U.U.D.M. Project Report 2016:37

Examensarbete i matematik, 15 hpHandledare: Thomas KraghExaminator: Jörgen ÖstenssonAugusti 2016

Department of MathematicsUppsala University

Classification of Compact Orientable Surfacesusing Morse Theory

Johan Rydholm

Classi�cation of Compact Orientable Surfaces

using Morse Theory

Johan Rydholm

1

Introduction

Here we classify surfaces up to di�eomorphism. The classi�cation is done insection Construction of the Genus-g Toruses, as an application of the previouslydeveloped Morse theory. The objects which we study, surfaces, are de�nedin section Surfaces, together with other de�nitions and results which lays thefoundation for the rest of the essay.

Most of the section Surfaces is taken from chapter 0 in [2], and gives aquick introduction to, among other things, smooth manifolds, di�eomorphisms,smooth vector �elds and connected sums. The material in sectionMorse Theoryand section Existence of a Good Morse Function uses mainly chapter 1 in [5]and chapters 2,4 and 5 in [4] (but not necessarily only these chapters). In thesetwo sections we �rst prove Lemma of Morse, which is probably the single mostimportant result, even though the proof is far from the hardest. We prove theexistence of a Morse function, existence of a self-indexing Morse function, and�nally the existence of a good Morse function, on any surface; while doing thiswe also prove the existence of one of our most important tools: a gradient-likevector �eld for the Morse function.

The results in sections Morse Theory and Existence of a Good Morse Func-tion contains the main resluts and ideas. They have many generalizations andmany other applications; to mention one, in [4] Milnor goes on and prove thegeneralized Poincaré Conjecture1 in dimensions n ≥ 5, using similar results. Inthis essay, the Classi�cation Theorem, great as it may seem, is almost like onlya corollary of these two sections.2

1The theorem is: If M is a simply-connected smooth manifold having dimension n ≥ 5with the (integral) homology of the n-sphere, then M is homeomorphic to the sphere in thesame dimension Sn; if n = 5 or n = 6, then M is even di�eomorphic to Sn.

2A good popular account of the classi�cation of surfaces are given in chapter 3 in [7].

2

Contents

Introduction 2

Surfaces 4

Morse Theory 9

Existence of a Good Morse Function 21

Constrution of the Genus-g Toruses 34

3

Surfaces

This section introduces some basics of the theory of manifolds, speci�cally wede�ne the concept of a surface and a genus-g torus.

De�nition 1. A smooth manifold of dimension n is a topological space M to-gether with two families of open sets {Ui}i∈I , {Vi}i∈I , of Rn and M respectively,and a set of homeomorphisms {φi : Ui → Vi}i∈I - called parametrizations,whose inverses, are called coordinate system or just coordinates - such that⋃

i∈IVi = M,

and all transition functionsφ−1i ◦ φj

are smooth; we also require the triad ({Ui}, {Vi}, {φi}) to be maximal in thesense: given ({Ui}, {Vi}, {φi}), if φ : U → V satis�es the conditions abovetogether with the triad, then (U, V, φ) is in the triad. Lastly, a smooth manifoldis required to be hausdor� and second countable.

A surface S is a manifold of dimension 2.

Moreover, for brevity, we require a surface to be connected, compact andorientable (to be de�ned later). In other words, what we will call surfacesare connected, compact and orientable 2-dimensional smooth manifolds. Also,in our de�nition the manifolds (and thereforee also the surfaces) are withoutboundary; sometimes we will nevertheless talk about a boundary, formally thisis the set of all points where all neighbourhoods - instead of being homeomor-phic (and by the de�nitions below, also di�eomorphic) to U open in Rn - arehomeomorphic to Hn = {(x1, . . . , xn) ∈ Rn|xn ≥ 0}. Sometimes we will have towork with surfaces with boundary, when this is the case it will be obvious fromcontext, and it won't cause any special problems. Note also that if the over-lap for two parametrizations is empty, then the transition function is triviallysmooth.

De�nition 2. A mapping ϕ : M → M ′ between two smooth manifolds M andM ′, of dimension n and m respectively, is said to be smooth at a point p, if forany parametrizations φ : U → V ⊂ M,φ′ : U ′ → V ′ ⊂ M ′ around p and ϕ(p)respectively, the map φ′−1 ◦ ϕ ◦ φ is smooth at x0 = φ−1(p).

The mapping φ is said to smooth if it is smooth at all points in M .

In particular, a real valued function f : S → R from a surface S is said to besmooth, if for any parametrization φ : U → V ⊂ S, the function f ◦φ is smooth.

We can now de�ne a notion of equivalence of surfaces, and more generally,equivalence of smooth manifolds, that is when they from our viewpoint areconsidered the same: the notion of di�eomorphic smooth manifolds.

De�nition 3. Two smooth manifolds M and M ′ are said to be di�eomorphic,when there exists a smooth bijection

ϕ : M∼−→M ′,

such that its inverse ϕ−1 is smooth; such a map is called a di�eomorphism.

4

The goal is to classify surfaces up di�eomorphism; we will see that the clas-si�cation can be done by a class of surfaces to be de�ned later, called genus-gtoruses. Before that we will de�ne tangent spaces and di�erentials.

De�nition 4. A smooth map γ : (−ε, ε)→M from an open interval (−ε, ε) ofR to a smooth manifold M , is called a curve in M . Suppose that p ∈ M is apoint in the manifold, with γ(0) = p, for some curve γ. Let D denote the set ofall real valued functions on M, that are smooth at p. The tangent vector to thecurve γ at p is the function γ′ : D → R de�ned by

γ′(f) =d(f ◦ γ)

dt(t = 0).

A tangent vector at p is just the tangent vector of some curve. The set of alltangent vectors ofM at p is called the tangent space ofM at p, and it is denotedTMp.

A result from the theory of smooth manifolds says that TMp is a �nitedimensional vector space having the same dimension as M .

De�nition 5. Let M and M ′ be two smooth manifolds, and let ϕ : M → M ′

be a smooth map. For all p ∈M and all v ∈ TMp, by a well known result fromthe theory of smooth manifolds (which is not hard to prove), given any curve γfor which v is the tangent vector of, the linear map

dϕp : TMp → TM ′p, dϕp(v) = β′(0),

where β = ϕ ◦ γ, is independent of the choice of γ. This map is called thedi�erential of ϕ at p.

Next we de�ne the notion of a vector �eld on a manifold. A special typeof vector �elds called gradient-like vector �elds de�ned in the next section willbe one of the main tools used. Afterwards we de�ne the notion of a smoothmanifold being orientable, and give a citerion using möbius bands to determinewhether a surface is orientable or not.

The set TM = {(p, v)|p ∈ M, v ∈ TMp} is called the tangent bundle of M .It is well known that the tangent bundle is a smooth manifold of dimensiontwo times the dimension of M . Using this smooth manifold we can give thede�nition.

De�nition 6. A vector �eld on a smooth manifold M is a map X : M → TM ,such that for each p ∈M we assaign a vector v ∈ TMp: X : p 7→ (p, v).

Since TM is a smooth manifold we have an earlier de�nition that guaranteesthat it makes sense to ask wetherX is smooth or not. Hence we can simply de�nea smooth vector �eld to be a vector �eld X that is smooth by this de�nition(de�nition 2).

De�nition 7. We say that a manifold M is orientable if there are parametriza-tions {φi} covering all of M such that for every pair φi : Ui → Vi, φj : Uj → Vjwith Vi ∩ Vj 6= ∅, the change of coordinates φ−1j ◦ φi has positive determinant.

For surfaces: A surface S is orientable if and only if it is impossible toembedd a möbius band B in S.3

3An embedding of B in S is a map e : B → S such that (i) dϕp : TBp → TSp is injectivefor all p ∈ B, and (ii) ϕ is a homeomorphism from B to ϕ(B).

5

Now we give some facts and terminology about vector �elds which will beused in many of the proofs. We begin with trajectorys.

De�nition 8. A trajectory in a point p of a smooth vector �eld X is mapγ : (−ε, ε) → V , from an open interval (−ε, ε) to a neighbourhood V of p suchthat γ(0) = p and γ′(t) = X(γ(t)).

A trajectory of a smooth vector �eld X, from a point p to another point q,is a smooth curve γ : (0, 1) → S such that γ(t) → p as t → 0, γ(t) → q ast→ 1, and γ′(t) = X(γ(t)).

The curve in the de�nition above is known to exist and be unique for smallenough ε. Next we mention, a theorem from the theory of di�erential equationswhich will be needed.

Theorem 1. Suppose that X is a smooth vector �eld on a manifold M , and letp ∈M . Then there exists a neighbourhood V ⊂M of p, an interval (−ε, ε) anda smooth map ϕ : (−ε, ε)× V →M such that the curve

t 7→ ϕ(t, q),

t ∈ (−ε, ε), for any q ∈ V , exists and is the unique satisfying

∂ϕ

∂t= X(ϕ(t, q)), ϕ(0, q) = q.

The curves t→ ϕ(t, q) in the theorem above, for any �xed q, whose deriva-tives are the trajectories, are called integral curves of the vector �eld.

To de�ne a genus-g torus, we are going to need the notion of di�eomorphismsbeing isotopic. This is something that also will be used in the sections aboutMorse functions. We are also going to need connected sums. These two notionsallow us to past together surfaces in a unique way.

De�nition 9. Suppose that ϕ0 : M →M and ϕ1 : M →M are di�eomorphismson a smooth manifold M . We say that they are smoothly isotopic, if there is asmooth map (a smooth homotopy) F : [0, 1]×M →M such that:

1. F (0, p) = ϕ0(p), for all p ∈M ,

2. F (1, p) = ϕ1(p), for all p ∈M , and

3. F (t, p) : M∼−→M is a di�eomorphism for all t ∈ [0, 1].

Lemma 1. Any orientation preserving di�eomorphism ϕ : S1 ∼−→ S1 is smoothlyisotopic to the identity map.

Proof. Let S1 be given by R/2πZ, and let ϕ : R/2πZ ∼−→ R/2πZ be an ori-entation preserving di�eomorphism. It is clear that any orientation presevingdi�eomorphism of R/2πZ onto itself can be given as function on R onto it-self, which on [0, 2π] is a strictly increasing smooth function going from someθ ∈ [0, 2π) to θ + 2π, and which is 2π-periodic.

Suppose ϕ goes from θ to θ + 2π. Let the isotopy be de�ned by

F (t, x) = tϕ(x) + (1− t)x.

