Symposium on Symplectic Geometry 1997-2000 Report
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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 . -
MORSE-BOTT HOMOLOGY 1. Introduction 1.1. Overview. Let Cr(F) = {P ∈ M| Df P = 0} Denote the Set of Critical Points of a Smooth
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 362, Number 8, August 2010, Pages 3997–4043 S 0002-9947(10)05073-7 Article electronically published on March 23, 2010 MORSE-BOTT HOMOLOGY AUGUSTIN BANYAGA AND DAVID E. HURTUBISE Abstract. We give a new proof of the Morse Homology Theorem by con- structing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse- Smale and to the chain complex of smooth singular N-cube chains when the function is constant. We show that the homology of the chain complex is independent of the Morse-Bott-Smale function by using compactified moduli spaces of time dependent gradient flow lines to prove a Floer-type continuation theorem. 1. Introduction 1.1. Overview. Let Cr(f)={p ∈ M| df p =0} denote the set of critical points of a smooth function f : M → R on a smooth m-dimensional manifold M. A critical point p ∈ Cr(f) is said to be non-degenerate if and only if df is transverse to the ∗ zero section of T M at p. In local coordinates this is equivalent to the condition 2 that the m × m Hessian matrix ∂ f has rank m at p. If all the critical points ∂xi∂xj of f are non-degenerate, then f is called a Morse function. A Morse function f : M → R on a finite-dimensional compact smooth Riemann- ian manifold (M,g) is called Morse-Smale if all its stable and unstable manifolds intersect transversally. Such a function gives rise to a chain complex (C∗(f),∂∗), called the Morse-Smale-Witten chain complex. -
Appendix an Overview of Morse Theory
Appendix An Overview of Morse Theory Morse theory is a beautiful subject that sits between differential geometry, topol- ogy and calculus of variations. It was first developed by Morse [Mor25] in the middle of the 1920s and further extended, among many others, by Bott, Milnor, Palais, Smale, Gromoll and Meyer. The general philosophy of the theory is that the topology of a smooth manifold is intimately related to the number and “type” of critical points that a smooth function defined on it can have. In this brief ap- pendix we would like to give an overview of the topic, from the classical point of view of Morse, but with the more recent extensions that allow the theory to deal with so-called degenerate functions on infinite-dimensional manifolds. A compre- hensive treatment of the subject can be found in the first chapter of the book of Chang [Cha93]. There is also another, more recent, approach to the theory that we are not going to touch on in this brief note. It is based on the so-called Morse complex. This approach was pioneered by Thom [Tho49] and, later, by Smale [Sma61] in his proof of the generalized Poincar´e conjecture in dimensions greater than 4 (see the beautiful book of Milnor [Mil56] for an account of that stage of the theory). The definition of Morse complex appeared in 1982 in a paper by Witten [Wit82]. See the book of Schwarz [Sch93], the one of Banyaga and Hurtubise [BH04] or the survey of Abbondandolo and Majer [AM06] for a modern treatment. -
FLOER THEORY and ITS TOPOLOGICAL APPLICATIONS 1. Introduction in Finite Dimensions, One Way to Compute the Homology of a Compact
FLOER THEORY AND ITS TOPOLOGICAL APPLICATIONS CIPRIAN MANOLESCU Abstract. We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then describe Floer stable homotopy types, the related Pin(2)-equivariant Seiberg-Witten Floer homology, and its application to the triangulation conjecture. 1. Introduction In finite dimensions, one way to compute the homology of a compact, smooth manifold is by Morse theory. Specifically, we start with a a smooth function f : X ! R and a Riemannian metric g on X. Under certain hypotheses (the Morse and Morse-Smale conditions), we can form a complex C∗(X; f; g) as follows: The generators of C∗(X; f; g) are the critical points of f, and the differential is given by X (1) @x = nxy · y; fyjind(x)−ind(y)=1g where nxy 2 Z is a signed count of the downward gradient flow lines of f connecting x to y. The quantity ind denotes the index of a critical point, that is, the number of negative eigenvalues of the Hessian (the matrix of second derivatives of f) at that point. The homology H∗(X; f; g) is called Morse homology, and it turns out to be isomorphic to the singular homology of X [Wit82, Flo89b, Bot88]. Floer homology is an adaptation of Morse homology to infinite dimensions. It applies to certain classes of infinite dimensional manifolds X and functions f : X ! R, where at critical points of f the Hessian has infinitely many positive and infinitely many negative eigenvalues. -
