Morse and describing the relation between algebraictopological properties of topological spaces and extremal properties of functions functionals on them. An introduction to morse theory translations of mathematical monographs, vol. Critical point of smooth functions on a smooth manifold briefly, morse theory 1 is divided into two parts. Chapter iii introduces the theory discrete morse functions developed by robin forman 12 in the 1990s. Ams proceedings of the american mathematical society. Tromba, morse lemma in banach spaces, the euler characteristic of vector fields on banach manifolds and a globalization of lerayschauder degree, to appear. Morse theory and floer homology, audin, michele, damian, mihai. In particular, we find an example of a group that is of type fpz but is not finitely presented. This map is actually a ber bundle with ber f q 11 1, called the milnor ber.
Measuring the distance between merge trees springerlink. Our goal in the present paper is to present a morse theory for differentiable real valued functions on hilbert manifolds. Morse theory in the 1990s martin guest in tro duction since the publication of milnor s b o ok 47 in 1963, morse theory has b een a standard topic in the education of geometers and top ologists. M of the complement of the projectivized m mc, where the action is the. Morse theory has received much attention in the last two decades as a result of a famous paper in which theoretical physicist edward witten relates morse theory to quantum field theory. A brief history of morse homology yanfeng chen abstract morse theory was originally due to marston morse 5. Pdf jacobi metric and morse theory of dynamical systems.
Morse theory, manifold with boundary, cobordism, bifurcation of singular points. Chapter 4 morse theory morse theory is the study of the topology of differentiable manifolds by means of the analysis of the criticalpoint behavior of smooth functions. Morse homology is a special case for the oneform df. Inhiswork,letreatsa general hypersurface singularity.
Morse theory was developed in the 1920s by mathematician marston morse. Morse theory the simplest example of supersymmetric quantum mechanics is a system which is very well known in mathematics. To this end, we put combinatorial spin structures in a formalism perhaps most comparable to formans discrete morse theory 10. Morse theory and the euler characteristic 3 the points x2xat which df xfails to have full rank are called critical points of f. Matsumotos morse theory is one of few relatively recent treatments of finitedimensional morse theory, written at a level high enough to make it useful for advanced undergraduates or more likely graduate students, but with a slower pace than, say, one of milnor s books. For instance, since mis compact, then fattains its maximum value, and hence, fhas at least one critical point. The two segments at a reflection point either form a straight angle, or an angle whose bisector is normal to m. Bott, on the periodicity theorem for complex vector bundles, acts, mathematica, vol. The idea is torus provided by john milnor in his excellent book morse theory. Milnor constructed a smooth 7 manifold which is homeomorphic but not drawings in milnor s book on morse theory are excellent and hard to improve. Its main idea is to study the di erential topology of a manifold using the smooth functions living on it and their critical points. The lemma of morse shows that the behaviour of f at p can be completely described by this index. In order to measure the size of handles on the surface, we examine them one by one and consider cutting this region along a nonseparating cut.
This encompasses both forms of morse theory mentioned above in a unified way. Lazovskis university of waterloo april 18, 20 contents 1 background 2. In this paper we will study f via morse theory, following work of le2. In mathematics, specifically in the field of differential topology, morse homology is a homology theory defined for any smooth manifold. According to the basic insights of marston morse, a typical differentiable function on a. Morse theory, for the uninitiated, involves studying the behavior of.
The common name for three different theories based on ideas of m. Given a smooth cobordism between m and n, a morse function is a function f with nowhere vanishing gradient except at isolated points, where the hessian is nondegenerate. Brouwers definition, in 1912, of the degree of a mapping. Matsumotos morse theory is one of few relatively recent treatments of finitedimensional morse theory, written at a level high enough to make it useful for advanced undergraduates or more likely graduate students, but with a slower pace than, say, one of milnors books. Existence of perfect morse functions of spaces with semifree circle action kogan, mikhail, journal of symplectic geometry, 2003. Hence we can combine these equations and write fx1. The methods used, however, are those of differential topology, rather than the combinatorial methods of brouwer. Morse theory is a branch of variational calculus in the large calculus of variations in the large. Keywords embedded morse theory manifold with boundary cobordism.
An outofcore algorithm for isosurface topology simpli. The bott periodicity theorems were originally inspired by morse theory see part iv. Morse theory elias stein, john willard milnor, michael. Lecture 14 morse theory on loop spaces, geodesics and first variation. Morse theory could be very well be called critical point theory. A special case of novikovs theory is circlevalued morse theory, which michael hutchings and yijen lee have connected to reidemeister torsion and seibergwitten theory. The latter chapters are predominately focused on discussing discrete morse theory and its relationship with smooth morse theory. Forman 2002 studies discrete morse functions that assign a single value to eac h cell in a complex. They present some topics from the beginnings of topology, centering about l. Forman, morse theory for cell complexes advances in math. Morse, the calculus of variations in the large, american mathematical society, new york, 1934. Introductory references for morse theory mathematics stack.