6

It is clear that F (0, x) = id and F (1, x) = ϕ. We show that it is a di�eomor-phism going from tθ to tθ + 2π, for any �xed t. Firstly, F is clearly smooth.Moreover, strictly increasing:

F ′x(t, x) = tϕ′(x) + (1− t) > 0,

because ϕ′(x) > 0 (and either t > 0 or 1− t > 0, always t, 1− t ≥ 0). It startsat tθ, because F (t, 0) = tϕ(0) + (1 − t)0 = tθ, and ends at tθ + 2π, becauseF (t, 2π) = tϕ(2π) + (1− t)2π = t(θ + 2π) + (1− t)2π = tθ + 2π.

De�nition 10. Let S1 and S2 be two surfaces. The connected sum of S1 andS2, denoted S1#S2, is obtained in the follwing way. Begin by deleteing an openball from each surface, denote the balls B1 and B2; then both S1 − B1 andS2 − B2 have a boundary - denoted ∂B1 and ∂B2, respectively - di�eomorphicto S1. De�ne an orientation preserving di�eomorphism

g : ∂B1∼−→ ∂B2.

Then S1#S2 is obtaint by taking the disjoint union (S1−B1)∪g (S2−B2), wherewe identify the points on ∂B1 and ∂B2 by the di�eomorphism g. The obtainedsurface S1#S2 is by lemma 1 unique up to di�eomorphism.

The genus of a surface is the number of holes of a surface. Hence a genus-gtorus is like a torus but with g holes, instead of just one. The formal de�nitionis as follows.

De�nition 11. We de�ne a genus-g torus, denoted Tg, inductively. A 0-torus isany surface homeomorphic to a sphere. A 1-torus is any surface homeomorphicto the torus; we can take the following one as archetype:

T1 =

{(x, y, z) ∈ R3|

(b−

√x2 + y2

)2+ z2 = a2

},

where a, b > 0 are constants such that b > a. (a is the radius of the tube, b isthe radius from the center of the torus hole to the center of the tube.)

Suppose we have de�ned Tg, for all g ≤ g0. We then de�ne Tg0+1 as Tg0#T1.

Figure 1: The connected sum of a 1-torus and a 2-torus, obtaining a 3-torus.

7

We will show, in the last section, that the sphere and the torus both areunique up to di�eomorphism, and not only homeomorphism, from which it willfollow that all Tg are.

The great accomplishment of the classi�cation theorem is that it shows thatthe number of "holes" in a surface is enough to classify them: all surface withg holes are di�eomorphic to the genus-g torus, and hence to each other, and soup to di�eomorphism the genus-g toruses make up all possible surfaces.

8

Morse Theory

This section introduces some of the central notions of Morse theory. Morsetheory is about a class of smooth real valued functions on manifolds (surfaces),which can be seen as a way to de�ne and measure the height (possibly negative)of a manifold (surface), such that the changes as the height increases are good(in a sense given later). In the next section we will see how we can deformthe measurement - rather then deforming the surface - so that we get a goodway to measure height: we show that there exists a good Morse function. Onesection further we will connect this to the notion of genus. The theory here isonly given for surfaces; most results hold also for arbitrary manifolds, only theproofs have to be slightly changed.

We begin by de�ning the concepts of critical point and critical value, after-wards we de�ne the concept of index which allows us to compare di�erent typesof critical points.

De�nition 12. Let S be a surface, and let f : S → R be a real valued function.We say that p ∈ S is a critical point of f if

f ′(p) = 0,

where f ′ : TSp → TRf(p) is the induced map of the tangent spaces. In aparametrization φi, with φi(x0, y0) = p, this means that

∂(f ◦ φi)∂x

(x0, y0) =∂(f ◦ φi)

∂y(x0, y0) = 0.

A critical value x ∈ R of f is a real number that is the image of a criticalpoint. Any points which is not a critical point of f is called a regular point of f ,and any real number which is not a critical value of f is called a regular valueof f .

De�nition 13. Let S be a surface, let p ∈ S be a point, let f : S → R be areal valued function on S, and let φ be a parametrization of S around p withφ(x0, y0) = p. The Hessian of f with respect to φ is de�ned as

Hφ(f) =

(∂2(f◦φ)∂x2

∂2(f◦φ)∂x∂y

∂2(f◦φ)∂x∂y

∂2(f◦φ)∂y2

).

If φ and ψ are two di�erent parametrizations of S around p, with φ(x0, y0) =ψ(x0, y0) = p, then the Hessian of f with respect to φ and ψ are related by

Hψ(f) = JT (θ) ◦Hφ(f) ◦ J(θ),

where θ is the change of coordinates from ψ to φ, i.e. θ = φ−1 ◦ ψ, and J(θ)is the Jacobian of θ (JT (θ) is just its transpose). Because of this the followingde�nition is independent of parametrization.

De�nition 14. Let f : S → R be a real valued smooth function on a surface S.We say that a critical point p ∈ S is non-degenerate, if the matrix Hφ(f)(x0, y0)is non singular for any parametrization φ, with φ(x0, y0) = p.

Critical points with singular hessian are called degenerate.

9

Later we will show that there for any surface S exists a function f : S → Rwhere all critical points are non-degenerate, with terminology to be de�ned: onany surface there exists a Morse function. Hence to �nd important imformationof a surface one only needs to consider non-degenerate critical points (this alsodepends on a lemma roughly saying that we only need to consider regions aroundcritical points). To do so, we use the following de�nitions:

De�nition 15. Let f : S → R be a smooth real valued function on a surfaceS. If all critical points of f are non-degenerate, then we say that f is a Morsefunction on S.

De�nition 16. Let f : S → R be a Morse function on a surface S. Let p ∈ S bea critical point. The index of p with respect to f is de�ned to be the dimensionof the maximal subspace of TSp on which H(f) is negative de�nite.

Again using the Jacobian, one can show that this de�nition is independentof coordinates. Note that the index is 0, 1 or 2, depending on whether p is alocal minimum, a sadlepoint or a local maximum respectively.

The following theorem, due to Morse, provides the key tool when using theindex of a critical point to analyze a surface.

Theorem 2 (Lemma of Morse). Let f : S → R be a real valued smooth functionon a surface S. Let p ∈ S be a non-degenerate critical point. Then there existsa parametrization φ : U → V , with φ(x0, y0) = p ∈ V , such that:

Index(p) = 0 =⇒ f ◦ φ(x, y) = f(p) + x2 + y2,

Index(p) = 1 =⇒ f ◦ φ(x, y) = f(p) + x2 − y2,Index(p) = 2 =⇒ f ◦ φ(x, y) = f(p)− x2 − y2.

Proof. Let φ be a parametrization around p. We will construct new coordinatesφ−1 = (v1, v2) such that f ◦ φ−1 = f(p) ± v21 ± v22 . Then (by maybe switchingplace with v1 and v2) it is immidiate from

Hφ(f) =

(±2 00 ±2

)that if the index is 0 then f = f(p) + v21 + v22 , if the index is 1 then f =f(p) + v21 − v22 , and if the index is 2 then f = f(p)− v21 − v22 .

Firstly, we can assume (x0, y0) = 0, i.e. φ−1(p) = 0. Moreover we canassume that f(p) = 0. If we take U small enough we can assume U convex.Then we note that g : U → R de�ned by g = f ◦ φ can be written

g(x, y) = xh1(x, y) + yh2(x, y),

for some smooth functions h1, h2. Because, by using the fundamental theoremof analysis and the chain rule, we see that

g(x, y) =

∫ 1

0

g′t(tx, ty)dt =

∫ 1

0

(∂g

∂x(tx, ty)x+

∂g

∂y(tx, ty)y

)dt.

Hence we can let

h1(x, y) =

∫ 1

0

∂g

∂x(tx, ty)dt, h2(x, y) =

∫ 1

0

∂g

∂y(tx, ty)dt.

10

Since p = 0 is a critical point, h1(0) = ∂g∂x (0) = 0, h2(0) = ∂g

∂y (0) = 0. Therefore,by using the same reasoning as above on f , now on h1, h2, we can write

h1(x, y) = xk11(x, y) + yk12(x, y) h2(x, y) = xk21(x, y) + yk22(x, y),

and hence also:

g(x, y) = x2k11 + xy(k12 + k21) + y2k12.

We can assume that k12 = k21 (if k12 6= k21, replace k12, k21 with k′12 = k′21 =12 (k12 + k21)). Also note that(

k11(0) k12(0)k21(0) k22(0)

)=

1

2

(∂2g∂x2 (0) ∂2g

∂xy (0)∂2g∂xy (0) ∂2g

∂y2 (0)

),

hence the matrix (kij)(0) is non-singular. Now we only need to show that wecan pick coordinates so that k12 = k21 = 0, and k11 = ±1, k22 = ±1, in aneighbourhood around 0.

We will show this by doing two coordinate changes. The �rst coordinatechange is

u1(x, y) =√|k11(x, y)|

(x+

yk12(x, y)

k11(x, y)

)u2(x, y) = y.

(It is possible, by a linear change, to assume k11(x, y) 6= 0, maybe adding anextra coordinate change). Since, for the Jacobian J of the mapping (x, y) 7→(u1, u2) at 0, we have that

J(0, 0) = det

(∂u1

∂x∂u1

∂y∂u2

∂x∂u2

∂y

)(0) = det

(A B0 1

)(0) = A(0) =

√|k11(0, 0)|,

(A,B)4 and therefore J(0, 0) 6= 0, we can use the inverse function theorem toconclude that (x, y) 7→ (u1, u2) in a small neighbourhood of 0 actually is acoordinate change. Now we see that f expressed in coordinates u1, u2 becomes

(x2k11 + 2xyk12 + y2k22)(u1, u2) = ±u21 + l(u1, u2)u22,

where l(u1, u2) is a smooth function. The second coordinate change is just

v1(u1, u2) = u1

v2(u1, u2) =√|l(u1, u2)|u2.

The mapping (u1, u2) 7→ (v1, v2) is even simpler to show is a coordinate change.Finally we get that f expressed in v1, v2 is

f = ±v21 ± v22 .

This completes the proof.

4

A =

(k11k11x

2|k11|3/2

)(x+

yk12

k11

)+√|k11|

(1 + y

k12xk11 − k12k11xk211

)B =

(k11k11y

2|k11|3/2

)(x+

yk12

k11

)+√|k11|

(k12

k11+ y

k12yk11 − k12k11y

k211

)

11

The following result is now an immediate consequence of the above theorem.As an application on surfaces, by compactness there can only be �nitely manynon-degenerate critical points.

