A Quick View of Lagrangian Floer Homology Andr´es Pedroza Abstract In this note we present a brief introduction to Lagrangian Floer homology and its relation with the solution of Arnol’d conjecture, on the minimal number of non-degenerate fixed points of a Hamiltonian diffeomorphism. We start with the basic definition of critical point on smooth manifolds, in oder to sketch some aspects of Morse theory. Introduction to the basics concepts of symplectic geometry are also included, with the idea of understanding the statement of Arnol’d Conjecture and how is related to the intersection of Lagrangian submanifolds. 1 Introduction Many elegant results in mathematics have to deal with the fixed-point-set of a function. For ex- ample: Brouwer fixed-point theorem, Lefschetz fixed-point theorem, Banach fixed-point theorem and Poincar´e-Birkhoff theorem, just to name a few. Furthermore, these results are fundamental in their own area of mathematics and have interesting consequences in diverse areas of mathematics; differential equations, topology and game theory among others. Symplectic geometry has its own fixed-point theorem, which was conjectured by V. Arnol’d [1] in 1965. The Arnol’d Conjecture was motivated by Poincar´e-Birkhoff theorem: An area-preserving diffeomorphism of the annulus which maps the boundary circles to themselves in different direction, must have at least two fixed points. The generalization of Poincar´e-Birkhoff theorem fits in symplectic geometry and not in volume- preserving geometry. The Arnol’d Conjecture establishes a lower bound on the number of fixed points a Hamiltonian diffeomorphism in terms of the topology of the manifold. The fixed points of arXiv:1701.02293v1 [math.SG] 9 Jan 2017 a Hamiltonian diffeomorphism, (in fact any diffeomorphisms) can be seen as the intersection of its graph and the diagonal. In the context of symplectic geometry, is the intersection of two Lagrangian submanifolds. In 1987, A. Floer [10] developed a homological theory that focused on the intersection of La- grangian submanifolds. In particular, under some hypotheses, he proved the Arnol’d Conjecture for a particular class of closed symplectic manifolds. This theory is called Lagrangian Floer homology. In these notes we sketch how Lagrangian Floer homology is defined. In fact we review some aspects of Morse theory from its basics; like non-degenerate critical points, the Hessian, flow lines of the gradient vector field up to Morse homology. The reason being, that Lagrangian Floer homology emulates in many aspects Morse homology. Also we cover the basics of symplectic manifolds and Hamiltonian diffeomorphisms. The last section deals with Lagrangian Floer homology and how it is used to prove the Arnol’d Conjecture. For the basic notions of differential geometry the reader can look at [41]; for the aspects of symplectic geometry [7] and [23]; and also [24] where the analytical aspect of holomorphic curves Andr´es Pedroza Facultad de Ciencias, Universidad de Colima, Bernal D´ıaz del Castillo No. 340, Colima, Col., Mexico 28045 e-mail: [email protected] 1 2 Andr´es Pedroza is covered. For details and proofs on the construction of Lagrangian Floer homology see [3], [32] and [33]. For an excellent introduction to Fukaya categories see a[4]; and [40] for a detail treatment of the subject. These lecture notes are based on a course given at the 7th. Mini Meeting on Differential Ge- ometry held at CIMAT in February 2015. The author wishes thank the organizers and participants for the pleasent atmospere. Finally the author was partially supported by a CONACYT grant CB- 2010/151846. 2 Morse-Smale Functions Let M be a smooth manifold of dimension n and f : M → R a smooth function. A point p ∈ M is called a critical point of f if the differential d fp : TpM → R at p is the zero map. Denote by Crit( f ) the set of critical points of f . Notice that Crit( f ) can be the empty set, however if M is compact then it is not empty, since a smooth function on M has a maximum and a minimum. Let p ∈ M be a critical point of f and (x1,...,xn) a coordinate chart about p. The Hessian matrix of f at p relative to the chart (x1,...,xn), is the n × n matrix ∂ 2 f Hess( f , p)= (p) . ∂x ∂x i j A critical point p is said to be non-degenerate if the matrix Hess( f , p) is non-singular. Note that the Hessian matrix is symmetric, hence if it is non-singular its eigenvalues