Morse Theory and Thom-Smale Homology Qingyuan Bai Last update: May - PDF document

Morse Theory and Thom-Smale Homology Qingyuan Bai Last update: May 16, 2019 Abstract This survey focuses on the Morse theory of finite dimensional compact manifolds. We introduce Morse function and discuss its first applications. Then we turn to study the pseudo gradient field X adapted to some Morse function f . There is a complex ( C f ∗ , ∂ X ) associated to such pair ( f, X ) if X satisfies the so called ‘Smale property’. We prove that for different choice of ( f, X ) , the associated complexes are naturally chain homotopic. So the ‘Morse Homology’ is actually a smooth invariant of manifolds. Contents 1 Introduction 1 2 Basic Morse Theory 3 2.1 Definition and first examples . . . . . . . . . . . . . . . . . . . . 3 2.2 Genericness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Homotopy type and critical values . . . . . . . . . . . . . . . . . 6 3 Thom-Smale Complex and Morse Homology 9 3.1 Pseudo gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Smale condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 Thom-Smale complex . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4 Manifold with boundary (or corners) . . . . . . . . . . . . . . . . 22 4 A priori: the invariance of Thom-Smale complex 23 References 27 1 Introduction In this section, we briefly talk about the history of Morse theory. 1

In the original work of Marston Morse, he defined Morse function f on a manifold M and its index λ p at each critical point p (the exact definition of these objects will be given in the next section). His idea was that the ana- lytic behaviour of functions on a manifold M should reflect its topology. This connection was clarified by the following Morse’s inequality: p t λ p be the polynomial in Theorem 1.1 (Morse’s Inequality) . Let M t ( f ) := � t whose n -th coefficient is the number of critical points whose index is n . Let P t ( M ; K ) be the Poincar´ e polynomial of M taken over K . Then we have M t ( f ) − P t ( M ; K ) = (1 + t ) Q ( t ) (1) where Q ( t ) is a polynomial in t whose coefficients are nonnegative. The original proof works for functions on finite dimensional manifolds, but the same strategy can be applied to functionals on infinite dimensional spaces (like loop spaces). With this in mind, Morse proved his famous theorem assert- ing that for any Riemannian metric on S n , there are infinitely many geodesics connecting two arbitrarily chosen points on it. Moreover, Bott used the same strategy to prove his periodicity theorem concerning the homotopy groups of O ( n ) and U ( n ). We will not touch upon the infinite dimensional set up in our treatment, however. The curious reader might read [5]. It was Smale’s idea to focus on the (pseudo-)gradient flows on the manifold. By counting flow lines, we can define a complex ( C f ∗ , ∂ X ) freely generated by critical points of f . It was proved that the homology of ( C f ∗ , ∂ X ) is isomorphic to the singular homology of M , so the classical inequality follows as an algebraic fact. This construction, as Bott put it, is ‘the most beautiful formulation of nondegenerate Morse theory’. The Thom-Smale complex was famous again in 80s after Witten’s famous paper. In his work, Witten considered a Riemannian manifold with a Morse function. He used this model to describe physic phenomenon that were first proposed in inifnite dimensional settings of Yang-Mills theory. It turns out that this complex was exactly the Thom-Smale complex that we defined above. So the theory can serve as a toy model to help us understand the general theory. Similarly, it works as a toy model of Floer theory, in which we try to define ‘Thom-Smale complex’ in the infinite-dimensional spaces. The later theory has become a central tool in modern symplectic geometry. The organization of this survey is as follows. In section 2, we define Morse function and show its genericness. Then we develop basic Morse theory, culmi- nating in a proof of the Morse’s Inequality. In section 3, we introduce pseudo- gradient flow that satisfies Smale property and show its genericness. The rest of the section is devoted to the definition and basic properties of Thom-Smale complex. 2