Corollary 1. Non-degenerate critical points of a surface are isolated.

Figure 2: A torus standing on the xy-plane, the height along the z-axis shownby a Morse function f . There are four critical points, p0, p1, p2, p3, havingindices 0, 1, 1, 2 respectively. The lines mark the critical values f(pj).

Next we will de�ne, and prove the existence of, a gradient-like vector �eldfor a Morse function f on any surface S. Properties of these vector �elds willbe used later on.

De�nition 17. Let f : S → R be a Morse function on a surface S. A gradientlike-vector �eld for f , is a smooth vector �eld ξ such that

1. ξ(f)(q) > 0 for all regular points q, and

2. given a critical point p of f , there are coordinates φ−1 = (x, y) in a neigh-bourhood U of p, such that f ◦ φ = f(p) ± x2 ± y2 and ξ has coordinates(±x,±y), where the sign depends on index as above (either both positiveif index 0, x positive and y negative if index 1, or both negative if index2), in all of U .

Lemma 2. For every Morse function f on any surface S, there exists a gradient-like vector �eld ξ for f .

For the proof we will need a fact about partitions of unity. Partitions of unityprovides an excellent tool for extending local properties of neighbourhoods tothe whole surface.

12

De�nition 18. Let S be a surface. A family of open subsets {Ui}i∈I of S issaid to be locally �nite if every point p ∈ S has a neighbourhood W such thatW ∩Ui 6= ∅ for only a �nite number of indices in I. The support of a functionf : S → R is the closure of the set of points where f is non zero; we denote thesupport by supp(f).

Let {Vi}i∈I be an open cover of S. We say that a family of smooth functions{fi : S → R}i∈I is a partition of unity subordinate to the cover {Vi} if:

1. For all i ∈ I, fi ≥ 0 and supp(fi) ⊂ Vi.

2. The family {supp(fi)}i∈I is locally �nite.

3.∑i∈I fi(p) = 1, for all p ∈ S.

Theorem 3. For any surface S and any locally �nite cover of neighbourhoods{Vi} of S, there exists a partition of unity subordinate to the cover.

We will prove this theorem using a type of functions called bump functions.Bump functions will also be used in later proofs when constructing certain vector�elds.

De�nition 19. A bump function is a smooth nowhere negative real valuedfunction b : M → R≥0 from a manifold M (for us M will be a surface or somesubset of R2), having compact support.

Here is a proof of existence of a kind of bump functions on surfaces.

Lemma 3. Given any surface S, a parametrization φ : U → V ⊂ S, andany point p ∈ V . There exists a bump function b : S → R on S such thatf(S) ⊂ [0, 1], supp(S) ⊂ V and b(q) = 1 in a neighbourhood of p.

Proof. The function

h(x) =

{e−

1x , if x > 0,

0, if x ≤ 0

is smooth at all points in R. Let C(s) = {(x, y) ∈ R2||x|, |y| < s} denote thethe open square in R2 with side 2s; let C(s) denote its closure. We will beginby constructing a smooth function b0 : R2 → R such that b0(x, y) ∈ [0, 1],b0(x, y) = 1 for (x, y) ∈ C(1), and b0(x, y) = 0 for (x, y) 6∈ C(2).

Let

h0(x) =h(x)

h(x) + h(1− x),

so that h0(x) = 1 for x ≥ 1, and h0(x) = 0 for x ≤ 0. Now let

h1(x) = h0(x+ 2)h0(2− x).

Then h1(x) = 1 for |x| ≤ 1, and h1(x) = 0 for |x| ≥ 2. Thus we can let b0 bede�ned by

b0(x, y) = h1(x)h1(y).

Let U = φ−1(V ) and (x0, y0) = φ−1(p). Take a square neighbourhood C ′ ={(x, y) ∈ U ||x0 − x| < ε, |y0 − y| < ε} of φ−1(p) in U . Let

k(x, y) =2

ε(x− x0, y − y0);

13

this function maps C ′ di�eomorphically on C(2). Then the desired bump func-tion is given by

b(q) =

{b0 ◦ k ◦ ϕ−1(q), if q ∈ V

0, otherwise.

Proof of theorem 3. For each p ∈ S, let hp : S → R be a bump function so thathp = 1 in a neighbourhood Up ⊂ supp(hp) ⊂ Vi, for some i. Clearly {Up}p∈Sis an open cover of S, by compactness we can pick a �nite number of the Up's,U1, . . . , Un, that also cover S. We have that Ul ⊂ Vil , for some Vil ∈ {Vi},l = 1, . . . , n. For all i such that Vi 6∈ {Vi1 , . . . , Vin}, let fi = 0 constantly. ForVi ∈ {Vi1 , . . . , Vin} let

fi =hl1 + · · ·+ hlk∑n

j=1 hj,

where the hl1 , . . . , hlk comes from the Ulj 's such that Ulj ⊂ Vi = Vil . It is noweasy to check that {fi} is a partition of unity subordinate to the cover {Vi}.

When we further use bump functions we will sometimes assume they havesimilar extra properties. We will omit proving that these bump functions actu-ally exists, even though they of course do exists, and the proofs are not hard.Also the proof of lemma 3 contains the crucial ideas to how one does.

Proof of lemma 2. Let p1, . . . , pn be the critical points of f . By theorem 2, foreach pi there is a parametrization φi : U i → V i such that on U i, f ◦ φi(x, y) =

f(pi) ± x2 ± y2. Let V i0 be neighbourhoods of pi such that V i0 ⊂ V i. LetU0 =

⋃ni=0 U

i0 and V0 =

⋃ni=0 V

i0 .

Each q ∈ S−(⋃n

i=1 Vi)is a regular point of f , hence, by the implicit function

theorem, we can write f ◦ φq(x, y) = constant + x in Uq, where φq : Uq → Vq isa parametrization.

Note that S −(⋃n

i=1 Vi)is compact. Using this and the earlier �nding, we

can �nd neighbourhoods V1, . . . , Vk such that

1. S −(⋃n

i=1 Vi)⊂⋃ki=1 Vi,

2. V0 ∩ Vj = ∅, for 1 ≤ j ≤ k, and

3. there are parametrizations φj : Uj → Vj so that f ◦ φi = constant+ xi onUj , for j = 1, . . . k.

Now, for all 1 ≤ i ≤ n, on V i there are vector �elds vi with coordinates (±x,±y),and for each 1 ≤ j ≤ k, on Vj there is the vector �eld vj , which in coordinates is∂/∂x. We can now piece together these vector �elds using a partition of unitysubordinate to the cover V 1, . . . , V n, V1, . . . , Vk: let {f1, . . . , fn, f1, . . . , fk} besuch a partition of unity, now we can de�ne the smooth vector �eld by

ξ(q) =

n∑i=1

f i(q)vi(q) +

k∑j=1

fj(q)vj(q).

It is well de�ned because the sum is �nite, and smooth because all functionsf i, fj , and all locally de�ned vector �elds are. We now only have to check thatξ(f)(q) > 0 for any regular point q.

14

Let q ∈ S be a regular point. Then at least one of f1, . . . , fn, f1, . . . , fk isnon-zero at q, and all of them are non-negative. Moreover if q is in any of thesets Vj , then vj(f) = ∂/∂x(x + constant) = 1 > 0; if q is in any of the sets V i

then vi(f) = (±x,±y) · (±2x,±2y) = 2x2 + 2y2 > 0.Thus we obtain a gradient-like vector �eld ξ for f on S.

Now let S be a surface and f be a Morse function on S. For a ∈ R, letSa = {p ∈ S|f(p) ≤ a}, which is a compact submanifold of S, possibly withboundary. We now prove an important lemma about how Sa changes when aincreases. It shows that we only need to analyze a surface around the criticalpoints of a Morse function when classifying the surface.

Lemma 4. Let a < b be two real numbers, and suppose f−1[a, b] = {p ∈ S|a ≤f(p) ≤ b} ⊂ S contains no critical points of f . Then Sa is di�eomorphic to Sb.

For the proof we need a tool provided by 1-parameter groups of di�eomor-phisms. And a lemma about how to generate them, which we state only forsurfaces.

De�nition 20. A 1-parameter group of di�eomorphisms of a manifold M is asmooth map ϕ : R×M →M such that:

1. For each �xed t ∈ R, the map de�ned by M 3 p 7→ ϕ(t, p), denotedϕt : M

∼−→M , is a di�eomorphism.

2. For all s, t ∈ R, ϕs+t = ϕs ◦ ϕt.

Given a a 1-parameter group of di�eomorphisms ϕ on a smooth manifoldM , we de�ne a smooth vector �eld X on M by

X(f)(q) = limh→0

f(ϕh(q))− f(q)

h,

where f is any smooth real valued function on M . The vector �eld X is said togenerate ϕ.

Lemma 5. A smooth vector �eld X on a surface S, which vanishes outside ofa compact subset K of S, generates a unique 1-parameter group of di�eomor-phisms ϕ on S.

Proof. The �rst thing we prove is that ϕ - satisfying the equality above - as-suming it is a 1-parameter groupof di�eomorphisms, must be unique, then wego on and show that one exists.

Let ϕ be 1-parameter group of di�eomorphisms generated by X. For thecurve t 7→ ϕt(q) we have that

dϕt(q)

dt(f) = lim

h→0

f(ϕt+h(q))− f(ϕt(q))

h

= limh→0

f(ϕh(ϕt(q)))− f(ϕt(q))

h

= X(f)(ϕt(q)).

15

Hence, t 7→ ϕt(q) satisfy the di�erential equation with initial condition

dϕt(q)

dt= X(ϕt(q)),

ϕ0(q) = q,

because ϕ0 = id. This di�erential equation has a locally unique smooth solutionby theorem 1, q ∈ Vq, t ∈ (−εq, εq), for each q ∈ S which extends uniquelyglobally because R is connected. This proves the �rst part.

Cover the compact setK by a �nite number of open neighbourhoods V1, . . . , Vnfrom the family {Vq}q∈K . Let ε0 be the smallest of the numbers εi correspond-ing to Vi, 1 ≤ i ≤ n. If we let ϕt(q) = q for q 6∈ K, then the di�erential equationabove has a unique solution for t ∈ (−ε0, ε0) and all q ∈ S. This function issmooth, and it is obvious that if |s|, |t|, |s+ t| < ε0

2 , then ϕs+t = ϕs ◦ϕt. Henceeach ϕt, t ∈ (−ε0, ε0), is a di�eomorphism ϕt : S

∼−→ S.For |t| ≥ ε0, we write t = n ε02 + rt, n ∈ Z, rt ∈ (− ε02 ,

ε02 ). Then, if n ≥ 0 set

ϕt = ϕ ε02◦ . . . ϕ ε0

2◦ ϕrt ,

where ϕ ε02is iterated n times, and if n < 0 do as above but with ε0

2 replaced by− ε02 . Then ϕt is smooth, well de�ned and a di�eomorphism for all t ∈ R, andclearly ϕ satisfy ϕs+t = ϕs ◦ ϕt for all s, t ∈ R.

