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Morse-Conley-Floer Homology Morse-Conley-Floer Homology Contents 1 Introduction 1 1.1 Dynamics and Topology . 1 1.2 Classical Morse theory, the half-space approach . 4 1.3 Morse homology . 7 1.4 Conley theory . 11 1.5 Local Morse homology . 16 1.6 Morse-Conley-Floer homology . 18 1.7 Functoriality for flow maps in Morse-Conley-Floer homology . 21 1.8 A spectral sequence in Morse-Conley-Floer homology . 23 1.9 Morse-Conley-Floer homology in infinite dimensional systems . 25 1.10 The Weinstein conjecture and symplectic geometry . 29 1.11 Closed characteristics on non-compact hypersurfaces . 32 I Morse-Conley-Floer Homology 39 2 Morse-Conley-Floer homology 41 2.1 Introduction . 41 2.2 Isolating blocks and Lyapunov functions . 45 2.3 Gradient flows, Morse functions and Morse-Smale flows . 49 2.4 Morse homology . 55 2.5 Morse-Conley-Floer homology . 65 2.6 Morse decompositions and connection matrices . 68 2.7 Relative homology of blocks . 74 3 Functoriality 79 3.1 Introduction . 79 3.2 Chain maps in Morse homology on closed manifolds . 86 i CONTENTS 3.3 Homotopy induced chain homotopies . 97 3.4 Composition induced chain homotopies . 100 3.5 Isolation properties of maps . 103 3.6 Local Morse homology . 109 3.7 Morse-Conley-Floer homology . 114 3.8 Transverse maps are generic . 116 4 Duality 119 4.1 Introduction . 119 4.2 The dual complex . 120 4.3 Morse cohomology and Poincare´ duality . 120 4.4 Local Morse homology . 122 4.5 Duality in Morse-Conley-Floer homology . 122 4.6 Maps in cohomology . 123 4.7 Shriek maps in Morse-Conley-Floer homology . 125 4.8 Other algebraic structures . 125 4.9 Some identities . 127 4.10 Coincidence theory . 127 5 Spectral Sequence 129 5.1 The birth of a spectral sequence . 130 5.2 The Morse-Conley relations via the spectral sequence . 132 5.3 Interpretation of the spectral sequence for Morse-Bott functions 134 5.4 A counter example to misstated Morse-Bott inequalities. 136 II Periodic Orbits on Non-Compact Hypersurfaces 139 6 Periodic orbits 141 6.1 Introduction . 141 6.2 Contact type conditions . 144 6.3 The variational setting . 150 6.4 The Palais-Smale condition . 154 6.5 Minimax characterization . 161 6.6 Some remarks on the geometry and topology of the loop space . 164 6.7 The topology of the hypersurface and its shadow . 166 6.8 The link . 174 6.9 Estimates . 184 6.10 Proof of the main theorem . 187 ii Contents 6.11 Appendix: construction of the hedgehog . 187 6.12 Appendix: local computations . 196 7 Nederlandse samenvatting 199 7.1 Overzicht . 199 7.2 Deel I: Morse-Conley-Floer homologie . 199 7.3 Deel II: Periodieke banen op niet compacte hyperoppervlakken 200 Bibliography 203 iii Preface This thesis consists of two parts. In Part I a homology theory for flows on finite dimensional manifolds is developed, and in Part II criteria for the ex- istence of periodic orbits on non-compact hypersurfaces in Hamiltonian dy- namics are found. The main connection between the two parts are the use of Morse theory in the study of dynamical systems. In Chapter 1 we give an overview of the setting and the history of the subjects, and discuss the results of this thesis more in depth. In Chapter 2 we define Morse-Conley-Floer ho- mology and prove some of its basic properties. This chapter is based on the paper [RV14]. In Chapter 3 we treat functorial aspects of Morse homology, local Morse homology, and Morse-Conley-Floer homology. We apply Morse- Conley-Floer homology to the study Morse decompositions and degenerate variational systems in Chapter 5. In Part II, that consists of Chapter 6, we study the existence of periodic orbits on non-compact energy hypersurfaces. This chapter is based on the paper [VDBPRV13]. v Chapter 1 Introduction Voordat ik een fout maak Maak ik die fout niet Johan Cruijff 1.1 Dynamics and Topology In this thesis two separate problems are studied. We develop a homology the- ory to study differential equations in general, and we find periodic solutions to a particular differential equation in Hamiltonian mechanics. The main con- nection between the results is the use of topological tools –in particular Morse theory– in the study of dynamical systems. An overview of the results of thesis is given in Sections 1.6,1.8,1.7 and 1.11. Let us first comment on the symbiosis of dynamical systems theory and topology. 