FLOER THEORY and ITS TOPOLOGICAL APPLICATIONS 1. Introduction in Finite Dimensions, One Way to Compute the Homology of a Compact
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FLOER HOMOLOGY and MIRROR SYMMETRY I Kenji FUKAYA 0
FLOER HOMOLOGY AND MIRROR SYMMETRY I Kenji FUKAYA Abstract. In this survey article, we explain how the Floer homology of Lagrangian submanifold [Fl1],[Oh1] is related to (homological) mirror symmetry [Ko1],[Ko2]. Our discussion is based mainly on [FKO3]. 0. Introduction. This is the first of the two articles, describing a project in progress to study mirror symmetry and D-brane using Floer homology of Lagrangian submanifold. The tentative goal, which we are far away to achiev, is to prove homological mirror symmetry conjecture by M. Kontsevich (see 3.) The final goal, which is yet very § very far away from us, is to find a new concept of spaces, which is expected in various branches of mathematics and in theoretical physics. Together with several joint authors, I wrote several papers on this project [Fu1], [Fu2], [Fu4], [Fu5], [Fu6], [Fu7], [FKO3], [FOh]. The purpose of this article and part II, is to provide an accesible way to see the present stage of our project. The interested readers may find the detail and rigorous proofs of some of the statements, in those papers. The main purose of Part I is to discribe an outline of our joint paper [FKO3] which is devoted to the obstruction theory to the well-definedness of Floer ho- mology of Lagrangian submanifold. Our emphasis in this article is its relation to mirror symmetry. So we skip most of its application to the geometry of Lagrangian submanifolds. In 1, we review Floer homology of Lagrangian submanifold in the form intro- § duced by Floer and Oh. They assumed various conditions on Lagrangian subman- ifold and symplectic manifold, to define Floer homology. -
Ciprian Manolescu
A combinatorial approach to Heegaard Floer invariants Ciprian Manolescu UCLA July 5, 2012 Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 1 / 34 Motivation (4D) Much insight into smooth 4-manifolds comes from PDE techniques: gauge theory and symplectic geometry. Donaldson (1980’s): used the Yang-Mills equations to get constraints on the intersection forms of smooth 4-manifolds, etc. Solution counts −→ Donaldson invariants, able to detect exotic smooth structures on closed 4-manifolds. Seiberg-Witten (1994): discovered the monopole equations, which can replace Yang-Mills for most applications, and are easier to study. Solution counts −→ Seiberg-Witten invariants. Ozsv´ath-Szab´o (2000’s): developed Heegaard Floer theory, based on counts of holomorphic curves in symplectic manifolds. Their mixed HF invariants of 4-manifolds are conjecturally the same as the Seiberg-Witten invariants, and can be used for the same applications (in particular, to detect exotic smooth structures). Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 2 / 34 Motivation (3D) All three theories (Yang-Mills, Seiberg-Witten, Heegaard-Floer) also produce invariants of closed 3-manifolds, in the form of graded Abelian groups called Floer homologies. These have applications of their own, e.g.: What is the minimal genus of a surface representing a given homology class in a 3-manifold? (Kronheimer-Mrowka, Ozsv´ath-Szab´o) Does a given 3-manifold fiber over the circle? (Ghiggini, Ni) In dimension 3, the Heegaard-Floer and Seiberg-Witten Floer homologies are known to be isomorphic: work of Kutluhan-Lee-Taubes and Colin-Ghiggini-Honda, based on the relation to Hutchings’s ECH. -
Heegaard Floer Homology