Proof of lemma 4. Pick a gradient-like vector �eld ξ for f . Then, since by as-sumption f−1[a, b] contains no critical points of f , we have that ξ(f)(p) > 0 forall p ∈ f−1[a, b]. Multiplying ξ at each point in f−1[a, b] by 1

ξ(f) we obtain a

new gradient-like vector �eld ξ for f in f−1[a, b] - then ξ(f) is constantly equalto 1 in f−1[a, b] - which we can assume, using a bump function, vanishes outsidea compact neighbourhood of this set. Hence lemma 5 applies, and so we get a1-parameter group of di�eomorphisms ϕ on S. For any ϕt(p), with p ∈ f−1[a, b]we get that

df(ϕt(p))

dt= ξ(f)(p) = 1,

hence f(ϕt(p)) = f(p) + t, where f(ϕ0(p)) = f(p) comes from ϕ0 = id. We nowconclude that the di�eomorphism ϕb−a : S

∼−→ S sends Sa onto Sb, and henceinduces, by restriction, a di�eomorphism ϕ|Sa : Sa

∼−→ Sb. For this we only

need to show that ϕb−a(Sa) = Sb. To see this, if p ∈ Sa, then f(ϕb−a(p)) =(b − a) + f(p) ∈ Sb, because f(p) ≤ a, hence ϕb−a(p) ∈ Sb; if on the otherhand p ∈ Sb, then f(ϕ−1b−a(p)) = (a− b) + f(p) ∈ Sa, beacuase f(p) ≤ b, hence

ϕ−1b−a(p) ∈ Sa.

We conclude this section with the theorem mentioned above ensuring theexistence of a function with no degenerate critical points. In the followingsections all critical points of a real valued smooth function will therefore beassumed to be non-degenerate.

Theorem 4. For any surface surface S there exists a Morse function f : S → Ron S.

What we will do is to embedd the surface in Rm, which always is possible forsome large enough m 5, and then show that the function Lp : S → R, de�ned by

5See [3].

16

Lp(q) = ‖p−q‖2, where ‖, ‖ is the euclidean norm in Rm, contains no degeneratecritical points for almost all p ∈ Rm. (For almost all will here mean the sameas all but a set of measure zero).

Let N = {(p, v) ∈ S × Rm|v · p = 0}; this is the set of all pairs (p, v)with p ∈ S and v ∈ Rm perpendicular to p. It is easy to show that N is anm-dimensional manifold, sometimes called a normal bundle of S.

De�nition 21. Let E : N → Rm be de�ned by E(p, v) = p+ v, and let q ∈ S.We de�ne a focal point of (S, q), to be a point e ∈ Rm such that e = q + v, forsome v with (q, v) ∈ N , where the kernel of the jacobian JE(q, v) of E at (q, v)is non-trivial.

A focal point of S is a focal point of (S, q), for some q ∈ S.

The goal is to show that a point q ∈ S is a degenerate critical point of Lp, ifand only if p is a focal point of (S, q). Then theorem 4 follows from a corollaryof the following theorem due to Sard6.

Theorem 5 (Sard's theorem). Let f : U → Rm be a smooth map from an openset U ⊂ Rn, n and m are any natural numbers. Let C = {x ∈ U |rank(dfx) <m}. Then the set f(C) ⊂ Rm has measure zero in Rm.

Corollary 2. For almost all x ∈ Rm, x is not a focal point of S.

Now we de�ne the �rst and second fundamental form, they will help us locatethe focal points. Let (x, y) = φ−1 be coordinates of an open set V ⊂ S. Theinclusion map S → Rm determines m smooth functions u1(x, y), . . . , um(x, y),where we use the notation: u : R2 ⊃ U → Rm, u(x, y) = (u1(x, y), . . . , um(x, y)).And so to distinguish between S and S ⊂ Rm, we write q when we talk aboutpoints in the surface and q when we talk about points in the surface embeddedin Rm.

De�nition 22. The �rst fundamental form associated with the coordinate sys-tem (x, y) (and with the embedding into Rm) is the symmetric matrix of realvalued functions

(gi) =

(g1 g2g2 g3

)=

(∂u∂x ·

∂u∂x

∂u∂x ·

∂u∂y

∂u∂x ·

∂u∂y

∂u∂y ·

∂u∂y

).

The second fundamental form is the symmetric matrix of vector valued function

(li) =

(l1 l2l2 l3

)de�ned as follows. Let the vectors a1 = ∂2u

∂x2 , a2 = ∂2u∂x∂y and a3 = ∂2u

∂x2 , eachbe expressed as a sum of a vector tangent to S, and a vector normal to S. Wede�ne li as the normal component of ai, i = 1, 2, 37.

6See [6].7We can de�ne an embedding of S in Rm as a smooth map u : S → Rm, such that (i) the

di�erential dup is injective for all p ∈ S, and (ii) u is a homeomorphism from S to u(S). Thenthe tangent space of S in Rm, at any point p, can be de�ned as dup(TSp). Now it easy tode�ne tangential and normal components of vectors.

17

Given a vector n that is orthogonal to S at the point p with ‖n‖ = 1, thematrix (

n · a1 n · a2n · a2 n · a3

)=

(n · l1 n · l2n · l2 n · l3

)is called the second fundamental form in the direction n at the point p.

Now we assume that the coordinates are chosen so that (gi) at p is theidentity matrix. Let n be a unit vector orthogonal to S at p. Then we de�nethe principal curvatures K1 and K2 at the point p in the normal direction n asthe eigenvalues of the matrix (n · li); 1

K1and 1

K2are called the principal radii

of curvature. If the matrix (n · li) is singular, then at least one of K1 and K2

are zero, and hence the corresponding 1Ki

is not de�ned.8

Now let lp,n = p+ tn be the line in direction n, where n is a normal vectorwith ‖n‖ = 1, through the point p ∈ S.

Lemma 6. The focal points of (S, q) along ln,q are precisely the points q+ 1Kin.

Proof. Choosem−2 smooth vector �elds v1(x, y), . . . , vm−2(x, y), with ‖vi‖ = 1,along S in Rm, such that at each point, vi is orthogonal to S and each pair ofvectors vi, vj , i 6= j, are orthogonal to each other. We now introduce coordinatesφ−1N = (x, y, t1, . . . , tm−2) on N by

φN (x, y, t1, . . . , tm−2) = (u(x, y), t1v1(x, y) + · · ·+ tm−2vm−2(x, y)).

Then the function E : N → Rm induces a function

(x, y, t1, . . . , tm−2) 7→E∗ u(x, y) +

m−2∑i=1

tivi(x, y).

The partial derivatives of this function are

∂E∗

∂x=∂u

∂x+

m−2∑i=1

ti∂vi∂x

,

∂E∗

∂y=∂u

∂y+

m−2∑i=1

ti∂vi∂y

,

∂E∗

∂ti= vi.

Now consider the following matrix Q, obtained by taking the inner product ofthe partial derivatives of E∗ with the vectors ∂u

∂x ,∂u∂y , v1, . . . , vm−2:

Q =

∂E∗

∂x ·∂u∂x

∂E∗

∂x ·∂u∂y

∂E∗

∂x · v1 . . . ∂E∗

∂x · vm−2∂E∗

∂y ·∂u∂x

∂E∗

∂y ·∂u∂y

∂E∗

∂y · v1 . . . ∂E∗

∂y · vm−2∂E∗

∂t1· ∂u∂x

∂E∗

∂t1· ∂u∂y

∂E∗

∂t1· v1 . . . ∂E∗

∂t1· vm−2

......

.... . .

...∂E∗

∂tm−2· ∂u∂x

∂E∗

∂tm−2· ∂u∂y

∂E∗

∂tm−2· v1 . . . ∂E∗

∂tm−2· vm−2

.

8For a more detailed introduction to these concepts see for example [1].

18

To simplify, we can write this matrix as being constituted by four matricesA,B,C,D in the following way:

Q =

(A BC D

),

where we after also simplifying gets,

A =

(∂u∂x +∑m−2i=1 ti

vi∂x

)· ∂u∂x

(∂u∂x +

∑m−2i=1 ti

∂vi∂x

)· ∂u∂y(

∂u∂y +

∑m−2i=1 ti

∂vi∂y

)· ∂u∂x

(∂u∂y +

∑m−2i=1 ti

∂vi∂y

)· ∂u∂y

,

B =

(∑m−2i=1 ti

∂vi∂x · v1 . . .

∑m−2i=1 ti

∂vi∂x · vm−2∑m−2

i=1 ti∂vi∂y · v1 . . .

∑m−2i=1 ti

∂vi∂y · vm−2

),

C =

0 0...

...0 0

,

D = Im−2.

It is clear that the rank of this matrix is equal to the rank of the jacobian ofE (because ∂u

∂x ,∂u∂y , v1 . . . , vm−2 are linearly independent), at the corresponding

points (corresponding, of course, by the coordinates or parametrization). Alsowe see that the nullity of Q is determined only by the upper left corner matrixA; focusing on this matrix then, we see that by the identities

0 =∂

∂x

(vi ·

∂u

∂x

)=∂vi∂x

∂u

∂x+ vi

∂2u

∂x2,

0 =∂

∂x

(vi ·

∂u

∂y

)=∂vi∂x

∂u

∂y+ vi

∂2u

∂x∂y,

0 =∂

∂y

(vi ·

∂u

∂x

)=∂vi∂y

∂u

∂x+ vi

∂2u

∂x∂y,

0 =∂

∂y

(vi ·

∂u

∂y

)=∂vi∂y

∂u

∂y+ vi

∂2u

∂y2,

i = 1, 2, . . . ,m− 2, we have that

A =

(g1 −

∑m−2i=1 tivi · l1 g2 −

∑m−2i=1 tivi · l2

g2 −∑m−2i=i tivi · l2 g3 −

∑m−2i=1 tivi · l3

).