1.1.1 Topology Topology is the study of that part of shape which is invariant under continu- ous deformations. It should not be confused with geometry, although the two are intimately related. In Figure 1.1 three different spaces are sketched. The sphere and the heart can be deformed into each other and are topologically the same. The torus cannot be deformed into either one of them. A lot of effort is devoted to understand which spaces are essentially the same in this regard. It is hard to imagine all possible deformations of a given space, and 1 1. INTRODUCTION Figure 1.1: A sphere, a heart and a torus. The first two spaces are homeomor- phic (they have the same topology), but the torus has a different topology. To distinguish spaces with different topology, topological invariants are used. In this case the fundamental group distinguishes the spaces: The loop drawn on the right hand side of the torus cannot be shrunk to a point along the surface, while any loop drawn on the sphere or the heart can be contracted to a point. to show that two spaces are distinct one defines and computes invariants. In- variants are usually algebraic objects, such as numbers or groups, that do not change under any deformation. The sphere and the torus are distinguished by the fundamental group. Any loop on the sphere can be contracted along the surface to a point, but this is impossible for some loops on the torus. The fundamental group describes exactly which loops are contractible and which are not. Because the fundamental groups of the sphere and the torus differs, it follows that these spaces are inequivalent. The Euler characteristic is an- other invariant which distinguishes the spaces. The Euler characteristic of a 2-dimensional space is computed as follows c(M) = number of triangles ´ number of edges + number of vertices. (1.1) where the numbers are computed for a triangulation of the space, see Figure 1.2. For the sphere one computes, independent of the chosen tri- angulation, that the Euler characteristic is 2, while for the torus the Euler characteristic is 0, which also shows that the spaces differ. 2 1.1. Dynamics and Topology Figure 1.2: Triangulations of the sphere and the torus. From a triangulation the Euler characteristic can be computed, cf. Equation (1.1). The Euler char- acteristic does not depend on the triangulation, but only on the surface. The sphere has Euler characteristic 2 and the torus has Euler characteristic 0. 1.1.2 Topology and dynamical systems Many phenomena in mathematics, physics, biology, chemistry and the social sciences are described by differential equations. A differential equation relates a quantity with its various derivatives. For example the rate of growth of a population is proportional to the size of the population, and the acceleration experienced by the moon is related to the distance of the moon to the earth. Differential equations describing evolution in time lead to dynamical systems. The study of (algebraic) topology has its origins in dynamical systems the- ory [Poi10]. Any non-trivial dynamical system is too complex to understand in full detail. The exact evolution of a dynamical system is extremely sensi- tive to initial conditions and model parameters. We do not pretend we can measure the model parameters and initial conditions exactly, so it is impor- tant to understand which part of the dynamical behavior is preserved under small perturbations of the system in question. Topology is precisely about such robust information. In Part I we define topological invariants for dynam- ical systems, which capture qualitative information of the gradient behavior of arbitrary flows. In Part II we study a problem in conservative dynamics. Topological tools are used to detect periodic orbits of a specific system. In this thesis topology is used as a tool to study dynamical systems, but the synergy of topology and dynamical systems goes both ways: well chosen dynamical systems can give deep insight into pure topological problems, see 3 1. INTRODUCTION for example [Bot56, Mil63, Mil65]. 1.1.3 Outline of the introduction We first give some background material for Part I: In Section 1.2 we discuss classical Morse theory and Morse homology in Section 1.3. Conley theory is reviewed next in Section 1.4. We then proceed, using the notion of local Morse homology of Section 1.5 to Section 1.6 to the the the first result of the thesis, which is the definition of Morse-Conley-Floer homology. Functorial proper- ties and applications of Morse-Conley-Floer homology are given in Sections and 1.7 and 1.8. Part II is concerned with the search of periodic orbits of Hamiltonian sys- tems on non-compact energy hypersurfaces. We briefly discuss the setting and history of Hamiltonian dynamics and the Weinstein conjecture in Section 1.10 after which we describe the contribution of this thesis to this problem in Sec- tion 1.11. 1.2 Classical Morse theory, the half-space approach Morse realized [Mor25] that there are strong connections between the topol- ogy of a space and the number and type of critical points of functions defined on this space.
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