Heegaard Floer Homology Joshua Evan Greene Low-dimensional topology encompasses the study of man- This is a story about Heegaard Floer homology: how it ifolds in dimension four and lower. These are the shapes developed, how it has evolved, what it has taught us, and and dimensions closest to our observable experience in where it may lead. spacetime, and one of the great lessons in topology during the 20th century is that these dimensions exhibit unique Heegaard... phenomena that render them dramatically different from To set the stage, we return to the beginning of the 20th higher ones. Many outlooks and techniques have arisen in century, when Poincar´eforged topology as an area of in- shaping the area, each with its own particular strengths: hy- dependent interest. In 1901, he famously and erroneously perbolic geometry, quantum algebra, foliations, geometric asserted that, as is the case for 1- and 2-manifolds, any 3- analysis,... manifold with the same ordinary homology groups as the Heegaard Floer homology, defined at the turn of the 3-dimensional sphere 푆3 is, in fact, homeomorphic to 푆3. 21st century by Peter Ozsv´ath and Zolt´an Szab´o, sits By 1905, he had vanquished his earlier claim by way of amongst these varied approaches. It consists of a power- an example that now bears his name: the Poincar´ehomol- ful collection of invariants that fit into the framework ofa ogy sphere 푃3. We will describe this remarkable space in (3 + 1)-dimensional topological quantum field theory. It a moment and re-encounter it in many guises. -
Fall 2017 MATH 70330 “Intermediate Geometry and Topology” Pavel Mnev Detailed Plan. I. Characteristic Classes. (A) Fiber/Vec
Fall 2017 MATH 70330 \Intermediate Geometry and Topology" Pavel Mnev Detailed plan. I. Characteristic classes. (a) Fiber/vector/principal bundles. Examples. Connections, curvature. Rie- mannian case. Ref: [MT97]. (b) (2 classes.) Stiefel-Whitney, Chern and Pontryagin classes { properties, axiomatic definition. Classifying map and classifying bundle. RP 1; CP 1, infinite Grassmanians Gr(n; 1), CW structure, cohomology ring (univer- sal characteristic classes). Characteristic classes as obstructions (e.g. for embeddability of projective spaces). Euler class. Also: Chern character, splitting principle, Chern roots. Also: BG for a finite group/Lie group, group cohomology. Ref: [MS74]; also: [BT82, Hat98, LM98, May99]. (c) Chern-Weil homomorphism. Example: Chern-Gauss-Bonnet formula. Chern- Simons forms. Ref: Appendix C in [MS74]; [AM05, Dup78, MT97]. (d) Equivariant cohomology. Borel model (homotopy quotient), algebraic ver- sion { Cartan and Weil models. Ref: [GS99, Mei06, Tu13]. II. Bits of symplectic geometry. (a) Symplectic linear algebra, Lagrangian Grassmanian, Maslov class. Ref: [BW97, Ran, MS17]. (b) Symplectic manifolds, Darboux theorem (proof via Moser's trick). Dis- tinguished submanifolds (isotropic, coisotropic, Lagrangian). Examples. Constructions of Lagrangians in T ∗M (graph, conormal bundle). Ref: [DS00]. (c) Hamiltonian group actions, moment maps, symplectic reduction. Ref: [Jef] (lectures 2{4,7), [DS00]. (d) (Optional) Convexity theorem (Atyiah-Guillemin-Sternberg) for the mo- ment map of a Hamiltonian torus action. Toric varieties as symplectic re- ductions, their moment polytopes, Delzant's theorem. Ref: [Pra99, Sch], Part XI in [DS00], lectures 5,6 in [Jef]. (e) Localization theorems: Duistermaat-Heckman and Atiyah-Bott. Ref: [Mei06, Tu13], [Jef] (lecture 10). (f) (Optional) Classical field theory via Lagrangian correspondences. -
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 . -
[Math.SG] 22 Mar 2004
A SURVEY OF FLOER HOMOLOGY FOR MANIFOLDS WITH CONTACT TYPE BOUNDARY OR SYMPLECTIC HOMOLOGY ALEXANDRU OANCEA Abstract. The purpose of this paper is to give a survey of the various versions of Floer homology for manifolds with contact type boundary that have so far appeared in the literature. Under the name of “Symplectic homology” or “Floer homology for manifolds with boundary” they bear in fact common features and we shall try to underline the principles that unite them. Once this will be accomplished we shall proceed to describe the peculiarity of each of the constructions and the specific applications that unfold out of it: classification of