Given a morse cycle as a formal sum of critical points of a morse function, the unstable manifolds for the negative gradient flow are compactified in a suitable way, such that gluing them. On the floer homology of cotangent bundles alberto abbondandolo. When the second stiefelwhitney class of the underlying manifold does not vanish on 2tori, this isomorphism requires the use of. A morse theoretic description of string topology ralph l. To do this we rst explain the necessary background in morse theory.
Alberto abbondandolo and matthias schwarz, notes on floer homology and loop space homology, morse theoretic methods in nonlinear analysis and in symplectic topology, 10. American mathematical society 201 charles street providence, rhode island 0290422 4014554000 or 8003214267 ams, american mathematical society, the tricolored ams logo, and advancing research, creating connections, are trademarks and services marks of the american mathematical society and registered in the u. Morse theory and floer homology p p p p p 1 1 f 2 p 1 2 3 f 2 4 fig. Jun 26, 2018 american mathematical society 201 charles street providence, rhode island 0290422 4014554000 or 8003214267 ams, american mathematical society, the tricolored ams logo, and advancing research, creating connections, are trademarks and services marks of the american mathematical society and registered in the u. Floer homology is an infinitedimensional analogue of morse homology. I heard good things about the book of audin and damian, but i have not read it. Morse and morsebott homology pennsylvania state university. An explicit isomorphism between morse homology and singular homology is constructed via the technique of pseudo cycles. Given a morse cycle as a formal sum of critical points of a morse function, the unstable manifolds for the negative gradient flow are compactified in a suitable way, such that gluing.
Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, lie algebras, galois theory, and algebraic. The latter will be useda s an instrument to express the homology encoded in a morse complex associated to a fixed morse function independent of this function. Matthias schwarz, morse homology, progress in mathematics 111, 1993 there is also a variant due barannikov, and in more abstract form viterbo. I edward witten, supersymmetry and morse theory, j. I heard good things about the book of audin and damian, but i. Matthias schwarz february 24, 2004 abstract this paper concerns floer homology for periodic orbits and for a lagrangian intersection problem on the cotangent bundle t. An explicit isomorphism between morse homology and singular homology is constructed via the technique of pseudocycles. We apply morse conleyfloer homology to the study morse decompositions and degenerate variational systems in chapter 5. In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.
It really is a chain complex, or, index di erence two flows 6 manifold, dimension 7 compactness 7 the boundary is the brokens 7 4. Matthias schwarz now on we call a symplectic manifold m. Lectures on monopole floer homology francesco lin abstract. Lectures on morse homology augustin banyaga springer. The key idea of the proof of abbondandolo and schwarz is to construct an isomorphism between the floer homology of t mand a certain morse homology on a space of loops, which we already know to be isomorphic to the singular homology of m. In order for the statement to make sense, we can equip the. Morse theoretic methods in nonlinear analysis and in symplectic topology, p biran, o cornea, f lalonde, editors, nato sci. The morse homology theorem says that the homology of the morsesmalewitten chain complex is isomorphic to the singular homology of mwith integer coe. These theorems appear as the essence from the theory on compactness, gluing and orientation which has been developed throughout the last. Topics covered include the morsesmalewitten complex, some basic analysis and fredholm theory, the spectral flow and the maslov index, the compactness problem, the construction of floer homology.
This paper by cohen and norbury discussed steenrod operations, including adem relations and cartan formulae heres the abstract. Another approach t o morsebott homology can b e found in a pap er by f uk a y a 23, where the flo erbott homology of the chernsimons functional is studied. Given a morse cycle as a formal sum of critical points of a morse function, the unstable manifolds for the negative gradient. This is followed by an extensive example for m rpn. The idea of morse theory is to extract information about the global topology of x from the critical points off,i. We introduce morse homology theory by two main theorems about the existence of a canonical boundary operator associated to a given morse function f and about the existence of canonical isomorphisms between each pair of such morse complexes. In terms of floer theory, condition a implies that for.
We introduce morse homology for manifolds with boundary, and construct relative morse homology sequences. Wiley online library andres vina, a characteristic number of hamiltonian bundles over, journal of geometry and physics, 56, 11, 2327, 2006. Lecture notes on morse and conley theory stony brook. Morse homology also serves a simple model for floer homology, which is covered in the second part. Morse theory for periodic solutions of hamiltonian systems. But first, it seems worthwhile to outline the standard morse theory. Morse theory is the method of studying the topology of a smooth manifold m. An infinite dimensional analog of morse homology in symplectic geometry is known as floer homology. Morse homology michael landry april 2014 this is a supplementary note for a hopefully fun, informal one hour talk on morse homology, which i wrote after digesting the opening pages of michael hutchings great notes 2. We proof the morse inequalities, these give a relation between the critical points of a morse function and the betti numbers.
After developing the relation with the fourdimensional theory, our attention shifts to gradings and correction terms. We also deduce a new universal floer homology, defined without obstruction, for pairs of. On the action spectrum for closed symplectically aspherical manifolds. In chapter 3 we treat functorial aspects of morse homology, local morse homology, and morse conleyfloer homology. Morse homology progress in mathematics volume 111 9783034896887. Morse theory and floer homology university of texas. The latter will be useda s an instrument to express the homology encoded in a morse complex associated to a fixed morse. A relative morse index for the symplectic action floer. For very basic obstruction theory, these notes might be helpful. The second and main result is a new construction of the isomorphism between.