![Arxiv:1302.1025V3 [Math.SG] 22 Oct 2015](https://docslib.b-cdn.net/cover/4961/arxiv-1302-1025v3-math-sg-22-oct-2015-1534961.webp)
Arxiv:1302.1025V3 [Math.SG] 22 Oct 2015
HAMILTONIAN FLOER HOMOLOGY FOR COMPACT CONVEX SYMPLECTIC MANIFOLDS SERGEI LANZAT Abstract. We construct absolute and relative versions of Hamiltonian Floer ho- mology algebras for strongly semi-positive compact symplectic manifolds with con- vex boundary, where the ring structures are given by the appropriate versions of the pair-of-pants products. We establish the absolute and relative Piunikhin{Salamon{ Schwarz isomorphisms between these Floer homology algebras and the corresponding absolute and relative quantum homology algebras. As a result, the absolute and relative analogues of the spectral invariants on the group of compactly supported Hamiltonian diffeomorphisms are defined. 1. Introduction. In [14] U. Frauenfelder and F. Schlenk defined the Floer homology for weakly exact compact convex symplectic manifolds. The authors also established the Piunikhin{ Salamon{Schwarz (PSS) isomorphism between the ring of Floer homology and the ring of Morse homology of such manifolds. This in turn led to the construction of the spectral invariants on the group of compactly supported Hamiltonian diffeomor- phisms analogous to the spectral invariants constructed by M. Schwarz in [33] and by Y.- G. Oh in [23] for closed symplectic manifolds. We extend the definitions and the constructions of U. Frauenfelder and F. Schlenk to the case of strongly semi-positive compact convex symplectic manifolds. As a result, we get absolute and relative ver- sions of Hamiltonian Floer homology algebras, where the ring structures are given by the appropriate versions of the pair-of-pants products. We establish the absolute and relative Piunikhin{Salamon{Schwarz isomorphisms between these Floer homology al- gebras and the corresponding absolute and relative quantum homology algebras. -
Parameterized Morse Theory in Low-Dimensional and Symplectic Topology
Parameterized Morse theory in low-dimensional and symplectic topology David Gay (University of Georgia), Michael Sullivan (University of Massachusetts-Amherst) March 23-March 28, 2014 1 Overview and highlights of workshop Morse theory uses generic functions from smooth manifolds to R (Morse functions) to study the topology of smooth manifolds, and provides, for example, the basic tool for decomposing smooth manifolds into ele- mentary building blocks called handles. Recently the study of parameterized families of Morse functions has been applied in new and exciting ways to understand a diverse range of objects in low-dimensional and sym- plectic topology, such as Morse 2–functions in dimension 4, Heegaard splittings in dimension 3, generating families in contact and symplectic geometry, and n–categories and topological field theories (TFTs) in low dimensions. Here is a brief description of these objects and the ways in which parameterized Morse theory is used in their study: A Morse 2–function is a generic smooth map from a smooth manifold to a smooth 2–dimensional man- ifold (such as R2). Locally Morse 2–functions behave like generic 1–parameter families of Morse functions, but globally they do not have a “time” direction. The singular set of a Morse 2–function is 1–dimensional and maps to a collection of immersed curves with cusps in the base, the “graphic”. Parameterized Morse theory is needed to understand how Morse 2–functions can be used to decompose and reconstruct smooth manifolds [20], especially in dimension 4 when regular fibers are surfaces, to understand uniqueness statements for such decompositions [21], and to use such decompositions to produce computable invariants. -
2006-2007 Graduate Studies in Mathematics Handbook
