive real number.)

To complete the construction of the function f as in the statement ofthis Lemma, we only need to “patch” the function F+ ◦ h−1 and F− ◦ h−1

together. Choose sufficiently small δ > 0 such that f−1[−2−δ, 0] contains nocritical point of f . From Corollary 1.8 we know that the set f−1[−2− δ,−2]is diffeomorphic to f−1(−2) × [0, 1]. Using this fact, we can construct amodified Morse function f1 on M0 = f−1(−∞, 0] such that

1. f1 = f + 1 on the component of f−1[−2, 0] connected to C

2. f1 = f outside the component of f−1[−2− δ, 0] connected to C

3. the critical points of f1 coincide with those of f

(See Figure 2.25.) Note that we in particular have f1(p−) = +23> f(p+).

Finally, define f by

Figure 2.25: The picture of M0 when the height of a point corresponds tothe values of the functions f and f1 at the point. Outside the shaded region,the function f1 corresponds either to f or f + 1

f =

F+ ◦ h−1 on h(S1 × [0, t+ + ε])F− ◦ h−1 on h(S1 × [−t− − ε, 0])

f on f−1[0,+∞)− h(S1 × [0, t+ + ε])f1 on f−1(−∞, 0]− h(S1 × [−t− − ε, 0])

36

Figure 2.26: An example of how f looks like in comparison to f . The heightof a manifold M is taken to be the value of f in the left picture and of fin the right picture. The red, green, blue, and purple regions respectivelycorrespond to f−1[0,+∞)−h(S1×[0, t++ε]), h(S1×[0, t++ε]), f−1[0,+∞)−h(S1 × [0, t+ + ε]), and f−1(−∞, 0]− h(S1 × [−t− − ε, 0])

(See Figure 2.26 to compare the function f with f). One can check that fsatisfies the properties in the statement of this lemma, completing the proofof Lemma 2.4.

2

2.2 Proof of the Classification Theorem

We finally prove the classification of compact orientable connected two-manifolds. We will use the following terminology in the statements of theclassification theorem; a manifold M is a union of two spheres with k holes ifM is a union of manifolds with boundary M1 and M2 such that M1 ∩M2 =∂M1 = ∂M2 and that each Mi is diffeomorphic to a manifold obtained by

37

removing k disjoint embedded disks B2 × {1, . . . , k} from a sphere S2 (seeFigure 2.27 and 2.28).

Figure 2.27: A sphere with seven holes

Figure 2.28: A union of two sphereswith seven holes

Theorem 2.5. Any compact orientable two-manifold is diffeomorphic to aunion of two spheres with k-holes for some k ∈ N.

We note that Theorem 2.5 by itself falls short of the “complete” clas-sification. There are two issues. To see the first issue, let M1 and M2 betwo n-dimensional manifolds with diffeomorphic boundary and suppose twomanifolds M and N are both obtained by identifying the boundary compo-nents of M1 and M2. The identifications of the boundary components maybe done through two different diffeomorphisms. Then, somewhat surpris-ingly, M and N are in general not necessarily diffeomorphic to each other.In our special case where each Mi is a sphere with k-holes, then any twomanifolds obtained by identifying the boundary components of M1 and M2

are diffeomorphic to each other. Hence a union of two spheres with k-holesis uniquely defined up to diffeomorphism. This result depends heavily on thefact that the boundary of a two-manifold has a very simple structure and isa consequence of the following theorem, which we state without a proof.

Theorem 2.6. Any diffeomorphism from S1 to S1 is isotopic to an identitymap.

Theorem 2.6 implies that any two diffeomorphisms between the boundarycomponents of M1 and M2 are isotopic when each Mi is a sphere with k-holes. Using this fact, it is not difficult to show that a union of two sphereswith k-holes is uniquely defined up to diffeomorphism.

To see the second issue with Theorem 2.5, letM be a union of spheres with

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k-holes and N be a union of spheres with l-holes. If k 6= l, then it is intuitivelyclear that M and N cannot be diffeomorphic. However, Theorem 2.5 doesnot actually eliminate the possibility that M and N are diffeomorphic whenk 6= l. Hence we quote the following theorem for completeness.

Theorem 2.7. Suppose M is a union of two spheres with k-holes and N isa union of two spheres with l-holes. Then M and N are diffeomorphic if andonly if k = l.

