U.U.D.M. Project Report 2016:37 Classification of Compact Orientable Surfaces using Morse Theory Johan Rydholm Examensarbete i matematik, 15 hp Handledare: Thomas Kragh Examinator: Jörgen Östensson Augusti 2016 Department of Mathematics Uppsala University Classication of Compact Orientable Surfaces using Morse Theory Johan Rydholm 1 Introduction Here we classify surfaces up to dieomorphism. The classication is done in section Construction of the Genus-g Toruses, as an application of the previously developed Morse theory. The objects which we study, surfaces, are dened in section Surfaces, together with other denitions and results which lays the foundation for the rest of the essay. Most of the section Surfaces is taken from chapter 0 in [2], and gives a quick introduction to, among other things, smooth manifolds, dieomorphisms, smooth vector elds and connected sums. The material in section Morse Theory and 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 these two sections we rst prove Lemma of Morse, which is probably the single most important result, even though the proof is far from the hardest. We prove the existence 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 this we also prove the existence of one of our most important tools: a gradient-like vector 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 and many other applications; to mention one, in [4] Milnor goes on and prove the generalized Poincaré Conjecture1 in dimensions n ≥ 5, using similar results. In this essay, the Classication Theorem, great as it may seem, is almost like only a corollary of these two sections.2 1The theorem is: If M is a simply-connected smooth manifold having dimension n ≥ 5 with the (integral) homology of the n-sphere, then M is homeomorphic to the sphere in the n n same dimension S ; if n = 5 or n = 6, then M is even dieomorphic to S . 2A good popular account of the classication 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, specically we dene the concept of a surface and a genus-g torus. Denition 1. A smooth manifold of dimension n is a topological space M to- gether with two families of open sets , of n and respectively, fUigi2I ; fVigi2I R M and a set of homeomorphisms fφi : Ui ! Vigi2I - called parametrizations, whose inverses, are called coordinate system or just coordinates - such that [ Vi = M; i2I and all transition functions −1 φi ◦ φj are smooth; we also require the triad (fUig; fVig; fφig) to be maximal in the sense: given (fUig; fVig; fφig), if φ : U ! V satises the conditions above together with the triad, then (U; V; φ) is in the triad. Lastly, a smooth manifold is 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 and orientable (to be dened later). In other words, what we will call surfaces are connected, compact and orientable 2-dimensional smooth manifolds. Also, in our denition the manifolds (and thereforee also the surfaces) are without boundary; sometimes we will nevertheless talk about a boundary, formally this is the set of all points where all neighbourhoods - instead of being homeomor- phic (and by the denitions below, also dieomorphic) to U open in Rn - are homeomorphic to n n . Sometimes we will have to H = f(x1; : : : ; xn) 2 R jxn ≥ 0g work with surfaces with boundary, when this is the case it will be obvious from context, 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 trivially smooth. Denition 2. A mapping ' : M ! M 0 between two smooth manifolds M and M 0, of dimension n and m respectively, is said to be smooth at a point p, if for any parametrizations φ : U ! V ⊂ M; φ0 : U 0 ! V 0 ⊂ M 0 around p and '(p) 0−1 −1 respectively, the map φ ◦ ' ◦ φ is smooth at x0 = φ (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 be smooth, if for any parametrization φ : U ! V ⊂ S, the function f ◦φ is smooth. We can now dene a notion of equivalence of surfaces, and more generally, equivalence of smooth manifolds, that is when they from our viewpoint are considered the same: the notion of dieomorphic smooth manifolds. Denition 3. Two smooth manifolds M and M 0 are said to be dieomorphic, when there exists a smooth bijection ' : M −!∼ M 0; such that its inverse '−1 is smooth; such a map is called a dieomorphism. 4 The goal is to classify surfaces up dieomorphism; we will see that the clas- sication can be done by a class of surfaces to be dened later, called genus-g toruses. Before that we will dene tangent spaces and dierentials. Denition 4. A smooth map γ :(−, ) ! M from an open interval (−, ) of R to a smooth manifold M, is called a curve in M. Suppose that p 2 M is a point in the manifold, with γ(0) = p, for some curve γ. Let D denote the set of all real valued functions on M, that are smooth at p. The tangent vector to the curve γ at p is the function γ0 : D! R dened by d(f ◦ γ) γ0(f) = (t = 0): dt A tangent vector at p is just the tangent vector of some curve. The set of all tangent vectors of M at p is called the tangent space of M at p, and it is denoted TMp. A result from the theory of smooth manifolds says that TMp is a nite dimensional vector space having the same dimension as M. Denition 5. Let M and M 0 be two smooth manifolds, and let ' : M ! M 0 be a smooth map. For all p 2 M and all v 2 TMp, by a well known result from the theory of smooth manifolds (which is not hard to prove), given any curve γ for which v is the tangent vector of, the linear map 0 0 d'p : TMp ! TMp; d'p(v) = β (0); where β = ' ◦ γ, is independent of the choice of γ. This map is called the dierential of ' at p. Next we dene the notion of a vector eld on a manifold. A special type of vector elds called gradient-like vector elds dened in the next section will be one of the main tools used. Afterwards we dene the notion of a smooth manifold being orientable, and give a citerion using möbius bands to determine whether a surface is orientable or not. The set TM = f(p; v)jp 2 M; v 2 TMpg is called the tangent bundle of M. It is well known that the tangent bundle is a smooth manifold of dimension two times the dimension of M. Using this smooth manifold we can give the denition. Denition 6. A vector eld on a smooth manifold M is a map X : M ! TM, such that for each p 2 M we assaign a vector v 2 TMp: X : p 7! (p; v). Since TM is a smooth manifold we have an earlier denition that guarantees that it makes sense to ask wether X is smooth or not. Hence we can simply dene a smooth vector eld to be a vector eld X that is smooth by this denition (denition 2). Denition 7. We say that a manifold M is orientable if there are parametriza- tions fφig covering all of M such that for every pair φi : Ui ! Vi, φj : Uj ! Vj with , the change of coordinates −1 has positive determinant. Vi \ Vj 6= ; φj ◦ φi For surfaces: A surface S is orientable if and only if it is impossible to embedd a möbius band B in S.3 3 An embedding of B in S is a map e : B ! S such that (i) d'p : TBp ! TSp is injective for all p 2 B, and (ii) ' is a homeomorphism from B to '(B). 5 Now we give some facts and terminology about vector elds which will be used in many of the proofs. We begin with trajectorys. Denition 8. A trajectory in a point p of a smooth vector eld X is map γ :(−, ) ! V , from an open interval (−, ) to a neighbourhood V of p such that γ(0) = p and γ0(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 as t ! 1, and γ0(t) = X(γ(t)). The curve in the denition above is known to exist and be unique for small enough . Next we mention, a theorem from the theory of dierential equations which will be needed. Theorem 1. Suppose that X is a smooth vector eld on a manifold M, and let p 2 M.