Palais morse theory pdf

Both morse theory on hilbert manifolds, developed by palais and smale 3, 51, and the luisternikschnirelman theory on banach manifolds due to palais 2 are several years old. Palais, morse theory on hilbert manifolds, topology, vol. Pdf we generalize the classic morse theory of smooth functions with nondegenerate. The palaissmale condition and the existence of saddle points on a compact manifold, any continuous function assumes its minimum. In chapter 1 we introduce the basic notions of the theory and we describe the main properties of. Smale, morse theory and a nonlinear generalization of the dirichlet problem, annals of mathematics, vol. Here is a more detailed presentation of the contents. Introduction in classical morse theory, manifolds are smooth and. It dates from the 1930s, but achieved its greatest results in the late 50s and early 60s with its application to the classification. L x defines an index invariantly, and it is shown that this is an extension of the usual definition of.

The palaissmale conditions for the yangmills functional volume 108 issue 34 d. Morsetype information on palaissmale sequences 204 11. Morse theory and floer homology university of texas. The second part consists of applications of morse theory over the reals, while the last part describes the basics and some applications of complex morse theory, a. Smale conjectured that morse theory can be extended to banach manifolds. The axis of a rotation, with bob palais, the journal for fixed point theory and its applications, 2 2007 215220 an introduction to wave equations and solitons, in the princeton companion to mathematics. Morse theory and nonminimal solutions 571 it is the effectof the map i on the relevant homotopy groups of pairs which determines whether crit. Morse theory deals with both finitedimensional and infinitedimensional spaces. On the theories of morse and lusternikschnirelman for. Since palais and smale 39,41,46 generalized finitedimensional morse theory 36,35 to nondegenerate c 2 functionals on infinite dimensional hilbert manifolds and used it to study multiplicity. Morse theory could be very well be called critical point theory.

The only difference worth mentioning is that in morse theory 3 the compactness of is replaced by condition of palaissmale see morse function, which, besides, is not satisfied in. T x m t x m such that in the coordinate system of u, d. Part ii of the book is a selfcontained account of critical point theory on hilbertmanifolds. It is useful not only for studying manifolds, but also for studying infinite cwtype spaces. Morses theory of thc ca l cul us of var1ations i n the l arge. Duality and perturbation methods in critical point theory. As we will see in chapter 4, however, \most smooth functions are morse. Introduction to morse theory let mand nbe smooth manifolds, and let f. The bott periodicity theorems were originally inspired by morse theory see part iv. A critical point u with c 1pg,uq0 is called a mountain pass point.

This paper present a new approach to morse theory with the aim to give to the unexperienced reader an extra tool for working in the critical point theory. Palais 14 on morse theory on hilbert manifolds to the case of gmanifolds. The applications we have in mind involve cube complexes and simplicial complexes. The kinds of theorems we would like to prove in morse theory will typically only apply to morse functions. In 1970 he was an invited speaker banach manifolds of fiber bundle sections at the. Formans discrete morse theory 14 has been successfully used to perform cohomology computations not only in algebraic topology 11,30,31, but also in commutative algebra 21, topological combinatorics 37, algebraic combinatorics 35 and even geometric group theory 4. Morse theory and applications to variational problems. Richard sheldon palais born may 22, 1931 is a mathematician working in geometry who introduced the principle of symmetric criticality, the mostowpalais theorem, the liepalais theorem, the morsepalais lemma, and the palaissmale compactness condition from 1965 to 1967 palais was a sloan fellow. Milnor was awarded the fields medal the mathematical equivalent of a nobel prize in 1962 for his work in differential topology.

Pl morse theory we want to develop a similar theory that applies to nice cell complexes instead of manifolds see 1. Therefore, in its modern form, morse theory 3 is an almost verbatim reiteration of morse theory 1. Lusternikschnirelman theory on banach manifolds 121 proof. Dynamics of gradient flows in the halftransversal morse theory goda, hiroshi and pajitnov, andrei v. Of course this presentation depends on the taste of the writer and the applications are chosen among the ones more familiar to him. Palais longterm research interests have spanned several areas, including compact differentiable transformation groups, nonlinear global analysis, critical point theory in particular morse theory, submanifold geometry and integrable systems and. A short introduction to morse theory alessandro fasse email. Pdf morse theory on hilbert manifolds richard palais. In mathematics, the morsepalais lemma is a result in the calculus of variations and theory of hilbert spaces. In particular, it is believed that morse theory on infinitedimensional spaces will become more and more important in the future as mathematics advances. Palais the term morse theory is usually understood to apply to two analagous but quite distinct bodies of mathematical theorems.

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. Introduction to morse theory a new approach springerlink. A new cohomology for the morse theory of strongly indefinite functionals on hilbert spaces abbondandolo, alberto, topological methods in nonlinear analysis, 1997. Morse theory, for the uninitiated, involves studying the behavior of functions on a smooth manifold near the critical points of the functions in order to deduce information about the topology of the manifold. Morse inequalities for orbifold cohomology hepworth. The morse theory of critical points of a real valued functionf defined on a finite dimensional manifold m without boundary was generalized by palais and smale to the case where h4 is a hilbert manifold without boundary 8, lo. In particular if all critical points are nondegenerate and therefore. Thus in the hypothesis of the previous theorem, we could have said that fis a c1morse function. Starting from the concept of morse critical point, introduced in a. Finally, in chapter 8, we use the morse theory developed in part ii to study the homology of isoparametric submanifolds of hilbert space.

M\to\mathbbr and their associated gradient flows classical morse theory centered around simple statements like morse inequalities, concerning just the betti numbers. Critical point theory in particular morse theory submanifold geometry. Intuitively,the relevant ends of consistof approximate solutions to the eulerlagrange equations of. Essentially all my published and some unpublished books and papers are available online for download as pdf files at richard s. Morse theory and applications to variational problems 477 ingredient in their proofs is a suitable negative pseudogradient. Bradlow and graeme wilkin dedicated to professor dick palais on the occasion of his 80th birthday 1. The idea is torus provided by john milnor in his excellent book morse theory. The general philosophy of the theory is that the topology of a smooth manifold is intimately related to the number and type. Lmw17a, where the nonproper morse theory of palaissmale ps64 is used. Morse theory on hilbert manifolds, topology 2 1963, 299340. It is shown that morse functions are dense in the set of invariant real valued functions on m if m is finite dimensional. Roughly speaking, it states that a smooth enough function near a critical point can be expressed as a quadratic form after a suitable change of coordinates the morsepalais lemma was originally proved in the finitedimensional case by the american mathematician marston morse. On the one hand one considers a smooth, real valued functionfon a compact manifold m, defines m, f x, a, and given a closed interval.