2006-2007 GRADUATE STUDIES IN MATHEMATICS HANDBOOK TABLE OF CONTENTS 1. INTRODUCTION .................................................................................................................... 1 2. DEPARTMENT OF MATHEMATICS ................................................................................. 2 3. THE GRADUATE PROGRAM.............................................................................................. 5 4. GRADUATE COURSES ......................................................................................................... 8 5. RESEARCH ACTIVITIES ................................................................................................... 24 6. ADMISSION REQUIREMENTS AND APPLICATION PROCEDURES ...................... 25 7. FEES AND FINANCIAL ASSISTANCE............................................................................. 25 8. OTHER INFORMATION..................................................................................................... 27 APPENDIX A: COMPREHENSIVE EXAMINATION SYLLABI ............................................ 31 APPENDIX B: APPLIED MATHEMATICS COMPREHENSIVE EXAMINATION SYLLABI ............................................................................................................................................................. 33 APPENDIX C: PH.D. DEGREES CONFERRED FROM 1994-2006........................................ 35 APPENDIX D: THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES........................................................................................................................................ -
Morse Functions and Morse Homology
Morse Functions and Morse Homology Abhineet Agarwal/Sabrina Victor April 2019 1 Morse Functions Let X be a finite dimensional compact smooth manifold, and let f : X ! R be a smooth function. Definition 1.1. A critical point of f is a point p 2 X such that dfjp= 0 : TpX ! R: Also define Crit(f) to be the set of critical points of f. Figure 1: Critical Points of Torus and Sphere ∗ If p is a critical point, we also define the Hessian H(f,p): TpX ! Tp X as follows: Let r be any connection on TX and if v 2 TpX, define: H(f; p)(v) = rv(df) Recall that the connection defines the notion of parallel transport on a vector bundle. Also note that in our definition of the Hessian does not depend on our choice of connection. This is because df vanishes at p and the difference 1 between any two connections is a tensor. Next let x1; :::xn are local coordinates @ ∗ for X near p, then with respect to the bases f g and fdxig for TpX; T X @xi p 2 respectively, the Hessian is given by the matrix @ (f) . This is exactly what we @xi@xj expect from the formula for the hessian above. Remark. Note that the hessian matrix is obviously a symmetric matrix. Then it is a known fact that the hessian is a self-adjoint map from TpX to itself. Definition 1.2. A critical point is nondegenerate if the Hessian does not have zero eigenvalue. Definition 1.3. -
Morse Theory for Lagrange Multipliers and Adiabatic Limits
MORSE THEORY FOR LAGRANGE MULTIPLIERS AND ADIABATIC LIMITS STEPHEN SCHECTER AND GUANGBO XU Abstract. Given two Morse functions f; µ on a compact manifold M, we study the Morse homology for the Lagrange multiplier function on M ×R which sends (x; η) to f(x)+ηµ(x). Take a product metric on M ×R, and rescale its R-component by a factor λ2. We show that generically, for large λ, the Morse-Smale-Witten chain complex is isomorphic to the one for f and the metric restricted to µ−1(0), with grading shifted by one. On the other hand, for λ near 0, we obtain another chain complex, which is geometrically quite different but has the same singular homology as µ−1(0). Our proofs contain both the implicit function theorem on Banach manifolds and geometric singular perturbation theory. Contents 1. Introduction1 2. Morse homology and Morse-Smale-Witten complex of (F; gλ)4 3. Adiabatic limit λ ! 1 9 4. Fast-slow system associated to the λ ! 0 limit 16 5. Adiabatic limit λ ! 0 23 6. Basic Morse theory and the chain complex C0 34 Appendix A. Geometric singular perturbation theory 36 Appendix B. Choosing the center manifold 41 References 41 1. Introduction Let M be a compact manifold. Suppose f and µ are Morse functions on M, and 0 is a regular value of µ. Then we consider the Lagrange multiplier function F : M × R ! R; (x; η) 7! f(x) + ηµ(x): Date: November 10, 2012. 1991 Mathematics Subject Classification. 57R70, 37D15. Key words and phrases. -
Morse Homology