Thus we have shown that q + tn is a focal point of (S, q), if and only if A issingular. Now if (gi) is the identity matrix, i.e. g1 = 1, g2 = 0, g3 = 1, thenA is singular if and only if 1

t is an eigenvalue of the matrix (n · li); hence A issingular if and only if t = 1

K1or t = 1

K2, and this completes the proof.

Proof of theorem 4. Let p ∈ Rm be a �xed point. For the function Lp : S → Rm,which in coordinates is de�ned by

Lp(u(x, y)) = ‖u(x, y)− p‖2 = u(x, y) · u(x, y)− 2u(x, y) · p+ p · p,

19

we have that

∂Lp∂x

= 2∂u

∂x(u− p),

∂Lp∂y

= 2∂u

∂y(u− p).

Thus q = u(x, y) is a critical point of Lp if and only if q − p is orthogonal to Sat q.

Moreover we have

∂2Lp∂x2

= 2

(∂2u

∂x2· (u− p) +

∂u

∂x· ∂u∂x

),

∂2Lp∂x∂y

= 2

(∂2u

∂x∂y· (u− p) +

∂u

∂x· ∂u∂y

),

∂2Lp∂y2

= 2

(∂2u

∂y2· (u− p) +

∂u

∂y· ∂u∂y

).

Setting p = u+ tn as in lemma 6, we obtain(∂2Lp∂x2

∂2Lp∂x∂y

∂2Lp∂x∂y

∂2Lp∂y2

)= 2

(g1 − tn · l1 g2 − tn · l2g2 − tn · l2 g3 − tn · l3

).

Therefore we have that q ∈ S is a degenerate critical point of Lp if and only ifp is a focal point of (S, q). Now using corollary 2 we obtain the �nal result: Foralmost all p ∈ Rm the function Lp : S → R has no degenerate critical points.

20

Existence of a Good Morse Function

We will here use the properties of Morse functions on surfaces - and the fact thatwe always can �nd a Morse function - to show that we always can �nd a goodMorse function on a surfaces. We will proceed by �rst show that any Morsefunction can be turned into a self-indexing Morse function, and thereafter proveand use a cancelation theorem to cancel critical points against each other. Fromthis we will in next section show that the surface up to di�eomorphism can bedetermined by the number of critical points having index 1 of the good Morsefunction, and moreover that this number is the same as two times the genus ofthe surface.

De�nition 23. Let f : S → R be a Morse function. We say that f is self-indexing if, for any critical point p ∈ S, we have

f(p) = index(p).

De�nition 24. Let f : S → R be a Morse function. We say that f is a goodMorse function if f has only one maximum and one minimum (one critical pointwith index 2, and one with index 0), and is self-indexing.

The goal of this section is the following theorem.

Theorem 6. Every surface S has a good Morse function f : S → R.

Proving this is the main step in the classi�cation of surfaces. We begin byintroducing some new theory.

Begin by observing that, by theorem 2, close to a critical point p, on everylevel set f = f(p) + ε there is an induced sphere. In local coordinates this is the1-sphere

x2 + y2 = ε

which goes upwards if the index of p is 0 and downwards if the index is 2. Ifthe index instead is 1 we get one 0-sphere which goes upwards,

x2 = ε

y = 0,

and one 0-sphere which goes downwards

x = 0

y2 = ε.

By lemma 4 this extends to an induced sphere on all level sets f = a, as long aswe do not pass a critical value on the way from p. Note that if we go upwardsfrom a critical point having index 0, we get an induced sphere having dimension1; if we go upwards or downwards from a critical point having index 1, we getan induced sphere having dimension 0 (that is, two disjoint points); if we godownwards from a critical point having index 2, we get an induced sphere havingdimension 1.

Now we �rst prove that we can turn any Morse funtion into a self-indexingone.

21

Lemma 7. Let f : S → R be a Morse function. Let p ∈ S be a critical pointof f with index 0 (index 2), such that f(p) = a. Then for all b ≤ a (all b ≥ a),there exists a Morse function g : S → R with the same critical points as f havingthe same indices, with g(p) = b, and such that f and g coincides except on aneighbourhood V of p.

Proof. We prove the case with indexf (p) = 0, the case indexf (p) = 2 is similar.If b = a we are done; so let b < a. Let V be a neibourhood centered at p, suchthat trough out U = φ−1(V ), we have f ◦ φ−1 = f(p) + x2 + y2; moreover, bypossibly making V smaller, we can assume that U = {(x, y) ∈ R2|x2 + y2 < c},where c is some small real number. Let γ : [a, a + c) → R be a smooth mapwith the properties:

1. dγdt (t) > 0, t ∈ [a, a+ c),

2. γ(a) = b,

3. γ(t) = t, t ∈ (a+ c− ε, a+ c), for some small ε (i.e. for t near a+ c).

Now we de�ne g : S → R by:

g(q) =

{γ(f(q)), if q ∈ V,f(q), if q 6∈ V.

Clearly g is smooth. Also, we have that g(p) = γ(f(p)) = b, and by the chainrule, since dγ

dt (t) > 0, in V we have

dg

dq=

dγ

d(t = f(q))f ′(q) = 0 =⇒ f ′(q) = 0 =⇒ q = p,

that is, p is the only critical point of g in V . Moreover,

g(p) = γ(f(p) + x2 + y2

), (x, y) ∈ U,

and so the hessian of g at p with respect to φ is given by(2γ′(a) 0

0 2γ′(a)

),

hence indexg(p) = 0. Finally near a+ c, γ(t) = t, hence for (x, y) close to c (i.e.close to the boundary of U),

g(x, y) = γ(f(x, y)) = f(x, y).

What is left to show is that such a γ exists. We proceed similarly to how weconstructed bump functions. Let

h0(t) =

{e−

1t , if t > 0,

0, if t ≤ 0,

which is a smooth function on all of R, with h0(t) > 0 for t > 0, and with

h0(t) = 0 for t ≤ 0. Also, for t > 0, dh0

dt (t) = 1t2 e− 1t > 0. Now let

h1(t) =h0(t)

h0(t) + h0(1− t).

22

Observe that

dh1dt

(t) =h′0(t)(h0(t) + h0(1− t))− h0(t)(h′0(t)− h′0(1− t))

(h0(t) + h0(1− t))2

=h′0(t)h0(1− t) + h0(t)h′0(1− t)

(h0(t) + h0(1− t))2,

so that dh1(t)dt (t) > 0 for t > 0. Also we have that h1(t) = 1 for t ≥ 1 and

h1(t) = 0 for t ≤ 0.Now we change the interval: let

h2(t) = h1

(t− ac− ε

).

Now h2 has the same properties as h1 above, only 0 is replaced by a and 1 bya+ c− ε. Now let

γ(t) = th2(t) + (t− (a− b))h2(2a+ c− ε− t),

where t ∈ [a, a + c). Clearly condition 2 and 3 holds for this function. For 3,

note that dh1(t)dt is symmetric in the sense that h′1(t) = h′1(1 − t). This implies

that h′2(t) = h′2(2a+ c− ε− t), hence

dγ

dt(t) = h2(t) + th′2(t) + h2(2a+ c− ε− t)− (t− (a− b))h′2(2a+ c− ε− t)

= h2(t) + h2(2a+ c− ε− t) + (a− b)h′2(t) > 0,

where the inequality comes from h2(t) ≥ 0, a−b > 0, h′2(t) ≥ 0, and in particularh2(2a+ c− ε− t) > 0 near and at t = a.

If b 6≤ a, then dγdt > 0 in all of [a, a+ c) might fail.

Now we will use lemma 7 to "push" all the critical points having index2 up, and all critical points having index 0 down. We will do this so thatfor some big A we have index(p) = 2 =⇒ f(p) = A, and for some smallB < A we have index(p) = 0 =⇒ f(p) = B, and moreover so that A >max ({f(p)|index(p) = 1}) and B < min ({f(p)|index(p) = 1}). Then f−1[A +ε, B−ε], for small ε > 0, contains only critical points having index 1. By addingsome big constant to f we can assume B = 0, and by dividing f by some bigconstant we can assume A = 2.

Now we need 2 lemmas, which will enable us to put all critical points havingindex 1 on any level set f = a ∈ (0, 2). Let 0 < a < b < 2. In the next lemmaswe will assume the reasoning above so that f−1[a, b] contains only critical pointsahaving index 1. Also, let ξ be any gradient-like vector �eld for f .

Lemma 8. Suppose that f−1[a, b] contains only two levels of critical points.That is p = {p1, . . . , pn}, p′ = {p′1, . . . , p′m}, with f(p) = c and f(p′) = c′, forsome c, c′ ∈ (a, b), and p ∪ p′ is the set of all critical points in f−1[a, b]. LetKp be the set of points on trajectories goint from or to any pi ∈ p, and Kp′

be de�ned the same way but with p′ instead of p (i.e. points on the inducedspheres). Suppose Kp and Kp′ disjoint. Then for any d, d′ ∈ [a, b] there existsa Morse function g such that:

23

1. ξ is a gradient-like vector �eld for g.

2. The critical points of g (in f−1[a, b]) are p∪p′, but with the levels changedto d and d': g(p) = d and g(p′) = d′.

3. g and f agrees near f−1(a) and f−1(b) (and outside of f−1[a, b]), andg = f + constant in neighbourhoods of the points in p and p′.

Note that Kp and Kp′ consists of levels of disjoint spheres having dimension0.

Lemma 9. Let W = f−1(c+c′

2

). It is possible to alter ξ in a neighbourhood V

of W so that Kp and Kp′ are disjoint, i.e. so that all spheres corresponding top1, . . . , pn, p

′1, . . . , p

′m are disjoint.

With these lemmas we simply take d = d′ and start just under 1 and startby pushing all levels of critical points greater then 1 down, and all levels smallerthen 1 up; the "pushing" is done by repeated use of lemma 8, and lemma 9guarantees that it is always possible. So proving these two lemmas is enough toprove the �rst step towards theorem 6.

Corollary 3 (Self-Indexing Theorem). Every surface admits a self-indexingMorse function.

Proof of lemma 8. Note that all trajectories through points outside (Kp ∪Kp′)go from f−1(a) to f−1(b). Let τ : f−1(a) → R be a smooth function whichis 0 near Kp ∩ f−1(a) and 1 near Kp′ ∩ f−1(a), constructed similarly to theconstruction of γ in lemma 7 or to the construction of bump functions; this isclearly possible since Kp and Kp′ are disjoint, hence also compact. The functionτ extends uniquely to a function τ : f−1[a, b] → R which is constant on eachtrajectory, and which is 0 near Kp and 1 near Kp′ .

