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