Morse theory in the 1990s martin guest in tro duction since the publication of milnors b o ok 47 in 1963, morse theory has b een a standard topic in the education of geometers and top ologists. Deac0205ch11231 berkeley lab, and the program 055 of the ministry of edu. That is, the complement of any complex hyperplane arrangement has the homotopy type of a cwcomplex in which the number of cells equals the. In sharp contrast, it is an unavoidable consequence of discretization that combinatorial gradient. Morse s theory of thc ca l cul us of var1ations i n the l arge. Wood, banach algebras and bott periodicity, topology, 4 196566, pp. For further information on morse theory, the following sources are extremely useful. In another context, a morse function can also mean an anharmonic oscillator see morse potential in mathematics, specifically in differential topology, morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. The local part is related to the idea of a critical point of a smooth function, the hessian of a function at its critical point, the morse index of a critical point, etc. Morse was on the faculty of the institute for advanced study, and princeton published his topological methods in the theory of functions of a complex variable in. Morse theory has provided the inspiration for exciting developments. Thus in the hypothesis of the previous theorem, we could have said that fis a c.
There is a natural generalization of the morse index of geodesics to variational calculus, which runs as follows. Morse theory is a method to determine the topology of a nite or in nite dimensional mani fold such as the space of paths or loops on a compact manifold from the critical points of only one suitable function on the manifold. Supersymmetry and morse theory 665 the results of 2 have an analogue for complex manifolds, which will be discussed in a separate paper. Once a morse function has been defined on a manifold, information about its topology can be deduced from its critical elements. Let be a realvalued smooth function on an open subset of and let be a smooth submanifold of. Let be the space of smooth curves for which the jet lies in and.
Morse theory on the milnor fiber we follow le 2 throughout this section. Dynamics of gradient flows in the halftransversal morse theory goda, hiroshi and pajitnov, andrei v. As we will see in chapter 4, however, most smooth functions are morse. We examine the finiteness properties of certain subgroups of right angled artin groups. This b o ok established suc h high standards for clarit y of exp osition and. In particular, it is believed that morse theory on infinitedimensional spaces will become more and more important in the future as mathematics advances. A short introduction to morse theory alessandro fasse email. From the discussion above, we are now interested in seeing exactly happens to the structure of the manifolds at these critical points. The dynamics of a nice enough f 2 c1m reconstructs msmoothly. Generalized billiard paths and morse theory for manifolds. A billiard path on a manifold m embedded in euclidean space is a series of line segments connecting reflection points on m. Based on the same idea, morse homology was introduced by thom, smale, milnor, and witten. This book describes morse theory for finite dimensions.
Combine lemma 9 with theorem 9 for the first statement. Embedded morse theory and relative splitting of cobordisms of. Then is a banach manifold, on which one has the smooth functional. Morse theory arises from the recognition that the number of critical points of fis constrained by the topology of m. Sketch of proof this is a standard result combining the fact that the. Threlfall, varlationsrechnung im grossen, published in the united states by chelsea, new york, 1951. Palais, morse theory on hilbert manifolds, topology, vol. 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. Our goal is to estimate the number of generalized billiard paths connecting fixed. This b o ok established suc h high standards for clarit y of exp osition and mathematical in uence that it has b een reprin ted sev. Morse theory and finiteness properties of groups springerlink. The structure of differential manifolds via morse theory 5 allows us to give a global characterization of the manifold. M will be called morse if all of its critical points are non degenerate.
Here is a list of books which i like other than milnor, of course. Milnor was awarded the fields medal the mathematical equivalent of a nobel prize in 1962 for his work in differential topology. Seminars were organized around john milnors morse theory. This theory extends many ideas from smooth morse theory to the context of cw complexes. In fact, in 15, milnor constructs smooth s3bundles p. Existence of perfect morse functions of spaces with semifree circle action kogan, mikhail, journal of symplectic geometry, 2003 a new morse theory and strong resonance problems li, shujie, topological methods in nonlinear analysis, 2003. The topology is coded in a reeb graph reeb 1946, where cycles in the reeb graph correspond to handles. By a cw structure on a space x, we shall mean a cwcomplex homotopy equivalent to x.
On the one hand, morse theory is extremely important in the classi cation programme of manifolds. Tromba, morse lemma in banach spaces, eulerpoincare index theory on banach manifolds, ann. Morse homology also serves as a model for the various infinitedimensional. Introduction to morse theory let mand nbe smooth manifolds, and let f. Smooth morse functions discrete morse functions applications references references. The kinds of theorems we would like to prove in morse theory will typically only apply to morse functions. Here is a more detailed presentation of the contents. In a generalized billiard path we also allow the path to pass through m. Morse theory has previously been adapted to discrete and applied settings.
A new cohomology for the morse theory of strongly indefinite functionals on hilbert spaces abbondandolo, alberto, topological methods in nonlinear analysis, 1997. Smale, morse theory and a nonlinear generalization of the dirichlet problem, annals of mathematics, vol. Discrete morse theory and localization 3 smooth case, every. Through the study of morse theory on the associated milnor fiber, we show that complex hyperplane arrangement complements are minimal. If x2xis not a critical point, it will be called a regular point. In chapter 1 we introduce the basic notions of the theory and we describe the main properties of morse functions. The before stating this lemma we first prove the lemma 2. Morse theory deals with both finitedimensional and infinitedimensional spaces. Milnor on differential topology university of texas at. One of the most cited books in mathematics, john milnors exposition of morse theory has been the most important book on the subject for more than forty years. Since f is linear, the set of its critical points is a subset of the vertices of m.
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