Morse Homology Trabajo de Tesis presentado al Departamento de Matem´aticas Presentado por: Juanita Pinz´on Caicedo Asesor: Bernardo Uribe Para optar por el t´ıtulo de Matem´atica Universidad de Los Andes Departamento de Matem´aticas Julio 2007 Contents Introduction ii 1 Preliminary Notions 1 2 Morse Functions 4 2.1 Gradient Flow Lines . 4 2.2 Stable, Unstable submanifolds . 6 2.3 Homotopy . 7 3 Morse-Smale Functions 15 4 The Morse Homology Theorem 18 5 Examples 26 5.1 Complex Projective Spaces . 26 5.2 Real Projective Spaces . 29 A Morse-Bott 36 i Introduction In the present work we study Morse-Smale functions over Riemannian manifolds and the Morse- Smale chain complexes that can be assigned to each one of them. Morse functions are real-valued functions with isolated and nondegenerate critical points, they are the starting point of Morse Theory. The main objective is to prove Morse Homology Theorem, a theorem that relates certain topological properties of a manifold to some information obtained from a real-valued function defined on it. The theorem says that the homology groups found using a triangulation of the manifold defined by the gradient flow lines of Morse-Smale functions are isomorphic to the singular homology groups. Therefore, Morse Homology Theorem allows us to calculate the homology groups of an unknown manifold by studying much simpler objects, namely, a Morse- Smale function and its critical points. Given their importance, Morse functions, critical points and gradient flow lines are the topic of section two. That section also contains a series of results which follow from a detailed observation of those functions. -
Morse-Conley-Floer Homology
Morse-Conley-Floer Homology T.O. Rot ([email protected]) and R.C.A.M. Vandervorst ([email protected])1 Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, the Netherlands. Abstract. The gradient flow of a Morse function on a smooth closed manifold generates, under suitable transversality assumptions, the Morse- Smale-Witten complex. The associated Morse homology is an invariant for the manifold, and equals the singular homology, which yields the classical Morse relations. In this paper we define Morse-Conley-Floer homology, which is an analogous homology theory for isolated invariant sets of smooth, not necessarily gradient-like, flows. We prove invariance properties of the Morse-Conley-Floer homology, and show how it gives rise to the Morse-Conley relations. AMS Subject Class: 37B30, 37C10, 58E05 Keywords: Morse homology, Lyapunov functions, Conley index theory 1. INTRODUCTION arXiv:1305.4074v2 [math.DS] 23 Apr 2014 The aim of this paper is to define an analogue of Morse homology for isolated invariant sets of smooth, not necessarily gradient-like, flows. We first recall how the gradient flow of a Morse gives rise to Morse homology. 1.1. Morse Homology. On a smooth, closed, m-dimensional manifold M the Morse-Smale pairs (f; g), consisting of a smooth function f : M ! R 1Thomas Rot supported by NWO grant 613.001.001. 1 2 T.O. ROT AND R.C.A.M. VANDERVORST and a Riemannian metric g, are characterized by the property that all critical points of f are non-degenerate and all their stable and unstable manifolds with respect to the negative g-gradient flow intersect transversally. -
J-Holomorphic Curves and Quantum Cohomology
J-holomorphic Curves and Quantum Cohomology by Dusa McDuff and Dietmar Salamon May 1995 Contents 1 Introduction 1 1.1 Symplectic manifolds . 1 1.2 J-holomorphic curves . 3 1.3 Moduli spaces . 4 1.4 Compactness . 5 1.5 Evaluation maps . 6 1.6 The Gromov-Witten invariants . 8 1.7 Quantum cohomology . 9 1.8 Novikov rings and Floer homology . 11 2 Local Behaviour 13 2.1 The generalised Cauchy-Riemann equation . 13 2.2 Critical points . 15 2.3 Somewhere injective curves . 18 3 Moduli Spaces and Transversality 23 3.1 The main theorems . 23 3.2 Elliptic regularity . 25 3.3 Implicit function theorem . 27 3.4 Transversality . 33 3.5 A regularity criterion . 38 4 Compactness 41 4.1 Energy . 42 4.2 Removal of Singularities . 43 4.3 Bubbling . 46 4.4 Gromov compactness . 50 4.5 Proof of Gromov compactness . 52 5 Compactification of Moduli Spaces 59 5.1 Semi-positivity . 59 5.2 The image of the evaluation map . 62 5.3 The image of the p-fold evaluation map . 65 5.4 The evaluation map for marked curves . 66 vii viii CONTENTS 6 Evaluation Maps and Transversality 71 6.1 Evaluation maps are submersions . 71 6.2 Moduli spaces of N-tuples of curves . 74 6.3 Moduli spaces of cusp-curves . 75 6.4 Evaluation maps for cusp-curves . 79 6.5 Proofs of the theorems in Sections 5.2 and 5.3 . 81 6.6 Proof of the theorem in Section 5.4 . 82 7 Gromov-Witten Invariants 89 7.1 Pseudo-cycles .