2 Basic Morse Theory In this section, we develop Morse theory on finite dimensional manifolds. Note that in our treatment, we always assume M to be a compact manifold of dimension m (with or without boundary). 2.1 Definition and first examples Given a smooth function f : M → R , x ∈ M is called a critical point of f if d f = 0 at x . In such case, there is a symmetric bilinear form H x f (so called Hessian form) defined on T x M given by f ( V 2 ) | x H x f ( v 1 , v 2 ) := V 1 d (2) where v i ∈ T x M and V i is arbitrary locally defined smooth vector field that coincides with v i at x . Proposition 2.1. The bilinear form H x f is well defined and symmetric. Proof. For any local vector field V 1 , V 2 , we have: V 1 ( V 2 f ) − V 2 ( V 1 f ) = [ V 1 , V 2 ] f (3) where [ V 1 , V 2 ] is the Lie bracket of two vector fields. Since d f = 0 at p , we see that V 1 ( V 2 f ) | p = V 2 ( V 1 f ) | p (4) Now that this bilinear form will be symmetric if it is well-defined. Note that V 1 ( V 2 f ) | p = V 1 | p ( V 2 f ) = v 1 | p ( V 2 f ) (5) So H ( v 1 , v 2 ) doesn’t depend on the choice of V 1 . Same argument shows that it doesn’t depend on the choice of V 2 . So it is well defined. Under local charts ( x 1 , ..., x m ), we have a basis for T x M = span { ∂ x 1 , ...∂ x n } . The matrix of H x ( f ) with respect to this basis is just the ‘Hessian’ matrix: ∂ 2 f ( ∂ xi ∂x j ). If at each critical point of f , the associated bilinear form is nondegen- erate, we say f is a Morse function. In such case, the index of f at x is defined to be the number of negative eigenvalues (counted with multiplicities) of H x f . The following lemma tells us Morse functions behave well locally. Lemma 2.1. Let f be a smooth function on M . If d f = 0 at x and H x f is nondegnerate. Then we can find some neighbourhood x ∈ U and φ : U → R m such that in the chart f takes the form k n f ◦ φ − 1 = c − � � x 2 x 2 i + (6) j i =1 j = k +1 3

Proof. This is a local statement, so we might assume p = 0 ∈ R m and ∂ x i f = 0 at 0. Then locally f − c can be written as � f − c = x i x j g ij (7) i,j where g ij = g ji are smooth functions. The symmetric matrix ( g ij ) | 0 is nonde- generate by Morse condition. Since the diagonalizing process is C ∞ , we can find a diffeomorphism defined near 0 such that f takes the required form un- der this chart. More precisely, there is a GL( R , n ) valued smooth function A defined near 0 such that A t ( g ij ) A is of the form diag { 1 , 1 , ..., − 1 , ... − 1 } . Then x �→ A − 1 x is a diffeomorphism near 0. Name the new coordinate y , then f − c = x t ( g ij ( x )) x = y t A t ( g ij ( x ( y )) Ay = � k i − � n i =1 y 2 j = k +1 y 2 j as promised. For further use, we will call the chart given by Lemma 2.1 a Morse chart. Corollary 2.1. A Morse function’s critical points are discrete. If M is compact, then f has only finitely many critical points. Example 2.1. The first example is sphere in R 3 : S 2 := { ( x, y, z ) : x 2 + y 2 + z 2 = 1 } equipped with the ‘height function’ f = z . Then f has a critical point of index 0 at its minimum and a critical point of index 2 at its maximum. Example 2.2. The (probably) most famous example of Morse theory is the embedded Torus T 2 in R 3 equipped with the ‘height function’ f = z . f has a critical point of index 0 at its minimum and a critical point of index 2 at its maximum. Moreover, it has 2 critical points of index 1 corresponding to two ‘saddle points’ of T 2 . Example 2.3. A less trivial Morse function on T 2 is given by f = sin 2 πx + sin 2 πy . This function can be viewed as obtained from ( S 1 , sin 2 πx ) via product construction. Likewise, this function has 1 critical point of index 0, 2 critical points of index 1 and 1 point of index 2. 2.2 Genericness However, so far we haven’t been assured that there exists Morse function on arbitrary manifold. We intend to prove the following theorem. Theorem 2.1. Let M be a compact manifold. The set of Morse function is C 0 dense in C ∞ ( M ) . In other words, any smooth function on M can be uniformly approximated by Morse function. By the following lemma, we reduce problem to embedded manifolds in R n . Lemma 2.2. Any compact manifold M (with or without boundary) admits an embedding into some Euclidean space R N . 4