We will not prove Theorem 2.7 here because a proof requires tools fromalgebraic topology that are quite different from the techniques developedin this paper. A proof is usually done by considering a homeomorphisminvariant such as Euler characteristic or genus and showing that the twomanifolds have different Euler characteristics or genera when k 6= l.

Theorem 2.5 is actually the harder part of the classification theorem;however, given the Morse theoretic tools we have developed, the proof willbe rather straightforward.

Proof of Theorem 2.5. Let M be a compact orientable two-manifold and fbe a Morse function on M . By the virtue of Theorem 1.16, we can withoutloss of generality assume that f has only one minimum and one maximum.By applying Lemma 2.4 to each index 1 critical point of f if necessary, wecan also assume that all the disconnecting modifications are done before theconnecting modifications. In other words, if the critical points {p1, . . . , pk}of f are labeled in a way that f(pi) < f(pj) whenever i < j and if we, asbefore, let N (c) denotes the number of disjoint circle components in the levelset f−1(c), then there is m ∈ N such that

{N (c0), . . . , N (cm−1), N (cm), N (cm+1), . . . , N (ck)}= {0, . . . , m− 1, m, m− 1, . . . , 0}

for c0 < f(p1) < c1 < f(p2) < . . . < ck−1 < f(pk) < ck. Note that wenecessarily have k = 2m because of the fact that the number of disjointcircle components changes exactly by one when passing through a criticalpoint (Lemma 2.2).

Let M1 = M cm and M2 = f−1[cm,+∞). We claim that M1 is a union oftwo spheres with m-holes. By our construction M1 is a manifold obtainedby applying (m− 1) disconnecting modifications to a disk B2 = M c1 . HenceTheorem 1.13 and Corollary 1.11 tell us that the set f−1[c1, c2] is a cylinder

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Figure 2.29: A pictureof M before modifica-tions

Figure 2.30: After ap-plying Theorem 1.16

Figure 2.31: After ap-plying Lemma 2.4

with a 1-handle attached. By induction we obtain that, for each i = 1, . . . ,m,f−1[ci, ci+1] is a disjoint union of (i − 1) cylinders and one cylinder with a1-handle attached. Now using Lemma 2.3, it is not difficult to show thatM1 = M cm is diffeomorphic to a sphere with m-holes.

Considering −f on M , we can similarly argue that M2 is diffeomorphicto a sphere with m-holes. This proves Theorem 2.5.

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Remark. A union of two spheres with k-holes in the statement of Theo-rem 2.5 is often called a genus k surface. The name comes from the factthat the number k corresponds to the genus of the manifold. Intuitively, thegenus of a two-dimensional orientable closed connected manifold is a maxi-mum number of circles that can be embedded into the manifold so that themanifold remains connected after removing the images of the circles. Fromthis viewpoint, it should be intuitively obvious that a union of two sphereswith k-holes indeed has genus k.

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3 Further Directions

In the last section, we established the complete classification of compact con-nected orientable two-manifolds. In fact, we can classify compact connectedtwo-manifolds (including non-orientable ones) using the Morse theoretic han-dle decomposition if we work a little bit harder. A natural thing to ask nextis: “can we obtain a similar classification theorem for higher dimensionalmanifolds?” It turns out classifying manifolds becomes much more difficultin higher dimensions. In this section, we give a brief account of what hasbeen proven and what questions remain open about three and four dimen-sional manifolds. For the following discussion, it is useful to introduce a moreabstract and general definition of smooth manifolds than the one we used inthe previous sections.

Definition 3.1. A smooth n-dimensional manifold is a pair (M, {xα}α∈Γ)where M is a Hausdorff compact topological space M and {xα}α∈Γ is acollection of maps for some indexing set Γ such that

1. each xα is a homeomorphism from an open set of M to an open set ofHn

2. y ◦ x−1 is a diffeomorphism between two open sets of Hn wheneverx,y ∈ {xα}α∈Γ and the domains of x and y overlap

3. M ⊂ (∪α∈Γdomain(xα)) i.e. every point of M possesses a neighborhoodhomeomorphic to an open set of Hn.

As before, we call the homeomorphisms xα coordinate functions. Underthis abstract definition, we always define differentiability using coordinatefunctions.