De�ne the Morse function g : S → R by

g(q) =

{f(q), if q 6∈ f−1[a, b],

G(f(q), τ(q)), if q ∈ f−1[a, b],

where G : [a, b]× [0, 1]→ [a, b] is any smooth function such that:

1. For all x, y, we have ∂G∂x (x, y) > 0, and G(a, y) = a, G(b, y) = b.

2. G(f(p), 0) = G(c, 0) = d and G(f(p′), 1) = G(c′, 1) = d′.

3. G(x, y) = x for x near 0 and 1 and all y, ∂G∂x (x, 0) = 1, for x in a neigh-

bourhood of f(p) = c, and ∂G∂x (x, 1) = 1, for x in a neighbourhood of

f(p′) = c′.

We verify that g satisfy the three conditions. The critical points of g are givenby g′(p) = 0, where

g′(p) =∂G

∂x(f(p), τ(p))f ′(p) +

∂G

∂y(f(p), τ(p))τ ′(p).

We omit proving that ξ is a gradient-like vector �eld also for g; what actuallyis important about this criterion is that the index of the critical points don'tchange (and that they are still non-degenerate), but this is straight forward.

24

Since τ ′(p) = 0, near the critical points of f , all critical points of f arealso critical points of g. Also, if q is a regular point of f , then it is a regularpoint of g. To this �rst observe that f ′(q) 6= 0 and ∂G

∂x 6= 0, so if τ ′(q) = 0we are done. Hence we can assume both f ′(q) 6= 0 and τ ′(q) 6= 0. Note thatf ′(q) and τ ′(q) are maps from TSq to R, and that by de�nition of τ , we havethat τ ′(ξ(q)) = 0, while f ′(ξ(q)) 6= 0; therefore we can see f ′(q) and τ ′(q) astwo linearly independent vector. So for g′(q) to be equal to zero we must have∂G∂x = ∂G

∂y = 0, which is a contradiction.

Near f−1(a) and f−1(b) we have that G(x, y) = x, hence f and g agree nearthese levels. Clearly g = f + constant near the points in p and p′ (because∂G∂x (x, 0) = 1 for x near c, ∂G

∂ (x, 1) = 1 for x near c′ and t′ = 0 near p andp′).

Proof of lemma 9. Let Sp = Kp ∩W and Sp′ = Kp′ ∩W (the notation is moti-vated by the fact that the sets are unions of spheres of critical points having thesame level). The proof will be in two steps: First we show that there exists adi�eomorphism h : W

∼−→W , smoothly isotopic to the identity, such that h(Sp)and Sp′ are disjoint. Then we use the isotopy to alter ξ on V .

Note that we can �nd small neighbourhoods Np1 , . . . , Npn inW of the pointsin this level on the trajectories going from p1, . . . , pn, respecively, each di�eo-morphic to R. Let Np = Np1 ∪ · · · ∪Npn ; this is a set di�eomorphic to Sp × R,where we moreover can assume, letting k : Sp × R ∼−→ Np be such a di�eomor-phism, that k(Sp × {0}) = Sp. Also let N0 = Np ∩ Sp′ , and g = π ◦ k−1|N0

, where

π : Sp×R→ R is the natural projection, so that q ∈ N0 if and only if x ∈ g(N0).Since g(N0) is a set consisting of �nitely many points in R, we can easily

pick a point z ∈ R − g(N0). Now take any isotopy H∗ : [0, 1] × R → R - usingthe notation, for �xed t, h∗t (x) = H∗(t, x) - such that:

1. h∗0 = id and h∗1(0) = z,

2. h∗t (x) = x for all |x| ≥ A, and all t ∈ [0, 1], where A is some large number.

Such an isotopy is easily seen possible to construct.Now we we de�ne the isotopy H : [0, 1]×W →W : let

ht(q) =

{k(p,H(t, x)), if q = k(p, x) ∈ Npq, if q ∈W −Np.

(Note that it is important that h∗t eventually becomes the identity, for H to bea smooth isotopy.) The desired di�eomorphism is h = h1.

Let e = c+c′

2 , and let ε be so small so that f−1[e − ε, e] ⊂ V . The integralcurves of ξ/ξ(f) determine a family of di�eomorphism Φt(q) from f−1(e− ε) tof−1(t), for 0 ≤ t ≤ ε, where Φ0 goes fromW to itself, and Φε goes from f−1(e−ε)ontoW . Using this we can de�ne a di�eomorphism ϕ : [e−ε]×W ∼−→ f−1[e−ε, e],such that ϕ({e− ε} ×W ) = f−1(e− ε) and ϕ({e} ×W ) = f−1(e) = W , by

ϕ(t, q) = Φ−1e−t(q).

That it really is a di�eomorphism depends on that Φ(t, q) is a 1-parameter groupof di�emorphisms (see proof of lemma 4).

25

We now de�ne a di�emorphism H : [e− ε, e]×W ∼−→ [e− ε, e], by

H(t, q) = (t, ht(q)),

where we have changed H slightly so that ht = id, for t near e− ε, and ht = h,for t near e. Then

ξ0(q) =ξ

ξ(f)(ϕ ◦ H ◦ ϕ−1(q))

de�nes a new smooth vector �eld. The sought after, altered, vector �eld is thengiven by

ξ(q) =

{ξ(f)(q)ξ0(q), if q ∈ f−1[e− ε],ξ(q), elsewhere.

It can easily be checked that it is a new gradient-like vector �eld for f . Wewill show that the trajectories going to and from p1, . . . , pn, p

′1, . . . , p

′m are all

disjoint. But this follows from that the new sphere Sp of p at the level e is h(Sp)and the new sphere Sp′ of p′ at the same level is the same as Sp′ , hence they aredisjoint.

The next step is to cancel out critical points so that, from a self-indexingMorse function, we obtain a new self-indexing Morse function having only 1maximum point and one minimum point.

Let ξf be a gradient-like vector �eld for a Morse function f . Let p0 be acritical point of f having index 0, and let p1 be a critical point of f havingindex 1. The sphere induced by going upwards from p0 will be denoted Sp0 , thesphere induced by going downwards from p1 will be denoted Sp1 . We begin byproving the cancelation theorem, then we continue by applying it to reduce thenumber of critical points having index 0, and �nally we reduce the number ofcritical points having index 2.

With theorem 1, we can conclude that, given the gradient-like vector �eldξ on a surface S, for any point p ∈ S there passes a unique trajectory T . Inparticular if p0 and p1 are critical points having indices 0 and 1 respectively, suchthat Sp0 and Sp1 intersects in one point, then there exists exactly one trajectoryT from p0 to p1.

We begin by assuming that f is a Morse function, which is not self-indexingbut instead that we have altered f so that there is some small ε > 0, withε < f(p0) and 1 − ε > f(p1), so that S∗ = f−1[ε, 1 − ε] contains only p0and p1 as critical points. Then S∗ will be a two dimensional compact smoothmanifold with boundary; we denote the boundary components by B0 = f−1(ε)and B1 = f−1(1− ε). This is easily seen to be possible in a way, very similar tohow we proved the existence of a self-indexing Morse function, using the lemmas8 and 9.

Theorem 7 (Cancelation Theorem). Let f be the Morse function describedabove with critical points p0 and p1 having index 0 and 1, respectively, andlet ξf be the gradient-like vector �eld for f above. Suppose that Sp0 and Sp1intersects in one point. Then it is possible to alter the gradient-like vector �eldξf on a small neighbourhood V of the only trajectory T from p0 to p1, producinga, on this neighbourhood, nowhere zero vector �eld ξg that is a gradient-likevector �eld for a Morse function g, that coincides with ξf outside of V .

26

This means that g is a Morse function with the same critical points as f ,having the same indices, except that p0 and p1 are regular points of g; if this istrue, then we say that p0 and p1 are cancelable.

We begin by proving some lemmas.

Lemma 10. We can alter the gradient-like vector �eld ξf in V to a newgradient-like vector �eld for f , so that for a smaller neighbourhood VT ⊂ V ofT , we have that T still is in VT , and there is a coordinate chart φ−1T : VT → R2

such thatφ−1T (p0) = (0, 0), φ−1T (p1) = (0, 1),

andφT−1∗ ξf (q) = η(x, y) = (x, v(y)),

where v is a smooth function of y, such that∣∣∣dvdy (y)

∣∣∣ = 1, for y near 0 and 1,

and

v(y) =

0, if y = 0, 1,

> 0, if y ∈ (0, 1),

< 0, elsewhere.

Proof. We can parameterize the closure of the trajectory by a smooth bijectivefunction γ : [0, 1]→ T , such that γ(0) = p0 and γ(1) = p1. Put this function onthe y-axis in a coorrinate system, so that T corresponds to {0}× [0, 1]. We can�nd neighbourhoods U0 and U1 of (0, 0) and (0, 1) respectively, together withparametrizations φ0 : U0 → V0 and φ1 : U1 → V1 onto neighbourhoods V0 andV1 of p0 and p1 respectively, such that ξf = (x, y) in U0 and ξf = (x, 1 − y) inU1 (and f = f(p0) + x2 + y2 in U0, f = f(p1) + x2 − (1 − y)2). By possiblyrotating U0 or U1 (or both) we can assume that T in U0 corresponds to φ0(0, y),for y ≥ 0, and T in U1 corresponds to φ1(0, y), for y ≤ 0. Moreover we see thatwe can make γ coincide with the y-coordinates of φ0(0, y) and φ1(, y) in T ∩ V0and T ∩ V1 respectively. Hence we can parametrize ξf along T by φ2|T

φ2−1∗ ξ(γ(y)) = (0, v(y)),

for a smooth positive function v, such that v satis�es the criterions speci�ed inthe lemma.

We can take δ > 0 so small so that (δ, 1 − δ) intersects with U0 and U1;and using theorem 1 we can extend φ2|T around {0} × [δ, 1 − δ] to a samllneighbourhood U2 that corrsponds, by a parametrization φ2, to a neigbourhoodV2 of the respriction of T to γ[δ, 1−δ], such that φ2 coincide with φ0 on U0 (andV0) and φ1 on U1 (and V1). And we can, by compactness, take V2 so small sothat all trajectories from U2 ∩ U0 (V2 ∩ V0) go to U2 ∩ U1 (V2 ∩ V1).

