Morse theory and floer homology p p p p p 1 1 f 2 p 1 2 3 f 2 4 fig. Introduction the aim of this paper is to give an introduction to heegaard floer homology 24 for closed oriented 3manifolds. The resulting homology, introduced by andreas floer, spectacularly revolutionized the area of symplectic topology and leaded to a proof of the famous arnold conjecture. Chapter 8 morse theory and floer homology springerlink. The idea of morse theory is to extract information about the global topology of x from the critical points off,i. Instanton floer homology is the \morse theory on the space of connections on smooth oriented 4manifolds of the yangmills. Morse theory and floer homology universitext by michele audin, mihai damian morse theory and floer homology universitext by michele audin, mihai damian pdf, epub ebook d0wnl0ad. Floer homology is an in nitedimensional type of morse homology where the periodic orbits are described as critical points of the symplectic action functional a h on loop space. Floer homology is a novel invariant that arises as an infinitedimensional analogue of finitedimensional morse homology. Floer homology is an adaptation of morse homology to in nite dimensions. Its involvement has been crucial in the recent achievements in symplectic geometry and in particular in the proof of the arnold conjecture. We prove that this homology is wellde ned and does not depend on the choice. The associated gradient flow equation corresponds to the seibergwitten equations on. Outline 1 morse theory from handles to ow lines 2 lagrangian and heegaard floer homology 3 combinatorial description 4 applications and problems peking u jiajun wang on combinatorial heegaard floer homology.
In part ii, that consists of chapter 6, we study the existence of periodic orbits on noncompact energy hypersurfaces. Thus, morse theory and floer homology is a relatively highlevel introduction to, and in fact a full account of, the extremely elegant and properly celebrated solution to the arnold problem by the prodigious and tragic andreas floer a victim in 1991, by suicide, to devastating depression. Let x be a complete riemannian manifold, not necessarily of finite. Morse theory for trajectories of lagrangian systems on riemannian manifolds with convex. May 17, 20 morseconleyfloer homology article pdf available in journal of topology and analysis 63 may 20 with 63 reads how we measure reads. The small mcduffsalamon book on holomorphic curves jholomorphic curves and quantum cohomology, available on mcduffs webpage has a small chapter on floer homology. An introduction to floer homology daniel ruberman floer homology is a beautiful theory introduced in 1985 by andreas floer 8. View the article pdf and any associated supplements and figures. Darko milinkovic, floer homology in classical mechanics and quantum field theory, ps.
It applies to certain classes of in nite dimensional manifolds xand functions f. Morse theory and floer homology mathematical association. By continuing to use our website, you are agreeing to our use of cookies. The generalization to situations in which both index and coindex are infinite, but the relative index of any pair of critical points is finite, is known as floer homology. Morse theory and floer homology is a relatively highlevel introduction to, and in fact a full account of, the extremely elegant and properly celebrated solution to the arnold problem by the prodigious and tragic andreas floer.
This book is an introduction to modern methods of symplectic topology. Pdf floers infinite dimensional morse theory and homotopy. This kind of homology theory is the central topic of this book. Pdf what you need to know about symplectic geometry. Contributions of this paper include 1 a monotonicity lemma for the behavior of the nonlin. Morse theory is the method of studying the topology of a smooth manifold m m by the study of morse functions m. Then we shall apply this theory to obtain some topological invariants of 3manifolds. Floer homology is an infinitedimensional analogue of morse homology. Rather surprisingly, the resulting homology is proved to be independent of the choice of the realvalued function and metric defining the negative gradient.
It may be viewed as the morse homology of the chernsimonsdirac functional on u 1 connections on the threemanifold. But first, it seems worthwhile to outline the standard morse theory. Morse theory, the conley index and floer homology bulletin. These lecture notes are a friendly introduction to monopole floer homology. All of this is closely related to ideas from quantum. Morse homology also serves a simple model for floer homology, which is covered in the second part. The proposal was to attach a stable homotopy type to the geometric data given in a version of floer homology. Morse homology also serves as a model for the various infinitedimensional. We take a 3dimensional manifold, construct a functional on some in. Morse theory, the conley index and floer homology, bulletin of the london mathematical society, volume 22, issue 2, 1 march 1990, pages 114 we use cookies to enhance your experience on our website. Rather than starting to introduce the general theory from the very beginning, we have chosen to adopt a di.
Witten floer homology 22, ech theory 20, 21, has become tools of understanding lower dimensional topology. Morse theory and floer homology michele audin, mihai damian. Kronheimer notes by dongryul kim spring 2018 the course was taught by peter kronheimer, on mondays, wednesdays, and. Math 283 instanton floer homology taught by peter b. Morse homology studies the topology of smooth manifolds by examining the critical points of a realvalued function defined on the manifold, and connecting them with the negative gradient of the function. Floer trajectories with immersed nodes and scaledependent gluing oh, yongguen and zhu, ke, journal of symplectic geometry, 2011. We will also discuss a related floer homology invariant for knots in s3, 31, 34. Floer theory still is the subject of a lot of active and exciting research and is nowadays an essential technique in symplectic topology. In this framework, the homological variant of morse theory is also known as floer homology. Displacement of polydisks and lagrangian floer theory fukaya, kenji, oh, yonggeun, ohta, hiroshi, and ono, kaoru, journal of symplectic geometry, 20. It is constructed using the smooth structure and an auxiliary metric on the manifold, but turns out to be topologically invariant, and is in fact isomorphic to singular homology. Changing the point of view 9 historical notes and bibliography 11 3.