ellipsoids and polydiscs in Cn, stability of the action spectrum for contact type boundaries of symplectic manifolds, existence of closed characteristics on contact type hypersurfaces and obstructions to exact Lagrange embeddings. The computation of the Floer cohomology for balls in Cn is carried by explicitly perturbing the nondegenerate Morse-Bott spheres of closed characteristics. Contents 1. Constructions of Floer homological invariants 2 1.1. Morse homology 2 1.2. Floer homology for closed manifolds 3 1.3. Floer homology for manifolds with contact type boundary or Symplectic homology 6 2. Comments and further properties 15 2.1. C0 bounds. 15 2.2. Relations between different symplectic homologies. 15 2.3. Invariance through isotopies. 16 3. A computation: balls in Cn 17 3.1. Recollections on the Robbin-Salamon index 18 3.2. Direct computation of the cohomology of a ball. 19 3.3. Alternative point of view: perturbation of critical Morse-Bott manifolds. 20 4. -
AN INTRODUCTION to INSTANTON KNOT FLOER HOMOLOGY These
AN INTRODUCTION TO INSTANTON KNOT FLOER HOMOLOGY NIKOLAI SAVELIEV These notes are intended to serve as an informal introduction to instan- ton Floer homology of knots. We begin by developing a version of instanton Floer homology for closed 3-manifolds [2], [5] associated with admissible SO(3) bundles, and then briefly review three instanton knot homology the- ories due to Floer [5] and to Kronheimer and Mrowka [6], [7]. 1. Stiefel–Whithey classes Let Y be a finite CW-complex of dimension at most three and P → Y an SO(3) bundle. Associated with P are its Stiefel–Whitney classes wi(P ) ∈ Hi(Y ; Z/2), i = 1, 2, 3. Only one of them, namely, 2 w2(P ) ∈ H (Y ; Z/2), will be of importance to us, for the reason that w1(P ) = 0 since P is ori- 1 entable, and w3(P )=Sq (w2(P )) by the Wu formula. The class w2(P ) can be defined as follows. Let { Uα } be a trivializing cover for the bundle P then the gluing functions ϕαβ : Uα ∩ Uβ −→ SO(3) (1) define a class [P ] ∈ H1(Y ; SO(3)) in Cechˇ cohomology. Conversely, a co- homology class in H1(Y ; SO(3)) defines a unique SO(3) bundle up to an isomorphism. The short exact sequence of Lie groups 1 −−−−→ Z/2 −−−−→ SU(2) −−−−→Ad SO(3) −−−−→ 1, where Z/2 = {±1} is the center of SU(2), gives rise to the exact sequence in Cechˇ cohomology, . → H1(Y ; SU(2)) −−−−→ H1(Y ; SO(3)) −−−−→δ H2(Y ; Z/2) → . 1 1 The class w2(P ) is the image of [P ] ∈ H (Y ; SO(3)) under the connecting homomorphism δ. -
Floer Homology, Gauge Theory, and Low-Dimensional Topology
Floer Homology, Gauge Theory, and Low-Dimensional Topology Clay Mathematics Proceedings Volume 5 Floer Homology, Gauge Theory, and Low-Dimensional Topology Proceedings of the Clay Mathematics Institute 2004 Summer School Alfréd Rényi Institute of Mathematics Budapest, Hungary June 5–26, 2004 David A. Ellwood Peter S. Ozsváth András I. Stipsicz Zoltán Szabó Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 57R17, 57R55, 57R57, 57R58, 53D05, 53D40, 57M27, 14J26. The cover illustrates a Kinoshita-Terasaka knot (a knot with trivial Alexander polyno- mial), and two Kauffman states. These states represent the two generators of the Heegaard Floer homology of the knot in its topmost filtration level. The fact that these elements are homologically non-trivial can be used to show that the Seifert genus of this knot is two, a result first proved by David Gabai. Library of Congress Cataloging-in-Publication Data Clay Mathematics Institute. Summer School (2004 : Budapest, Hungary) Floer homology, gauge theory, and low-dimensional topology : proceedings of the Clay Mathe- matics Institute 2004 Summer School, Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary, June 5–26, 2004 / David A. Ellwood ...[et al.], editors. p. cm. — (Clay mathematics proceedings, ISSN 1534-6455 ; v. 5) ISBN 0-8218-3845-8 (alk. paper) 1. Low-dimensional topology—Congresses. 2. Symplectic geometry—Congresses. 3. Homol- ogy theory—Congresses. 4. Gauge fields (Physics)—Congresses. I. Ellwood, D. (David), 1966– II. Title. III. Series. QA612.14.C55 2004 514.22—dc22 2006042815 Copying and reprinting. Material in this book may be reproduced by any means for educa- tional and scientific purposes without fee or permission with the exception of reproduction by ser- vices that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. -
