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
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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 . -
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
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. -
African Mathematical Union Amuchma-Newsletter-30
AFRICAN MATHEMATICAL UNION Commission on the History of Mathematics in Africa (AMUCHMA) AMUCHMA-NEWSLETTER-30 _______________________________________________________________ Special Issue: Over 600 Examples of African Doctorates in Mathematics TABLE OF CONTENTS page 1. Objectives of AMUCHMA 2 2. Examples of African Doctorates in Mathematics 2 3. Examples of African Mathematical Pioneers in the 20th 28 Century 4. Do you want to receive the next AMUCHMA-Newsletter 30 5. AMUCHMA-Newsletter website 30 _______________________________________________________________ Maputo (Mozambique), 29.04.2005 AMUCHMA 1. OBJECTIVES The A.M.U. Commission on the History of Mathematics in Africa (AMUCHMA), formed in 1986, has the following objectives: a. To improve communication among those interested in the history of mathematics in Africa; b. To promote active cooperation between historians, mathematicians, archaeologists, ethnographers, sociologists, etc., doing research in, or related to, the history of mathematics in Africa; c. To promote research in the history of mathematics in Africa, and the publication of its results, in order to contribute to the demystification of the still-dominant Eurocentric bias in the historiography of mathematics; d. To cooperate with any and all organisations pursuing similar objectives. The main activities of AMUCHMA are as follows: a. Publication of a newsletter; b. Setting up of a documentation centre; c. Organisation of lectures on the history of mathematics at national, regional, continental and international congresses and conferences. 2. OVER 600 EXAMPLES OF AFRICAN DOCTORATES IN MATHEMATICS (compiled by Paulus Gerdes) Appendix 7 of the first edition of our book Mathematics in African History and Culture: An Annotated Bibliography (Authors: Paulus Gerdes & Ahmed Djebbar, African Mathematical Union, Cape Town, 2004) contained a list of “Some African mathematical pioneers in the 20th century” (reproduced below in 3). -
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. -
The Abdus Sallam International Titn) Centre for Theoretical Physics
the II abdus sallam wited m1o.. III Hill IIII IIII educational, scientific and cultural international XA0404087 Organization titN) centre international ammic e,.era ag.!ncy for theoretical physics ON SYMPLECTOMORPHISMS OF THE SYMPLE CTISATION OF A COMPACT CONTACT MANIFOLD Augustin Banyaga Available at: http: //wvw. ictp. trieste. W-pub-of f IC/2004/9 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ON SYMPLECTOMORPHISMS OF TIE SYMPLECTISATION OF A COMPACT CONTACT MANIFOLD Augustin Banyaga Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA and The Abdus Salam International Centre for Theoretical Physics, 7Weste, Italy. Abstract Let (N, a) be a compact contact manifold and (N x K d(e'a)) its symplectisation. We show that the group G which is the identity component in te group of symplectic diffeomorphisms of (N x , d(eta)) that cover diffeomorphisms of N x S' is simple, by showing that G is isomorphic to the kernel of the Calabi homomorphism of the associated locally conformal symplectic structure. MIRAMARE - TRIESTE March 2004 1. Introduction and statements of the results The structure of the group of compactly supported symplectic diffeomorphisms of a symplec- tic manifold is well understood [1], see also 2 For instance, if M, Q) is a compact symplec- tic manifold, the commutator subgroup [Diffn(M)oDiffn(M)o] of the identity component Dif fo (M)o in the group of all symplectic diffeomorphisms, is the kernel of a homomorphism from Dif fn(M)o to a quotient of H'(M, R) (the Calabi homomorphism) and it is a simple group. -
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. -
Toward Hamiltonian Dynamics with Continuous Hamiltonians
Research by KIAS Visitors Research by KIAS Visitors Newsletter Vol 3 Vol Newsletter The KIAS Research by KIAS Visitors Research Toward Hamiltonian Dynamics (2008), 217-229, [arXiv:grqc/0608118]. strongly coupled quantum field theory and with continuous Hamiltonians [9] S. Hollands, A. Ishibashi and R. M. Wald, A higher condensed matter which rely on the fascinating dimensional stationary rotating black hole must be subject of black holes in higher dimensions. axisymmetric, Commun. Math. Phys. 271, 699 (2007) [10] R. Emparan, T. Harmark, V. Niarchos and N. A. Obers, Blackfolds, Phys. Rev. Lett. 102, 191301 (2009) Augustin Banyaga [1] R. Penrose, Gravitational Collapse and Space- [arXiv:0902.0427 [hep-th]]. KIAS Visitor (Pennsylvania State University) Time Singularities, Phys.Rev.Lett. 14, 57 (1965). [11] R. Emparan, T. Harmark, V. Niarchos, N. A. [2] Roy P. Kerr, Gravitational Field of a Spinning Mass Obers and M. J. Rodriguez, The Phase Structure of as an example of Algebraically Special Metrics, Higher-Dimensional Black Rings and Black Holes, Phys.Rev.Lett. 11, 237 (1963). JHEP 0710, 110 (2007) [arXiv:0708.2181 [hep-th]]. [3] J. Michell, Letter by Rev. John Michell to Herny [12] M. Caldarelli, R. Emparan and M. J. Rodriguez, Abstract. In this expository note, we define continuous Hamiltonian dynamical systems Cavendish, Phil.Trans. Royal Soc. 75, 35 (1784). Black Rings in (Anti)-deSitter space, JHEP 0811, 011 Pierre S. Laplace, Exposition du systeme du Monde, (2008) [arXiv:0806.1954 [hep-th]]. and formulate some basic notions in their study. (1796). [13] H. Elvang and P. Figueras, Black Saturn, JHEP [4] Karl Schwarzschild, On the gravitational field 0705, 050 (2007) [arXiv:hep-th/0701035]. -
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.