Then we can take V = V0 ∪ V1 ∪ V2 as a neighbourhood of T , with cor-responding coordinate neighbourhood U = U0 ∪ U1 ∪ U2 and parametrizationφ : U → V . We now de�ne the altered vector �eld of ξf , which we for now

denote ξ: Begin by take z1 and z2, so that 0 < z1 < z2; preliminary we alsorequire that {(x, y)||x| ≤ y ∈ [0, 1]} ⊂ U , we will make this stronger later. Letη(x, y) = (x, v(y)) be a smooth vector �eld on R2. Note that η coincides withφ0−1∗ (ξf ) in U0, with φ1

−1∗ (ξf ) in U1 and with φ−1∗ (ξ) on T ; i.e. η behaves locally

27

like a gradient-like vector �eld for f . Now let

ξ(q) =

φ∗(η(x, y)), if q = φ(x, y), |x| ≤ z1,(|x|−z1)ξf (q)+(z2−|x|)φ∗

z2−z1 , if q = φ(x, y), z1 ≤ |x| ≤ z2ξf (q), if q = φ(x, y), x ≥ z2.

Hence ξ is a smooth gradient-like vector �eld on all of S which coincides with ξfoutside of a small neighbourhood V of T , and with coordinates as in the lemmaon a smaller neighbourhood VT ⊂ V , where φT is given by the restriction of φto VT , φ|VT .

Now note that, in {0} × [δ, 1 − δ], ∂f∂x = 0, and ∂f

∂y > ε > 0, for some

positive constant ε0 (otherwise ∂f∂y would converge to 0, by compactness). Also

v(y) > ε1 > 0 on [δ, 1 − δ] for some constant, by the same reasoning. Henceletting z2 be small (and we can of course still always take z1 > 0 smaller) wecan make sure that

(x, v(y)) ·(∂f

∂x,∂f

∂y

)= x

∂f

∂x+ v(y)

∂f

∂y> 0.

Hence ξ is a gradient-like vector �eld for f on all of S.

From now on, our gradient-like vector �eld ξf will be assumed to be ξ fromthe previous lemma.

Lemma 11. Given an open neighbourhood V of T one can always �nd a smallerneighbourhood V ′ of T with the property that no trajectory leads from V ′ outsideof V and back again into V ′.

Proof. Towards a contradiction, now suppose that the statement is false. Thenwe would have a sequence {Ti}∞i=1 so that each Ti goes from a point ri to a pointsi outside V and then to a point ti, such that the sequences ri and ti tends toT as i goes to in�nity. S − V is compact, hence si → s 6∈ V as i→∞.

The curve from theorem 1 ϕ(t, s) must go to to B1, otherwise we wouldget a new trajectory from p0 to p1 (since the curve must go to p0). Then byusing the continuous dependence of ϕ(s′, t) on s′ we see that for all s′ in someneighbourhood Vs the trajectory through s′ goes to B1. The partial trajectoriesTs′ from s′ to B1 are compact; hence, in any metric, the least distance d(s′)from T to Ts′ depends continuously on s′ and will be bounded from 0 for all s′.Therefore, since ti ∈ Tsi , it is impossible that ti → T as i → ∞. The desiredcontradiction.

Now let V be an open neighbourhood such that V ⊂ VT , where VT the neigh-bourhood from lemma 10, and let V ′ be the neighbourhood from the previous

lemma - so that T ⊂ V ′ ⊂ V ⊂ V ⊂ VT .

Lemma 12. It is possible to alter the gradient-like vector �eld ξf on a compactsubset Vc of V

′ producing a on S∗ nowhere zero vector �eld ξg, such that every

integral curve of ξg that passes through a point in V was outside V at some time

t0 < 0 and will again be outside V at some time t1 > 0.

28

Proof. Start by replacing η(x, y) by η(x, y) de�ned by

η(x, y) = (x, v(|x|, y)),

where, v = v outside of a compact neighbourhood of φ−1(T ) in U ′ = φ−1(V ′)and v(0, y) < 0. This determines a nowhere zero vector �eld ξg on S∗. In ourcoordinate system, the integral curves of ξg in VT satisfy

dx

dt= x,

dy

dt= v(|x|, y).

Consider the integral curve t 7→ (x(t), y(t)) with initial value (x0, y0) as t in-creases. We show that t 7→ (x(t), y(t)) have to leave U = φ−1(V ). The casewhere t decreasees is similar.

We have that dxdt = x, which if we solve the di�erential equation becomes

x(t) = x0et, for some constant x0. If x0 6= 0, then, since U = φ−1(V ) is

compact, hence bounded, (x(t), y(t)) will eventually leave U . If instead x0 = 0,then we have dy

dt = v(0, y) which is everywhere negative; hence (x(t), y(t)) will

leave U .

In the next we will keep the notations from previous lemmas, in particularξg will denote our obtained new vector �eld.

Lemma 13. Every trajectory in S∗ of ξg goes from B0 to B1.

Proof. By previous lemmas: If an integral curve of ξg is ever in V′ it eventually

gets outside V , and since ξg agrees with ξf outside of Vc ⊂ V ′ it will remainouside V ′. Hence it will follow a trajectory of ξ to B1. Likewise one concludethat any trajectory must come from B0.

Lemma 14. The smooth vector �eld ξg is a gradient-like vector �eld of a Morsefunction g on S that agrees with f outside of S∗ and inside S∗ in neighbourhoodsof B0 and B1.

Proof. The proof will be in two steps: First we show how ξg determines a

di�eomorphism ϕ : [0, 1]×B0∼−→ S∗, such that ϕ({0} ×B0) = B0 and ϕ({1} ×

B0) = B1. Then we construct a Morse function g0 : [0, 1] × B0 → R such that∂g∂t > 0, and such that g0 agrees with f0 = f ◦ ϕ near B0 and B1. By thedi�eormorphism ϕ we will obtain the desired Morse function g = g0 ◦ϕ−1 fromS∗ to R with no critical points, which moreover will agree with f near B0 andB1.

Let ψ(t, p) : R × S∗ → S∗ be the family of integral curves of ξg, fromtheorem 1. On S∗, ξg is nowhere zero and all curves go from B0 to B1, henceξg is nowhere tangent to B0 or B1 (that is: nowhere tangent to the boundaryof S∗). Let τ0 : S∗ → R be the function that for each q ∈ S∗ assigns the t ∈ Rwhere the integral curve ψ(t, q) reaches B0; and likewise let τ1 send q to thet ∈ R where ψ(t, q) reaches B1. Note that τ0 and τ1 are smooth, which is aneasy consequence of the implicit function theorem. Therefore the projectionπ : S∗ → B0, π(p) = ψ(τ0(p), p) is smooth. Now consider the smooth vector�eld

τ1(π(p))ξg(p).

29

It's integral curves go from B0 to B1, and they do so in unit time. Hence wecould just as well assume that this property were already satis�ed by ξg. Now

the di�eomorphism ϕ : [0, 1]×B0∼−→ S∗ is given by

ϕ(t, p0) = ψ(t, p0)

were ψ is the family of integral curves for ξg, now assumed to have integralcurves that go from B0 to B1 in unit time.

We can choose δ > 0, such that for all p0 ∈ B0 and all t < δ or t > 1− δ wehave that

∂f0∂t

> 0.

Let α : [0, 1]→ [0, 1] be any smooth function that is equal to 0 in [δ, 1− δ] andequal to 1 near 0 and 1. Now we de�ne g0: First let k : B0 → R be given by

k(p0) =1−

∫ 1

0α(t)∂f0∂t (t, p0)dt∫ 1

0(1− α(t))dt

.

By choosing δ small enough, we may assume k(p0) > 0. Now let g0 : [0, 1]×B0 →R be given by

g0(s, p0) =

∫ s

0

(α(t)

∂f0∂t

(t, p0) + (1− α(t))k(p0)

)dt.

Since g0 clearly is smooth we only have to verify that it agrees with f0 for snear 0 and 1, and that g0 has no critical points. Using that α(s) = 1, for s near0 and 1: For these s near 0 we have that

g0(s, p0) =

∫ s

0

∂f0∂t

(t, p0)dt = f0(s, p0)− f0(0, p0),

where f0(0, p0) = 0, which shows the agreement. For s near 1 we have that

g0(s, p0) =

∫ s

0

(α(t)

∂f0∂t

(t, p0) + (1− α(t))k(p0)

)dt

=

∫ 1

0

(α(t)

∂f0∂t

(t, p0) + (1− α(t))k(p0)

)dt

−∫ 1

s

(α(t)

∂f0∂t

(t, p0) + (1− α(t))k(p0)

)dt

= 1− (f0(1, p0)− f0(s, p0)) = f0(s, p0),

because f0(1, p0) = 1, and because∫ 1

0

α(t)∂f0∂t

(t, p0)dt+

∫ 1

0

(1− α(t)) dtk(p0)

=

∫ 1

0

α(t)∂f0∂t

(t, p0)dt+

∫ 1

0(1− α(t))dt

(1−

∫ 1

0α(t)∂f0∂t (t, p0)dt

)∫ 1

0(1− α(t))dt

= 1.

Moreoverdg0ds

(s, p0) = α(s)∂f0∂s

(s, p0) + (1− α(s))k(p0) > 0.

30

This completes the proof of Cancelation Theorem.From now on in this section, let f : S → R denote a self indexing Morse

funcion, ξ a gradien-like vector �eld for f , and let Sa = {p ∈ S|f(p) ≤ a} bethe set of all points less then or equal to some "height" a.

We will now use Cancelation Theorem together with two lemmas connectingcriticial points to attachings of rectangles and discs, and an induction argument,to reduce the number of critical points of a self-indexing Morse function, so thatwe obtain a Morse function with only one minimum and one maximum.

De�nition 25. By an attaching of a rectangle R to Sa we will mean takingthe disjoint union

R ∪g Sa,

where we identify points on the boundary of Sa (∂Sa = f−1(a)) with points ontwo opposite sides of the rectangle by an injective continuous map g : R∗ → ∂Sa.For example R∗ can be given by {x = 0}∪{x = 1}, if R = {(x, y) ∈ R2|0 ≤ x ≤1, 0 ≤ y ≤ 1}.

Likewise, by an attching of a disc D, say D = {(x, y) ∈ R2|x2 + y2 ≤ 1}, wemean taking the disjoint union

Sa ∪g D,

where we again identify the points on ∂D = S1 by a injective continuous functiong : S1 → f−1(a).

We will show that the identi�cation, when attaching rectangles, can be doneby a smooth map g, where we will not use the rectangle R from the de�ni-tion above, but a smooth rectangle homeomorphic to R. This implies that inthe lemmas below, we will not only get homeomorphisms, but the stronger:di�eomorphisms.