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. -
Math 283 - Instanton Floer Homology
Math 283 - Instanton Floer Homology Taught by Peter B. Kronheimer Notes by Dongryul Kim Spring 2018 The course was taught by Peter Kronheimer, on Mondays, Wednesdays, and Fridays from 11am to 12pm. There were three assignements throughout the course. Contents 1 January 22, 2018 5 1.1 Topology of smooth 4-manifolds . .5 1.2 Connections . .7 2 January 24, 2018 8 2.1 Curvature . .8 3 January 26, 2018 11 3.1 Moduli spaces of connections . 11 3.2 Elliptic operators . 11 3.3 Linear gauge-fixing . 12 4 January 29, 2018 14 4.1 Sobolev embeddings . 14 4.2 Moduli space of anti-self-dual connections . 15 5 January 31, 2018 17 5.1 Reducibles on M ........................... 18 6 February 2, 2018 19 6.1 Uhlenbeck's compactness theorem . 19 7 February 5, 2018 22 7.1 Uhlenbeck compactification . 22 7.2 Three-manifolds and the Chern{Simons functional . 23 1 Last Update: August 27, 2018 8 February 7, 2018 25 8.1 Chern{Simons functional under gauge transformations . 25 8.2 Representation variety . 26 9 February 9, 2018 28 9.1 Gradient flows and PDE . 28 10 February 12, 2018 30 10.1 SO(3)-bundle as a SU(2)-bundle . 30 10.2 Morse theory . 31 11 February 14, 2018 33 11.1 Operator on a cylinder . 33 12 February 16, 2018 35 12.1 Spectral flow . 35 12.2 Negative gradient flow . 35 13 February 21, 2018 37 13.1 Morse homology . 37 13.2 Morse theory on the space of connections . 38 14 February 23, 2018 40 14.1 Instanton Floer homology . -
EMS Newsletter September 2012 1 EMS Agenda EMS Executive Committee EMS Agenda
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Editorial Obituary Feature Interview 6ecm Marco Brunella Alan Turing’s Centenary Endre Szemerédi p. 4 p. 29 p. 32 p. 39 September 2012 Issue 85 ISSN 1027-488X S E European M M Mathematical E S Society Applied Mathematics Journals from Cambridge journals.cambridge.org/pem journals.cambridge.org/ejm journals.cambridge.org/psp journals.cambridge.org/flm journals.cambridge.org/anz journals.cambridge.org/pes journals.cambridge.org/prm journals.cambridge.org/anu journals.cambridge.org/mtk Receive a free trial to the latest issue of each of our mathematics journals at journals.cambridge.org/maths Cambridge Press Applied Maths Advert_AW.indd 1 30/07/2012 12:11 Contents Editorial Team Editors-in-Chief Jorge Buescu (2009–2012) European (Book Reviews) Vicente Muñoz (2005–2012) Dep. Matemática, Faculdade Facultad de Matematicas de Ciências, Edifício C6, Universidad Complutense Piso 2 Campo Grande Mathematical de Madrid 1749-006 Lisboa, Portugal e-mail: [email protected] Plaza de Ciencias 3, 28040 Madrid, Spain Eva-Maria Feichtner e-mail: [email protected] (2012–2015) Society Department of Mathematics Lucia Di Vizio (2012–2016) Université de Versailles- University of Bremen St Quentin 28359 Bremen, Germany e-mail: [email protected] Laboratoire de Mathématiques Newsletter No. 85, September 2012 45 avenue des États-Unis Eva Miranda (2010–2013) 78035 Versailles cedex, France Departament de Matemàtica e-mail: [email protected] Aplicada I EMS Agenda .......................................................................................................................................................... 2 EPSEB, Edifici P Editorial – S. Jackowski ........................................................................................................................... 3 Associate Editors Universitat Politècnica de Catalunya Opening Ceremony of the 6ECM – M.