These notes are rather ancient, but maybe still helpful. In morse theory, see 20, 21, one studies smooth functions on ndimensional manifolds f. Morse theory, the conley index and floer homology salamon. Floer homology has been partly motivated by arnolds conjecture on periodic trajectories in classical mechanics. Yonggeun oh, symplectic topology and floer homology, pdf. It is devoted to explaining the solution of an important problem originating from classical mechanics.
He uses this theory to give obstructions to a knot being. Morse theory and floer homology michele audin springer. Persistent homology and floernovikov theory 1003 and can still speak of the. Homology, morse theory, floer theory, periodic orbits, fixed points, arnold conjecture. Floer also developed a version of this homology called relative which is. Founders of morse theory were marston morse, raoul bott and albert schwarz. Morse homology also serves as a simple model for floer homology, which is covered in the second part. Morse theory and floer homology university of texas. After developing the relation with the fourdimensional theory, our attention shifts to gradings and correction terms. All these di erent types of floer theory, are all certain in. Morse theory and floer homology mathematical association of. Morse theory relates the topology of xand critical points and gradient. Despite the mistakes in these notes, some people have found them useful as an introduction, so i will keep them up here while hoping to revise them sometime.
Wittens complex and infinite dimensional morse theory, j. Seibergwitten floer homology or monopole floer homology is a homology theory of smooth 3manifolds equipped with a spin c structure. The main theme in section 2 is contact geometry and its. It defines the morse complex and the morse homology, and develops some of their applications. It combined new ideas about morse theory, gauge theory, and cassons approach 1, 14 to homology 3spheres and the representations of their fundamental groups into lie groups such as su2 and so3. A very basic introduction to mathematical proofs pdf, 27 pages. Lecture 1 the aim of this series of seminars is to give an account of floer homology. The building blocks of floer homology are more intricate and imply the use of more sophisticated analytical. Download the ebook floer homology, gauge theory, and low dimensional topology. For example, if f is a morse function, then the number of critical points is at least. Proceedings of the clay mathematics institute 2004 summer school, alfred renyi institute of mathematics, budapest, hungary, june 526, 2004 clay mathematics proceedings, vol. Stanford libraries official online search tool for books, media, journals, databases, government documents and more. In mathematics, floer homology is a tool for studying symplectic geometry and lowdimensional topology.
The main theme in section 2 is contact geometry and its interplay with floer homology. Morse theory, the conley index and floer homology people. Andreas floer introduced the first version of floer homology, now called hamiltonian floer homology, in his proof of the arnold conjecture in symplectic geometry. The ideas in the rest of the book are also useful for floer theory. Morse theory and floer homology in searchworks catalog. Morse theory has historically served as an important intermediary between differen tial geometry and algebraic topology, and it has gained further importance in. From morse homology to floer homologya rapid course in morse homology a rapid course in morse homology let x be a smooth compact manifold of dimension d, and let f. The first part is a thorough introduction to morse theory, a fundamental tool of differential topology. On combinatorial heegaard floer homology jiajun wang 1 1lmam, school of mathematical sciences peking univeristy knots and low dimensional manifolds. Morse theory, the conley index and floer homology, bulletin of the london mathematical society, volume 22, issue 2, 1 march 1990, pages 114. Morse theory and floer homology ebok michele audin. Classical morse theory centered around simple statements like morse inequalities, concerning just the betti number s.
Persistent homology and floernovikov theory michael usher jun zhang we construct barcodes for the chain complexes over novikov rings that arise in novikovs morse theory for closed oneforms and in floer theory on notnecessarilymonotone symplectic manifolds. The symplectic variant of floer cohomology is related to quantum cohomology. Introduction to floer homology and its relation with tqft. Sergei novikov generalized this construction to a homology theory associated to a closed oneform on a manifold. We apply morse conley floer homology to the study morse decompositions and degenerate variational systems in chapter 5. In this expository paper, we will discuss where this theory comes from and what it is as well as its relation with tqft. In 1995 the author, jones, and segal introduced the notion of floer homotopy theory. Simon donaldson, floer homology groups in yangmills theory cambridge university press. Heegaard floer homology is an invariant of closed, connected, oriented, and smooth 3manifolds which appears as a version of lagrangian floer homology arising from a heegaard decomposition of the 3manifold. Heegaard diagrams correspond to realvalued morse theory in three dimensions, in these lectures, goda considers circlevalued morse theory for link complements. Lecture notes on morse homology with an eye towards floer theory and pseudoholomorphic curves, fall 2002 postscript, 63 pages.
We apply morseconleyfloer homology to the study morse decompositions and degenerate variational systems in chapter 5. The rst goal of this thesis is to prove the arnold conjecture using floer homology. Basics on hamiltonian dynamics and symplectic geometry, i 12 4. Floer theory for lagrangian cobordisms chantraine, baptiste, dimitroglou rizell, georgios, ghiggini, paolo, and golovko, roman, journal of differential geometry, 2020. Despite the mistakes in these notes, some people have found them useful as an introduction. Floer homology, gauge theory, and lowdimensional topology. Morse homology is a special case for the oneform df. Morseconleyfloer homology article pdf available in journal of topology and analysis 63 may 20 with 63 reads how we measure reads.
Introduction to floer homology and its relation with tqft qingtao chen nov. Lectures on monopole floer homology francesco lin abstract. The associated morse homology is an invariant for the manifold, and equals the singular homology, which yields the classical morse relations. In mathematics, specifically in the field of differential topology, morse homology is a homology theory defined for any smooth manifold. Morse functions and heegaard diagrams in this section we study a morse theoretical approach to heegaard decompositions.
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