are real and non-zero. The index of f at a non-degenerate critical point p, which is denoted by ind( f ; p), is defined as the number of negative eigenvalues of the Hessian matrix at p. The definition of the index at a non-degenerate critical point given above depends on the coordi- nate system; however it can be shown that the is independent of the coordinate system about the the critical point.There is an alternative definition of the index of a function at a non-degenerate critical point, that does not needs a coordinate system. For a critical point p ∈ M of f define the bilinear form 2 d fp : TpM × TpM → R 2 as d fp (X,Y ) := X(Y f ), where Y is any vector field on M whose value at p is Y. Notice that since p 2 is a critical point of f , the bilinear form d fp is symmetric, e e 0 = d fp[X,Y ] = [X,Y ]p( f )= Xp(Y f ) −Yp(X f ). 2 In this context, p is called non-degeneratee e eife the bilineare symmetrice e e form d fp is non-degenerate. The index of f at p is defined as the number of negative eigenvalues of the symmetric bilinear form 2 d fp . The two definitions given of non-degenerate critical point agree. The same applies for the two definitions of the index of a non-degeneratecritical point. For further details, see [3, Ch. 1] and [29]. Definition 1. A smooth function f : M → R for which all of its critical points are non-degenerate is called a Morse function. Now we consider some examples in the case when M = R2. The origin is the only critical point of the function f (x,y)= x2 + y2. Moreover is a non-degenerate critical point and its index is zero. The origin is also the only non-degenerate critical point of the functions g(x,y)= x2 − y2 and h(x,y)= −x2 −y2. In these cases the index at the origin is 1 and 2 respectively. These three examples describe the general behavior of a function on R2 near the origin when it is a non-degenerate critical point. The precise statement on the behavior of a function near a non-degenerate critical point is given by Morse lemma. Theorem 1 (Morse Lemma). Let f : Rn → R be a smooth function such that the origin is a non- degenerate critical point of index λ. Then there exists a coordinate chart (u1,...,un) about the origin A Quick View of Lagrangian Floer Homology 3 such that 2 2 2 2 f (u1,...,un)= f (0) − u1 −···− uλ + uλ +1 + ··· + un. It goes without saying that Morse lemma also holds for smooth functions defined on arbitrary manifolds. A consequence of Morse lemma, as stated above, is that there exists a neighborhood about the origin in Rn so that it is the only critical point in such neighborhood. Corollary 1. Non-degenerate critical points of a smooth function are isolated. Note that a Morse function defined on a compact manifold has finitely many critical points. The main reason behind the study of Morse functions is to understand the topology of the mani- fold. Thus for a smooth function f : M → R and a ∈ R define the level set Ma := {x ∈ M| f (x) ≤ a}⊂ M. Notice that when a0 is the absolute minimum of f , then Ma is empty for every a < a0. And in the case when a1 is the absolute maximum of f , then Ma = M for every a1 ≤ a. Now we explain what we mean by understandingthe topology of the manifold; one aspect is that the manifold can be constructed from information from a fixed Morse function on it. Consider a compact manifold M, a smooth Morse function f : M → R and for simplicity assume that p0,..., pk are all the critical points, with λi = ind( f ; pi) and λi < λi+1 for i ∈{0,...,k−1}. Thus f achieves its minimum at p0 and ind( f ; p0)= 0; and it achieves its maximum at pk and ind( f ; pk)= n. In orderto build the manifold M from the critical points of f , one starts with the point Mf (p0) = {p0}. Then from Theorem 2 below, it follows that Ma has the same homotopy type has Mf (p0) for a ∈ ( f (p0), f (p1)). By an λ-cell we mean a space homeomorphic to the closed ball of dimension λ. Hence, Ma is homeomorphic to the n-cell for a ∈ ( f (p0), f (p1)). The next step is to analyze the next non-degenerte critical point p1 ∈ M. In this case for a ∈ λ ( f (p1), f (p2)), it follows that Ma has the same homotopytype as Mf (p0) with a 1-cell attached. That ∂ is Ma ≃ Mf (p0) ∪g eλ1 , where g : (eλ1 ) → Mf (p0) is a gluing function. This process continues at every critical point. That is for a ∈ ( f (pi), f (pi+1)) the space Ma as the same homotopytype has to Mf (pi−1) λ λ with an attached i-cell.