Cambridge part iii course on morse homology lent term 2011 lecture notes i would be very grateful for your feedback in the form of typos, suggestions or mistakes you found, because these notes will be published as a book. Morse homology also serves as a simple model for floer homology, which is covered in the second part. In mathematics, floer homology is a tool for studying symplectic geometry and lowdimensional topology. We will discuss four types of floer homology that can be associated to a threemanifold each coming with its own di erent subtypes. Contributions of this paper include 1 a monotonicity lemma for the behavior of the nonlin. Floer homology, gauge theory, and lowdimensional topology. In the present work we generalize the construction of monopole floer homology due to kronheimer and mrowka to the case of a gradient flow with morsebott singularities. In this paper, we consider a new type of floer homology theory proposed in 3 and motivated from dostoglousalamons study of atiyah. Floer homology as an in nite dimensional morse theory, as we explain in the next section.
Morse theory and floer homology has the same index as d2fx. The three appendices present a few prerequisites in differential geometry, algebraic topology and analysis. These lecture notes are a friendly introduction to monopole floer homology. The treatment of homology and cohomology in this report primarily follows algebraic topology by allen hatcher.
In mathematics, specifically in the field of differential topology, morse homology is a homology theory defined for any smooth manifold. In this manuscript we will explain the theory with respect to the. The exact triangle is a key calculational tool in heegaard floer homology. The small mcduffsalamon book on holomorphic curves jholomorphic curves and quantum cohomology, available on mcduffs webpage has a small chapter on floer homology. Classical morse theory centered around simple statements like morse inequalities, concerning just the betti number s. Morse theory and floer homology will be particularly helpful for graduate and postgraduate students. Beyond them there exists morse theory or morse homology for in.
Relative floer and quantum cohomology and the symplectic. If the symplectomorphism is hamiltonian, the homology arises from studying the symplectic action functional on the universal cover of the free loop space of a symplectic manifold. Floer theory for lagrangian cobordisms chantraine, baptiste, dimitroglou rizell, georgios, ghiggini, paolo, and golovko, roman, journal of differential geometry, 2020. This book is an introduction to modern methods of symplectic topology.
The first part is a thorough introduction to morse theory, a fundamental tool of differential topology. Thus in order to construct the morsefloer homology with coe. Morse theory, the conley index and floer homology, bulletin of the london mathematical society, volume 22, issue 2, 1 march 1990, pages 114. The book originated in a graduate course given at strasbourg university, and contains a large range of figures and exercises. Discrete morse theory persistent homology persistence vs. In 1995 the author, jones, and segal introduced the notion of floer homotopy theory. The kam theorem established that periodic orbits can follow complex trajectories.
Morse theory, then, relates the topology of the manifold with the critical points of a function defined over it. Lecture notes on morse homology with an eye towards floer theory and pseudoholomorphic curves, fall 2002 postscript, 63 pages. In this paper we survey floer theory as it is relevant to low dimensional topology. Floer homology in symplectic geometry and in mirror symmetry. Symplectic topology and floer homology is a comprehensive resource suitable for experts and newcomers alike. Thirty years of floer theory for 3manifolds jennifer hom georgia tech, robert lipshitz university of oregon, daniel ruberman brandeis university july 31, 2017august 4, 2017 1 overview of the field floer homology, introduced by andreas floer in 1988, has had a dramatic impact on the. Chapter 8 morse theory and floer homology springerlink. 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. Po was partially supported by nsf grant number dms 0234311. Morse theory relates the dynamics of a gradient flow on a manifold to, for example, its homology. Morse theory, the conley index and floer homology salamon.
We then proceed, using the notion of local morse homology of section 1. Mathematics institute, university of warwick, coventry cv4 7al. This kind of homology theory is the central topic of this book. Morsefloer theory for superquadratic dirac equations, i. Floer homology extended this to infinitedimensional manifolds. Morse theory and floer homology michele audin springer. 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. To state our results, let y be a compact oriented 3manifold whose boundary by. View the article pdf and any associated supplements and.
Morse theory and floer homology mathematical association of. Notes on homology theory mcgill university school of. All of this is closely related to ideas from quantum. 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. That question can then be answered using floer theory. Morse theory for trajectories of lagrangian systems on. Floer homology for lagrangian intersections 1, floer homology for. Matumoto gs80, mat78, who reduced the problem to a question about homology cobordism in three dimensions. Another approach to morse theory is morse homology.
Interested readers are referred to this excellent text for a comprehensive introduction. Ono for a fruitful collaboration on the lagrangian intersection floer theory where some part of this survey is based on. Seibergwitten floer theory of 3manifolds we will try to understand as much as we can, but certainly not all, of the book by kronheimermrowka. Symplectic floer homology sfh is a homology theory associated to a symplectic manifold and a nondegenerate symplectomorphism of it. Michael hutchings, lecture notes on morse homology with an eye towards floer theory, matthias schwarz, morse homology. Morse theory has historically served as an important intermediary between differen tial geometry and algebraic topology, and it has gained further importance in. In this framework, the homological variant of morse theory is also known as floer homology. The ideas in the rest of the book are also useful for floer theory. A simplex is regular if it has a face coface with higher lower value.