Lemma 15. Let 0 < a < 1 and 1 < b < 2. Then Sb is di�eomorphic to Sa

with a smooth rectangle Ri for each critical point pi of index 1 attached.

Proof. For simplicity we will suppose that there only is one critical point havingindex 1, the general case is similar. Denote this point by p0.

Let V be a neighbourhood of p0 such that, in local coordinates, f = 1+x2−y2through out V . Let ε > 0 be so small so that f−1(1 − ε) and f−1(1 + ε) bothhave two components in V .

Let V ′ ⊂ V be a smaller neighbourhood of p0. We will now use a smoothfunction k : S → R on S, such that k is constanty zero on a neighbourhoodVk ⊂ V ′ of p - where Vk is so small so that it does not intersect with f−1(1 −ε) ∪ f−1(1 + ε) - is constantly 1 outside of V ′, and increases strictly in between(meaning that in the local polar coordinates (θ, r), ∂k

∂r is strictly greater thenzero). (k is 1 minus a bump function). Then using k and ξ we de�ne a newsmooth vector �eld ξ on S by:

ξ(q) =

{k(q) ξ(q)

ξ(f)(q) , if q ∈ f−1[1− ε, 1 + ε] ∪ V − {p},0, if q = p,

and ξ vanishes outside a compact neighbourhood of f−1[1−ε, 1+ε]∪V . Applyinglemma 5, we see that ξ generates a 1-parameter group of di�eomorphisms ϕ(t, q).

31

Consider ϕ2ε. We will argue that this di�emorphism sends f−1(1− ε) unionthe boundary of a subset R of V to f−1(1 + ε), and that R is homeomorphic toa rectangle. All p ∈ f−1(1 − ε) whose integral curves are always outside of V ′

will be pushed up by unit time (as in lemma 4), and hence be sent to f−1(1+ε).Those points that go through V ′ will be slowed down, hence they will reach alevel set less then 1 + ε.

Now we consider ϕ−12ε , and speci�cally what happens when applied to thepart of f−1(1 + ε) that is not hit by parts of f−1(1− ε). By continuity of ϕ2ε,we can choose V ′ and k so that this is one segment of each of the componentsof f−1(1+ ε) in V . By the continuity of ϕ2ε, and by smoothness of k going from0 to 1, ϕ−12ε of the segments will be two curves connecting the two compnentsof f−1(1 − ε). These curves together with the parts of f−1(1 − ε) that haveintegral curves through V ′ enclose a region R be homeomorphic to a rectangle(See picture).

Figure 3: Describes how ϕ2ε sends S1−ε ∪R to S1+ε.

From now on, whenever we talk about attaching of a rectangle we will meana smooth attaching of the smooth rectangle from the proof above.

Lemma 16. Let 1 < b < 2, and suppose that S has m maximum points. ThenS2 = S is di�eomorphic to Sb with m closed discs D1, . . . , Dm attached.

Proof. This follows from that (S − Sb) ∪ f−1(b) is di�eomorphic to m disjointdiscs.

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Lemma 17. Suppose that S has has n minima, that is, suppose that there existsn critical points p1, . . . , pn of f having index 0, n ≥ 2. Then there exists criticalpoints r1, . . . , rn−1 of f having index 1 such that that pi and ri are cancelable,for 1 ≤ i ≤ n− 1.

Proof. By observing how Sa changes as a increases, we will by reconstructingthe surface see that if this is not true, then we could not have a connectedsurface.

Let 0 < a < 1 and 1 < b < 2. First note that Sb must be connected. Thisis because passing from Sb to the full surface S is done by attaching discs, andattaching discs are done by di�eomorphisms of connected boundary components,hence these attchings can't connect disconnected components; only attachmentsof rectangles can do that.

By theorem 2 and lemma 4, Sa is di�eomorphic to the union of n disjointdiscs D1, . . . Dn. As we have noted earlier Sb must consist of one connectedcomponent, i.e. be connected. Take any of Di's, with corresponding criticalpoint pi. Di must be attached by some rectangle, corresponding to a criticalpoint rj - by possibly renamin we can take ri - having index 1 that is alsoattached to some other disc Dl, l 6= i, with corresponding critical point pl. Ifnot: then Sb could not be connected, since there would be nothing connectingDi to any of the other discs.

Moreover, the induced spheres - Spi , going upwards from pi, and Sri goingdownwards, from ri - must intersect in only one point, because the trajectoriesfrom ri must go down to both points pi and pl. Hence pi and ri are cancleable.

Corollary 4. On any surface S, there exists a self-indexing Morse functionhaving only 1 minimum point.

The cancelation of the maximum points is just �ipping the surface upsidedown - using the new Morse function 2−f - and then using the previous lemmaagain. Thus we obtain: Any surface admits a self-indexing Morse function withjust one maximum and one minimum, that is, any surface admits a good Morsefunction.

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Constrution of the Genus-g Toruses

In this section we let S denota a surface, f : S → R denota a good Morsefunction on S, and as always, Sa = {p ∈ S|f(p) ≤ a}. From the previoussection we know that to classify surfaces, we only need to look at how we canattach rectangles to a disc, and then attach one disc to to the obtained "discwith rectangles" in the end.

The �rst observation to be made is the following: Since the surface is con-nected and without boundary, there can only be one boundary component ofSb, for 1 < b < 2.

We next focus on the rectangles: The attaching of a rectangle can be madein 2 ways: the �rst is with no twist, the second with a twist (more twists arehomeomorphic to no twist or one twist). However, since S must be orientablethere can be no twist, since if there were we would have a homeomorphic copyof a Möbius band inside S, contradicting the assumtion that S is orientable.

One way to visualize the attachment of rectangles is to think of it as aprocedure, attaching one at a time; but as long as all attachements are todi�erent parts of the original disc, even if we see how the boundary componentschanges as we attach one rectangle at a time, this procedure is equivalent toattach all at the same time at di�erent places. We now make the followingobservation, based on the fact that boundary components here must be closedpaths.

Lemma 18.

1. If a rectangle is attached to two di�erent boundary components, then thenumber of boundary components reduces by one.

2. If a rectangle is attached to one boundary component, then the numer ofboundary components increase by one.

Proof. Obvious from picture.

Figure 4: Illustration of the previous lemma.

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Corollary 5. The number of critical points having index 1 of f must be even.

Proof. We start with one boundary component, and we must end with one;hence we must attach an even number of rectangles, and so the number ofcritical points having index 1 must be even.

Now we prove the two starting cases of classi�cation. For this we begin byobserving that there is only one way to attach a disc. This is a consequence oflemma 1, because the attaching is done by a di�eomorphism from S1 onto itself.

Theorem 8. Suppose there are no critical points having index 1 of f , then Sis di�eomorphic to a sphere.

Proof. By lemmas above S must be a di�eomorphic to two discs attached to eachother smoothly along their boundaries, which is a construction of the sphere.And moreover there is only one way to do this by lemma 1.

If lemma 1 were not true, there might have been two di�erent surfaces, notdi�eomorphic to each other, but both homeomorphic to the sphere (and hencehomeomorphic to each other).

Theorem 9. Suppose there are two critical points having index 1, then S isdi�eomorphic to a torus.

Proof. We will think of this as attaching one rectangle at a time. After attachingthe �rst rectangle we get a cylinder. The cylinder has two boundary components.The second rectangle must be attached to both these boundary components forSb, 1 < b < 2, to be di�eomorphic to a surface with one boundary component.Finally, attaching a disc along the boundary gives us a torus; the hole can bevisualized as the region between the second rectangle and the cylinder. Theattachments of the two rectangles and the sphere is done in a unique way (upto di�eomorphism), and smoothly.

Figure 5: Illustrates the reasoning of the previous proof.

Another way to see it, is to attach one rectangle to each disc, obtaining twocylinders, which obviously forms a torus when attached together.

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Thus we the torus T1 and the sphere S2 are unique up to di�eomorphism,hence all Tg are, g ≥ 0. Now we use induction to prove the �nal theorem of theclassi�cation.

Theorem 10 (Classi�cation of Surfaces). Suppose f has 2g critical points hav-ing index 1. Then S is di�eomorphic to the genus-g torus Tg.

The proof will be in three main steps: lemma A, lemma B and lemma C.

Lemma 19 (A). There is an attachement of two of the rectangles, such that oneis inside the other, so that removing this attachment (or doing the attaching)does not change the number of boundary components. Call such attachments forpretszel attachments.

Proof. If this wasn't true there could clearly not be only one boundary compo-nent.

Lemma 20 (B). After attaching all rectangles the boundary must be connected.We can now slide the pretzel attachment to the left of some point on the disc -the disc to which the rectngles are attached - or to the left of a previously slidedpretzel attachment so that:

1. All rectangles are still attached to disjoint parts of the original disc.

2. The slided pretzel attachment together with the originl disc makes a pathalong the boundary that does not intersect the boundary of any other rect-angles (i.e. we hve slided the pretzel out of the region of the boundry ofthe disc were the other rectangles are attached).

Proof. Clearly such slides can be done smoothly on the boundary, hence givesus the same surface up to di�eomorphism. The two criterions is easily seen topossible to satisfy, by picture similar to �gure 5.

Lemma 21 (C). Using lemma A and B inductively we get a disc with g disjointpretzel attachments.

Proof. Obvious.

Proof of Classi�cation of Surfaces. Attaching a single disc to the disc having gpretzels clearly gives us a genus g-torus.

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References

[1] Manfredo Perdigão do Carmo, Di�erential Geometry of Curves and Surfaces,Upper Saddle River, New Jersey 07458, Prentice-Hall, Inc., 1976.

[2] Manfredo Perdigão do Carmo, Riemannian Geometry, Boston, Birkhäuser,1992.

[3] Ib Madsen, Jørgentornehave, From Calculus to Cohomology, Cambridge,Cambridge University Press, 1997.

[4] John Milnor, Lectures on the H-Cobordism Theorem, Princeton, New Jersey,Princeton University Press, 1965.

[5] John Milnor, Morse Theory, Annals of Mathematical Studies 51, Princeton,New Jersey, Princeton University Press, 1963.

[6] John Milnor, Topology from the Di�erentiable Viewpoint, Revised Edition,Princeton, New Jersey, Princeton University Press, 1997. (Orginially pub-lished by The University Press of Virginia, 1965).

[7] Donal O'shea, The Poincaré Conjecture : In Search of the Shape of theUniverse, Penguin Books, 2008.

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