Simplicial lusternikschnirelmann category 10 references 12 1. If x2xis not a critical point, it will be called a regular point. Introduction since its inception by robin forman 9, discrete morse theory has been a powerful and versatile tool used not only in diverse elds of mathematics, but also in applications to other areas 14 as well as a computational tool 6. Existence of perfect morse functions of spaces with semifree circle action kogan, mikhail, journal of symplectic geometry, 2003.
Infinite dimensional morse theory and multiple solution. This is somewhat analogous to the fermi theory of weak. Morse theory and infinite familiesof harmonic maps between spheres article pdf available in communications in mathematical physics 2153 january 2000 with 29 reads how we measure reads. So the lefschetz fixed point formula can be written in the form and this identity follows from 1.
M\to\mathbbr and their associated gradient flows classical morse theory centered around simple statements like morse inequalities, concerning just the betti numbers. The structure of differential manifolds via morse theory 5 allows us to give a global characterization of the manifold. Bott, on the periodicity theorem for complex vector bundles, acts, mathematica, vol. The positive mass conjecture in general relativity the positive mass conjecture asserts that under appropriate hypotheses the total energy of a gravitating system is positive and can only be zero for flat minkowski space. Pdf morse theory and infinite familiesof harmonic maps. On the instanton complex of holomorphic morse theory siye wu consider a holomorphic torus action on a complex manifoldwhich lifts to a holomorphic vector bundle. Pdf supersymmetry and morse theory semantic scholar. Unlike the standard model, general relativity is a nonrenormalisable theory. One classical application of morse theory includes the.
Martin department of physics, northern illinois university, dekalb il 60115 i provide a pedagogical introduction to supersymmetry. Morse theory has provided the inspiration for exciting developments. Morse theory is a method to determine the topology of a. Witten writes down the relevant supersymmetric lagrangian and then states.
Supersymmetry and morse theory 665 the results of 2 have an analogue for complex manifolds, which will be discussed in a separate paper. Morse theory and finiteness properties of groups springerlink. The book is based on my lecture notes infinite dimensional morse theory and its applications, 1985, montreal, and one semester of graduate lectures delivered at the university of wisconsin, madison, 1987. Instantons in wittens supersymmetry and morse theory.
In section 3 we describe the construction of the seibergwitten floer stable homotopy type for rational homology. Morse theory and wittens proof of the morse inequalities. As we will see in chapter 4, however, most smooth functions are morse. 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. 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 the annals of mathematics studies series in 1947. In particular, its feature is to look at the critical points of a function, and to derive information on. Thus in the hypothesis of the previous theorem, we could have said that fis a c. We shall work with a banach space y of socalled abstract perturbations v. In the present paper we construct another in nite dimensional cohomology theory and a morse theory associated with it. We can compare the morsetheoretic and topological reidemeister torsion using the natural inclusion qtqt.
Wood, banach algebras and bott periodicity, topology, 4 196566, pp. Here is a more detailed presentation of the contents. This theory extends many ideas from smooth morse theory to the context of cw complexes. The idea of morse theory is to extract information about the global topology of x from the critical points off,i. Instanton solutions or tunneling paths in this theory would be extrema of this lagrangian. On the work of edward witten university of california. In this thesis, the profound connection between morse theory and supersymmetry will be explored along the lines of wittens 1982 paper. First applications a calculations b hcobordism theorem c lefshetz hyperplane theorem 3. For example, a morse theory of piecewise linear functions appears in 26 and the very powerful stratied morse theory was developed by goresky and macpherson 19,20. Seminars were organized around john milnors morse theory. In particular, we find an example of a group that is of type fpz but is not finitely presented. Division algebras and supersymmetry ii baez, john c. In chapter 1 we introduce the basic notions of the theory and we describe the main properties of morse functions.
Supersymmetry and morse theory school of natural sciences. They are independent of a by the second deformation lemma and the homotopy invariance of the homology groups. Morse homology also serves a simple model for floer homology, which is covered in the second part. From the discussion above, we are now interested in seeing exactly happens to the structure of the manifolds at these critical points. In section 2 we give an overview of conley index theory, focusing on gradient ows and the relation to morse theory. In chapter 4, we step aside from milnors account of morse theory and take a look at edward wittens article supersymmetry and morse theory, published in 1982. A brief history of morse homology yanfeng chen abstract morse theory was originally due to marston morse 5. Let w be a manifold with two boundary components, m 1. Morse theory deals with both finitedimensional and infinitedimensional spaces. We describe an extension of morse theory to smooth functions on compact riemann.