Morse homology, connections on principal bundles, spinc structures and dirac operators, etc. The associated morse homology is an invariant for the manifold, and equals the singular homology, which yields the classical morse relations. 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. Let x be a complete riemannian manifold, not necessarily of finite. More precisely, heegaard floer homology is a type of lagrangian floer homology. The proposal was to attach a stable homotopy type to the geometric data given in a version of floer homology. It is devoted to explaining the solution of an important problem originating from classical mechanics. Floer homology and rabinowitz floer homology luuk hendrikx a thesis submitted to the department of mathematics at utrecht university in partial ful llment of the requirements for the degree of master of mathematical sciences supervisor. Its involvement has been crucial in the recent achievements in symplectic geometry and in particular in the proof of the arnold conjecture.
Morse theory and floer homology university of texas. In particular, it is devoted to the foundations and applications of homology. Rather surprisingly, the resulting homology is proved to be independent of the choice of the realvalued function and metric defining the negative gradient. Volume 2 provides a comprehensive introduction to both hamiltonian floer theory and lagrangian floer theory. An introduction to floer homology daniel ruberman floer homology is a beautiful theory introduced in 1985 by andreas floer 8. 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. Andreas floer introduced the first version of floer homology, now called lagrangian floer homology, in his proof of the arnold conjecture in symplectic. This is exactly the objective which floer homology accomplishes. Good references on this sort of thing of varying emphasis, rigor and applicability to this course include dietmar salamon, lectures on floer. We apply morse conley floer homology to the study morse decompositions and degenerate variational systems in chapter 5.
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. Floer homology is the homology of a chain complex generated by critical points of the functional. Morse theory and floer homology in searchworks catalog. Floer also developed a version of this homology called relative which is. This theory was designed to tackle problems in hamiltonian dynamics. Morse theory and floer homology universitext 2014, audin. It relates the heegaard floer homology groups of threemanifolds obtained by surgeries along a framed knot in a closed, oriented threemanifold. After developing the relation with the fourdimensional theory, our attention shifts to gradings and correction terms. These notes are rather ancient, but maybe still helpful. The aim of this project is to get an understanding of basic concepts of homology and co homology. Stanford libraries official online search tool for books, media, journals, databases, government documents and more. 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. But first, it seems worthwhile to outline the standard morse theory.
Download the ebook floer homology, gauge theory, and low dimensional topology. Kronheimer notes by dongryul kim spring 2018 the course was taught by peter kronheimer, on mondays, wednesdays, and. This was followed by the work of floer, who introduced the floer cohomology groups in both the the symplectic and yangmills settings 22, 23. Morse theory and floer homology mathematical association. It is devoted to explaining the solution of an important problem originating.
A very basic introduction to mathematical proofs pdf, 27 pages. Pdf what you need to know about symplectic geometry. 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. Introduction to floer homology and its relation with tqft qingtao chen nov. Floer theory for lagrangian cobordisms chantraine, baptiste, dimitroglou rizell, georgios, ghiggini, paolo, and golovko, roman, journal of differential geometry, 2020 lagrangian submanifolds in hyperkahler manifolds, legendre transformation leung, naichung conan, journal of differential geometry, 2002. Symplectic topology and floer homology by yonggeun oh. We provide a short introduction to the various concepts of homology theory in algebraic topology. Proceedings of the clay mathematics institute 2004 summer school, alfred renyi institute of mathematics, budapest, hungary, june 526, 2004 clay mathematics proceedings, vol. To avoid overuse of the word continuous, we adopt the convention that maps between. Introduction to floer homology and its relation with tqft. The building blocks of floer homology are more intricate and imply the use of more sophisticated analytical. The focus of this course will be the floer cohomology theory called symplectic cohomology, a form of the loopspace floer cohomology on noncompact symplectic manifolds with constrained geometry at infinity liouville manifolds. In part ii, that consists of chapter 6, we study the existence of periodic orbits on noncompact energy hypersurfaces. A crash course of floer homology for lagrangian intersections.
In this expository paper, we will discuss where this theory comes from and what it is as well as its relation with tqft. Here, gradf of course is the gradient off for allx. Lectures on monopole floer homology francesco lin abstract. The aim of this dissertationis to give an exposition on floer homology on symplectic manifolds. Morse theory and floer homology michele audin, mihai. We will also discuss a related floer homology invariant for knots in s3, 31, 34. In the second direction, the moment map point of view served as a guide to the introduction. In particular, it is reminiscent of the original construction of kronheimermrowka km07 for closed 3manifolds. Floer homology is an infinitedimensional analogue of morse homology. We prove invariance properties of the morseconleyfloer homology, and show how it gives rise to the morseconley relations. Heegaard floer homology and alternating knots request pdf. Yangmills theory and geometry imperial college london. Lectures on morse homology pennsylvania state university.
Introduction the aim of this paper is to give an introduction to heegaard floer homology 24 for closed oriented 3manifolds. Floer for putting everlasting marks on his mathematics. Floer homology is a novel invariant that arises as an infinitedimensional analogue of finitedimensional morse homology. Despite the mistakes in these notes, some people have found them useful as an introduction. Hamiltonians would be to look for some generalization of morse theory to the in. View the article pdf and any associated supplements. Math 283 instanton floer homology taught by peter b. It defines the morse complex and the morse homology, and develops some of their applications. An introduction to homology prerna nadathur august 16, 2007 abstract this paper explores the basic ideas of simplicial structures that lead to simplicial homology theory, and introduces singular homology in order to demonstrate the equivalence of homology groups of homeomorphic topological spaces. Morse theory is the method of studying the topology of a smooth manifold m m by the study of morse functions m.