Definition 3.2. A map f : M → N between two smooth manifolds M andN is called smooth if the function y◦f ◦x−1 is smooth for any local coordinatex of M and any local coordinate y of N . If g is a smooth map between Mand N having a smooth inverse, then g is called a diffeomorphism betweenM and N .

The abstract definition is more convenient for some purposes, such as “cut-ting” and “pasting” manifolds. Also, the definition leads to many differentways of constructing a new manifold from other known manifolds. An impor-tant example is a construction of an “exotic R4,” which we will discuss later

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in this section. Let us also note that our results from the previous sectionsare still true under this new definition by virtue of the following theorem.

Theorem 3.3 (Whitney Embedding Theorem). Any smooth compact n-manifold can be embedded into R2n i.e. if M is a smooth n-manifold, thenthere is a map f : M → R2n that is a diffeomorphism onto its image andwhose image is a smooth n-manifold.

3.1 Handle Decomposition of Three-manifold and Hee-gard Splitting

To start our study of the topology of three-manifolds, we first try the sameMorse theoretic technique we used to prove the classification of two-manifolds.The following discussion is only meant to give some intuition and not meantto be rigorous.

Let M be a smooth compact connected three-manifold without bound-ary. Let f be a Morse function on M having exactly one maximum and oneminimum and satisfying the condition f(p) < f(q) whenever λf (p) < λf (q).Let {p1, . . . , pk} be the index 1 critical points of f such that f(pi) < f(pj)whenever i < j. Choose c ∈ R so that all the index 1 critical points liein f−1(−∞, c) and all the index 2 critical points lie in f−1(c,+∞). For asufficiently small ε > 0, the sublevel set M f(p1)−ε is diffeomorphic to B3. Thechange in the topology of M at each pi corresponds to attaching a boundarycomponent {−1, 1}×B2 of 1-handle B1×B2 (solid cylinder). Hence the man-ifold M c is obtained by attaching cylinders k times to B3 (see Figure 3.32).

The manifolds obtained by attaching k 1-handles to B3 can be com-pletely classified by their diffeomorphism type. Actually, if the resultingmanifold is orientable, it must be diffeomorphic to a genus k surface with in-side filled. Thus Morse theory can completely determine the topology of thesublevel sets below all the index 2 critical points. After the index 1 criticalpoints, however, the handle decomposition of a three-manifold become muchmore complex. In fact, nobody has been able to classify the diffeomorphismtypes of three-manifolds using Morse theoretic handle decomposition. Thereaders should be able to convince themselves that the diffeomorphism typeof manifold obtained by attaching 2-handles to M c soon become intractable.

Some readers may have noticed that the index 2 critical points of fcan be viewed as the index 1 critical points of −f and that we can ap-ply Morse theory to −f to obtain the information about the topology of

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Figure 3.32: B3 with three solid cylinders attached

(−f)−1(−∞,−c] = f−1[c,+∞). Let M− = M c and M+ = f−1[c,+∞).Since the boundaries of M− and M+ coincide, the index 1 critical points of−f must have been the same as the number of index 1 critical points of f .This shows that M is a union of two manifolds M− and M+ with their bound-aries identified where the diffeomorphism type of M− and M+ is known. Thisdecomposition of a three-manifold is called a Heegaard Splitting.

One may wonder if we can prove the classification theorem for threemanifolds using Heegaard Splitting. There are two obstacles to the classi-fication theorem using Heegaard Splitting. The first is that we can alwaysperturb the function f to create a pair of an index 1 and index 2 criticalpoints. To state this more precisely, choose ε > 0 small enough that theregion f−1[c − ε, c + ε] contains no critical points of f . We can constructa new function f which agrees with f outside the region f−1[c − ε, c + ε]but has an index 1 critical point p and an index 2 critical point q suchthat c− ε < f(p) < c < f(q) < c+ ε. If we let M− = f−1(−∞, c] and M+ =f−1[c,+∞), we see that this is a new decomposition of M into two manifolds

where M− and M+ are obtained by attaching a 1-handle to M− and M+

respectively. Of course M− and M− are not diffeomorphic and neither M+

and M+. However, the union of M− and M+ and the union of M− and M+

are diffeomorphic to each other. Thus there are infinitely many HeegaardSplitting decompositions of the same manifold.