The first part is a thorough introduction to morse theory, a fundamental tool of differential topology. Chapter iii introduces the theory discrete morse functions developed by robin forman 12 in the 1990s. According to the basic insights of marston morse, a typical. The latter chapters are predominately focused on discussing discrete morse theory and its relationship with smooth morse theory. Im reading wittens paper on supersymmetry and morse theory and am confused about the details of the instanton calculation which he uses to define a morse complex beginning at page 11 of the pdf. There is an analogue of the morse complex which counts gradient. Based on the same idea, morse homology was introduced by thom, smale, milnor, and witten. Morse theory and supersymmetry uvafnwi universiteit van. We examine the finiteness properties of certain subgroups of right angled artin groups. Under certain hypotheses the morse and morsesmale conditions, we can form a complex c x. Mp,qintersectionofasectionofabanachvectorbundlewith0section nontransverse transverse perturb. It is useful not only for studying manifolds, but also for studying infinite cwtype. A new morse theory and strong resonance problems li, shujie, topological methods in nonlinear analysis, 2003. Help our scientists and scholars continue their fieldshaping work.
Introduction to supersymmetry university of oxford. 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. Morse theory is the method of studying the topology of a smooth manifold m m by the study of morse functions m. It was based on a suitably adapted version of an in nite dimensional cohomology theory due to g. Morse theory in the 10s martin guest introduction since the. Einsteins theory of general relativity is described, just like other theories, by a lagrangian l gr 16. It implies that minkowski space is a stable ground state.
Since the aim of this monograph is to give a unified account of the topics in critical point theory, a considerable amount of new materials. Morse theory was developed in the 1920s by mathematician marston morse. Morse theory 7 theorem b also implies the lefschetz fixed point formula. We provide the details of both of these proofs in sections 2 and 3 of the paper. Lectures on morse theory, old and new morse theory is. The relative theory then leads to forcing results of the type. On the instanton complex of holomorphic morse theory. Division algebras and supersymmetry iii huerta, john, advances in theoretical and mathematical physics, 2012. That question can then be answered using floer theory. Morse theory for periodic solutions of hamiltonian systems. In fact, the morsetheoretic torsion is not a topological invariant when d is not exact. Morse theory has been generalised in many different ways in the. Morse theory and applications to variational problems 479 where a is less than all critical values.
Although the role of morse theory in this area was quickly taken over by the new machinery of algebraic topology, the geometrical nature of botts proof of. Classical morse theory centered around simple statements like morse inequalities, concerning just the betti numbers. There are, however, deep relations between functions defined on a space and the shape of the space, and the study of these relations is the main theme of morse theory. Morse theory the simplest example of supersymmetric quantum mechanics is a system which is very well known in mathematics. 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. An inverse problem, area, and morse theory sciencedirect. Yukio matsumoto gakushuin university tokyo may 2007 proposed schedule friday 11. Morse theory and floer homology p p p p p 1 1 f 2 p 1 2 3 f 2 4 fig. Morse theory is a beautiful and natural extension of the minimum principle for a continuous function on a compact space. Bott, morse theory and its application to homotopy theory.
In a very broad sense, spaces are objects of study in geometry, and functions are objects of study in analysis. To resolve this type of noncompactness, we assume that the dynamics. 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. An introduction to floer homology daniel ruberman floer homology is a beautiful theory introduced in 1985 by andreas floer 8. Morse theory and applications to variational problems. Lazovskis university of waterloo april 18, 20 contents 1 background 2. Basic stu on morse functions a normal form b relationship between handle attachments 2. The present article aims to close this gap and to introduce and work out in full analytical detail a yangmills morse homology theory over let us now brie. This book describes morse theory for finite dimensions. Morse theory and the euler characteristic 3 the points x2xat which df xfails to have full rank are called critical points of f. In this thesis, the profound connection between morse theory and supersymmetry will be explored. It defines the morse complex and the morse homology, and develops some of their applications. Im reading wittens paper on supersymmetry and morse theory and am confused about the details of the instanton calculation which he uses to define a morse complex beginning at page 11 of the pdf witten writes down the relevant supersymmetric lagrangian and then states.
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