Equivalences for morse homology department of mathematics. For details of the analysis see matthias schwarz, morse homology. Given a morse cycle as a formal sum of critical points of a morse function, the unstable manifolds for the negative gradient flow are compactified in a suitable way, such that gluing them appropriately leads to a pseudocycle and a welldefined integral. Buy morse homology by matthias schwarz online at alibris. We associate, to a lagrangian submanifold l of a symplectic manifold, a new homology, called the cluster homology of l, which is invariant up to ambient symplectic diffeomorphisms. Aug 20, 2004 this paper concerns floer homology for periodic orbits and for a lagrangian intersection problem on the cotangent bundle of a compact orientable manifold m. Ams transactions of the american mathematical society. The notion of a morse function can be generalized to consider functions that have nondegenerate manifolds of critical points. This kind of homology theory is the central topic of this book. Alberto abbondandolo and matthias schwarz abstract we. Based on the same idea, morse homology was introduced by thom, smale, milnor, and witten. Schwarz, on the floer homology of cotangent bundles, comm. Gm1 mark goresky and robert macpherson, morse theory and intersection homology theory. Lectures on morse homology pennsylvania state university.
Symplectic floer homology sfh is a homology theory associated to a symplectic manifold and a nondegenerate symplectomorphism of it. In this paper we define and study the moduli space of metricgraphflows in a manifold m. An application of the morse theory to the topology of liegroups, bull. Morse homology can also be formulated for morsebott functions. One such procedure, following 5, is outlined here, leading to the socalled morse homology of m. Department of mathematics, tokyo metropolitan university, minamiohsawa 11, hachioji, tokyo 1920397, japan. Sorry, we are unable to provide the full text but you may find it at the following locations. Topics covered include the morse smalewitten complex, some basic analysis and fredholm theory, the spectral flow and the maslov index, the compactness problem, the construction of floer homology. The floer homology of cotangent bundles mathematics. In this framework, the homological variant of morse theory is also known as floer homology. Equivalences for morse homology matthias schwarz abstract. Transversality for local morse homology with symmetries and applications, mathematische zeitschrift, 10. Geometrically, it can be thought of as the complete graph on. Morse complex of the lagrangian action functional, which is given by l, to the.
We give a new proof of the morse homology theorem by constructing a chain complex associated to a morse bottsmale function that reduces to the morse smalewitten chain complex when the function is morse smale and to the chain complex of smooth singular cube chains when the function is constant. Its involvement has been crucial in the recent achievements in symplectic geometry and in particular in the proof of the arnold conjecture. Morse homology also serves as a model for the various infinitedimensional. Schwarz, notes on floer homology and loop space homology, morse theoretic methods in nonlinear analysis and in symplectic topology.
The first result is a new uniform estimate for the solutions of the floer equation, which allows to deal with a larger and more natural class of hamiltonians. The main goal of the course was to give a complete and detailed proof of the morse homology theorem theo rem 7. Morse homology by schwarz, matthias, 1967publication date 1993 topics homology theory, morse theory. In mathematics, specifically in the field of differential topology, morse homology is a homology theory defined for any smooth manifold. Matthias schwarz, equivalences for morse homology, geometry and topology in. Mp,qintersectionofasectionofabanachvectorbundlewith0section nontransverse transverse perturb. Morse theory for periodic solutions of hamiltonian systems and the maslov index. Alberto abbondandolo and matthias schwarz, on the floer homology of cotangent bundles, communications on pure and applied mathematics, 59, 2, 254316, 2005. Matthias schwarz this book presents a link between modern analysis and topology. Morse homology of rpn sebastian p oder june 20 abstract given a realvalued function f on a manifold m, one can deduce topological information about m if f is a morse function. Frederic bourgeois sketched an approach in the course of his work on a morsebott version of symplectic field theory, but this work was never published due to substantial analytic difficulties. What are good morse theory lecture notes and books.
Gabor pete, section 2 of morse theory, lecture notes 19992001 pdf. This is a space of smooth maps from a finite graph to m, which, when restricted to each edge, is a gradient flow line of a smooth and generically morse function on m. This book is based on the lecture notes from a course we taught at penn state university during the fall of 2002. 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. Matthias schwarz, morse homology, progress in mathematics 111, 1993. Michael farber, topology of closed oneforms, mathematical surveys and monographs, vol. Part iii, morse homology, 2011 university of oxford. Based upon classical morse theory it develops the finite dimensional analogue of floer homology which, in the recent years, has come to. Citeseerx document details isaac councill, lee giles, pradeep teregowda. I matthias schwarz, morse homology, progress in mathematics 111, birkh.
Starting in the late 80s, due to the work of floer and. Morse theory the study of the critical points of a nondegenerate smooth function on a smooth manifold and in particular their topological implications is a classic subject whose standard texts, milnors morse theory and lectures on the hcobordism theorem and morses calculus of variations in the large, date from the 60s or earlier. A book that is tough to read, but is a gateway to floer theory is schwarz book. Given a morse cycle as a formal sum of critical points of a morse function, the unstable manifolds for the negative gradient flow are compactified in a suitable way. The morse chain complex 4 generators 5 di erential 5 stable and unstable manifolds 5 morsesmale pairs 5 the space of flow lines 6 index di erence one flows 6 3. Please contact me with questions, comments, errors, confessions, etc. Barannikov, the framed morse complex and its invariants, advances in soviet math. A brief history of morse homology yanfeng chen abstract morse theory was originally due to marston morse 5. We conclude by computing morse homology and cohomology of real projective space rp2 with coe. A abbondandolo, m schwarz, notes on floer homology and loop space homology, from. Schwarz, notes on floer homology and loop space homology, appeared in the volume morse theoretical methods in nonlinear analysis and symplectic geometry, nato summer school, montreal 2004, p.
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. Morse homology by schwarz, matthias, 1967publication date 1993 topics homology theory, morse theory publisher. These lecture notes are a friendly introduction to monopole floer homology. Cuplength estimates in morse cohomology request pdf. Sc matthias schwarz, morse homology, progress in mathematics.
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