To understand the second obstacle to the classification theorem, letM1 ∪M2 and N1 ∪N2 be decompositions of three-manifolds M and N usingHeegaard Splitting and suppose that Mi and Ni are diffeomorphic for each

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i = 1, 2. As mentioned earlier (see the discussion following Theorem 2.5),this does not imply that M and N are diffeomorphic. In the classification oftwo-dimensional manifolds, by virtue of Theorem 2.6 we could say that, giventwo two-dimensional manifolds with boundary, the diffeomorphism type of amanifold obtained by identifying their boundaries is independent of the dif-feomorphism used to identify the boundaries. However, two diffeomorphismsfrom the boundary of a three-manifold to that of another three-manifoldare not necessarily isotopic to each other. Consequently, given two three-dimensional manifolds with boundary, the diffeomorphism type of a mani-fold obtained by identifying their boundaries does depend on the choice of adiffeomorphism between their boundaries.

The two aforementioned obstacles make it extremely difficult to classifythe diffeomorphism types of a three-manifold using its Heegaard splitting.At least to date, nobody has managed to obtain a classification theorem forthree-manifold using a Heegaard splitting.

3.2 Poincare Conjecture and Classification of Three-manifolds

At the end of his 1904 paper, Poincare asked the following question (trans-lated into modern language):

If a connected closed three-manifold has trivial fundamental group,must it be diffeomorphic to S3?

The conjecture that the answer to this question is “yes” had been knownas Poincare Conjecture. The question remained open for about a hundredyears until in 2006 when the mathematical community came to agree onthe correctness of Perelman’s proof of Thurston’s Geometrization Conjecturewhich includes Poincare conjecture as a special case. Perelman’s proof wasactually published in 2003, but it took the mathematical community for afew years to verify and digest his proof because his proof was very intricateand contained strikingly original ideas.

Let us give a very rough idea of Perelman’s work. An important idea heused is the Ricci flow originally developed by Hamilton. The Ricci flow isan intrinsic geometric flow on a Riemannian manifold, and is analogous toa heat (diffusion) flow in a sense that it “smoothes out” the irregularities ina Riemannian metric. In other words, under certain conditions as the time

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tends to infinity, the Ricci flow transforms a Riemannian metric into anothermetric with respect to which the manifold has a constant curvature. How-ever, the Ricci flow in general develops singularities in a finite time, and therewere technical difficulties involved in taking the limit as the time tends toinfinity. Perelman managed to identify where the Ricci flow develops singu-larities, worked around the technical difficulties, and thus was able to proveThurston’s Geometrization Conjecture.

One consequence of Perelman’s work is that a three-manifold can alwaysbe decomposed into “elementary pieces” each of which we completely under-stand. However, those pieces can be “patched up” in many different ways,producing manifolds of different diffeomorphism types. Hence, though Perel-man’s contribution greatly enhanced our understanding of three-dimensionalmanifolds, a classification of three-manifolds by diffeomorphism type stillremains incomplete.

3.3 Exotic R4

Here we introduce the readers to a rather surprising behavior of four-dimensionalmanifolds. It might not be so surprising that manifolds of higher dimensionsbehave differently from those of lower dimension; however, four-dimensionalmanifolds sometimes behave even more strangely than the manifolds of di-mension higher than four.

We have been talking about the diffeomorphism types of manifolds, butwe can similarly ask about the homeomorphism types of manifolds. Sincethe difference between a homeomorphism type and a diffeomorphism type ofa manifold may be subtler than it appears, we first introduce the notion of adifferential structure of a manifold. Recall that an abstract manifold is notjust a set but a pair (M , {xα}α∈Γ) (see Definition 3.1). Now suppose thereis a homeomorphism x from an open set of M onto an open set of Rn suchthat the map x ◦ x−1

α is a diffeomorphism for each α ∈ Γ but x /∈ {xα}α∈Γ.By adding all such homeomorphisms to {xα}α∈Γ, we obtain a maximal col-lection {xβ}β∈I of coordinate functions of M . (It is maximal in a sense thatx ∈ {xβ}β∈I whenever x is a homeomorphism from an open set of M to anopen set of Rn such that the transition map x ◦x−1

β is diffeomorphic for eachβ ∈ I.) Actually, we often include in the definition of a smooth manifoldthat the collection {xα}α∈Γ is already maximal. This maximal collection ofcoordinate functions associated with M is called a differential structure ofM . Although we usually speak of the space M itself as a smooth manifold,

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one needs to keep in mind that an abstract manifold always comes with somecollection of coordinate functions.

A differential structure of a manifold is not necessarily unique. To seethis, suppose (M, {xα}α∈Γ) and (N, {yβ}β∈I) are homeomorphic but not dif-feomorphic (such two manifolds actually exist). If g : M → N is a home-omorphism (but not diffeomorphism) between M and N , then {yβ ◦ g}β∈Idefines a differential structure on M . However, a transition map yβ ◦g ◦x−1

α

is not diffeomorphic for any α ∈ Γ and β ∈ I because by our assumption gis not a diffeomorphism. Thus a differential structure {yβ ◦ g}β∈I on N isdistinct from {yβ}β∈I .

It turns out that, for n ≤ 3, two smooth n-dimensional manifold M andN are homeomorphic if and only if they are diffeomorphic. This statement,however, no longer holds true for n ≥ 4. An important counter-exampleis an exotic R4. The standard Rn is a smooth manifold (Rn, {idRn}) whereidRn is an identity map on Rn. An exotic R4 is a smooth four-manifold thatis homeomorphic to the standard R4 but not diffeomorphic to it. The firstexample of an exotic R4 was constructed by Gompf as an abstract manifold,and later on Taubes showed that there are uncountably many distinct exoticR4’s. This in particular means that R4 has uncountably many differentialstructures. Even before the first example of an exotic R4, it was known thatmanifolds of higher dimensions do not necessarily have a unique differentialstructure. However, strangely enough, it has also been shown that a Eu-clidean space Rn has a unique differential structure (namely the standardone) for any n 6= 4. In this sense, four-dimensional manifolds are weirderthan higher-dimensional manifolds.

There are a lot of open questions for four-dimensional manifolds. Onesuch example is the four-dimensional Poincare Conjecture. To clarify theparallel between the three-dimensional and four-dimensional Poincare Con-jecture, let us first formulate the three-dimensional Poincare Conjecture in adifferent but equivalent way.

Theorem (Three-dimensional Poincare Conjecture). If a three-manifold hasthe homotopy type of S3, then it is diffeomorphic to S3.

The above formulation of the three-dimensional Poincare conjecture can eas-ily be generalized to four-dimension.

Conjecture (Four-dimensional Poincare Conjecture). If a four-manifold hasthe homotopy type of S4, then it is diffeomorphic to S4.

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Currently, more people seem to consider the four-dimensional Poincare Con-jecture to be false.

The four-dimensional Poincare conjecture is just one open problem aboutfour-dimensional manifolds. There has been a great amount of study doneon the structure of four-dimensional manifolds, and given the complexity ofthe problems, it is expected to remain an active area of mathematics for along time.

4 Acknowledgment

I would like to thank Professor Eleny Ionel for her mentorship over a year.Her comments and advice were extremely valuable in the course of my under-graduate research. I would also like to extend my gratitude to the professorsin the Stanford math department for an exceptionally rewarding undergrad-uate experience.

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References

[DC] M. P. do Carmo, Riemannian Geometry. Boston: Birkhauser, 1992.

[GP] V. Guillemin, A. Pollack, Differential Topology. Englewood Cliffs:Prentice-Hall, Inc., 1974.

[M1] J. Milnor, Morse Theory. Princeton: Princeton University Press, No.51,1963.Figure 1.6, 1.7, and 1.8 are taken from this book.

[M2] J. Milnor, Lectures on the h-Corbordism Theorem. Princeton: Prince-ton University Press, 1965.

[N] L. Nicolaescu, An Invitation to Morse Theory. Springer, 2007.Figure 1.9 is taken from this book.

[W] A. Wallace, Differential Topology, First Steps. New York: Benjamin,1968.Figure 2.10, 2.14, 2.15, 2.29, 2.30, and 2.31 are taken from this book.

[GS] R. Gompf, A. Stipsicz, 4-Manifolds and Kirby Calculus. Providence:American Mathematical Society, 1999.Figure 3.32 is taken from this book.

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