This cited by count includes citations to the following articles in scholar. First we discuss the morse theory of euclidean space strati. In addition, it is an advantageous setting in which to integrate. 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. Milnor fibers and links of local complete intersections. Admissible morse functions are particular examples of stratified morse functions in the sense of the theory developed by goreskymacpherson in 4. Stratified morse theory due to the lack of proper bibliographical sources stratification theory seems to be a mysterious subject in contemporary mathematics. Introduction classical morse theory is a useful tool for the study of the topology of di. A standard reference is the book of goreskymacpherson which primarily treats the topological type. Due to the lack of proper bibliographical sources stratification theory seems to be a mysterious subject in contemporary mathematics. This observation relies on stratified morse theory due to goresky and macpherson, but is in a nutshell close to the elementary result used by connelly in his work on the existence of continuous unlocking deformations of hard ball configurations, see. Folge a series of modern surveys in mathematics 9783642717161.
Stratified morse theory classical morse theory establishes. This book contains a complete and elementary survey including an extended bibliography on. In mathematics, stratified morse theory is an analogue to morse theory for general stratified spaces, originally developed by mark goresky and robert macpherson. Theorems a, b, and c of classical morse theory 2 2. Stratified morse theory david masseys math page northeastern. Mark goresky, robert macpherson due to the lack of proper bibliographical sources stratification theory seems to be a mysterious subject in contemporary mathematics. Stiefelwhitney numbers for singular varieties volume 150 issue 2 carl mctague. This b o ok established suc h high standards for clarit y of exp osition and.
The main point is that in this book a similar degree of. There are versions for the topological type, homotopy type or cohomology. It is useful not only for studying manifolds, but also for studying infinite cwtype spaces. Stratified morse theory, pioneered by goresky and macpherson see, aims to study the topology of the pair x, s by means of a real c.
Suppose now x is a smooth manifold with a fixed whitney stratification s. Macpherson extends classical morse theory to compact strati. Macpherson, global questions in the topology of singular spaces, proc. Full text views reflects the number of pdf downloads, pdfs. Mintype morse theory for configuration spaces of hard spheres. Ams proceedings of the american mathematical society. We describe the basics of this theory and prove fundamental theorems relating the topology of a general simplicial complex with the critical. Thus in the hypothesis of the previous theorem, we could have said that fis a c. Pdf the goreskymacpherson formula for toric arrangements. As an application of their stratified morse theory they. This paper is a survey of our work based on the strati. Effective reduction of goreskykottwitzmacpherson graphs cochet, charles, experimental mathematics, 2005.
The results in this essay are due to goresky and macpherson gm83c. First we discuss the morse theory of euclidean space stratified by an arrangement. A short introduction to morse theory alessandro fasse email. Bott, on the periodicity theorem for complex vector bundles, acts, mathematica, vol. Ams transactions of the american mathematical society. As we will see in chapter 4, however, most smooth functions are morse.
Goresky and macpherson wished to have a stratified morse theory to allow them to prove a lefschetz hyperplane theorem for the intersection homology of. The following definitions go back to the original paper. Wood, banach algebras and bott periodicity, topology, 4 196566, pp. Cohen and peter orlik for robert macpherson on the occasion of his sixtieth birthday. The topology of complex analytic varieties and the lefschetz hyperplane theorem. The most important result in this direction is due to goresky and macpherson. Macpherson, intersection homology and perverse sheaves, 1990 notes pdf. Here is a more detailed presentation of the contents. The local structure of complex analytic varieties 6 4. Symplectic quotients have symplectic singularities. The definition of an admissible morse function is inspired by stratified morse theory as developed by goresky and macpherson. On stratified morse theory journal of singularities.
The morse theory we present here for intersection homology and for a compact whitney stratified singular complex analytic. This book contains a complete and elementary survey including an extended bibliography on stratification theory, including its historical development. As has been observed by morse 1, any generic vector field v on a compact smooth manifold x with boundary gives rise to a stratification of the boundary by compact submanifolds, where. Morse theory, singularities, transversality theory, complex analytic varieties, lefschetz theorems, connectivity theorems. Stratified morse theory mark goresky, robert macpherson. While it is complicated to apply this theory in general, this is quite simple for manifolds with corners. Even before the appearance of 16, goresky and macpherson published two papers, strati. Morse theory is the method of studying the topology of a smooth manifold m m by the study of morse functions m. Topologically stratified spaces provide a purely topological setting for the study of singularities analogous to the more differentialgeometric theory of whitney.
The witten complex for algebraic curves with conelike. The corresponding integral operator has some unusual defects it is not a linear operator. Compact lie groups acting on pseudomanifolds popper, raimund, illinois journal of mathematics, 2000. The homology groups of the smooth locus of a singular variety. The kinds of theorems we would like to prove in morse theory will typically only apply to morse functions. The goreskymacpherson formula for toric arrangements. The principal references used were the book gm88 and the article gm83b. Inspired by the works of forman on discrete morse theory, which is a combinatorial adaptation to cell complexes of classical morse theory on manifolds, we introduce a discrete analogue of the stratified morse theory of goresky and macpherson 1988. Because goresky and macpherson are proving such a general and precise version of the theorem, it is a bit difficult to parse. Robert mark goresky born 1950 is a canadian mathematician who invented intersection homology with robert macpherson he received his ph. His thesis, titled geometric cohomology and homology of stratified objects, was written under the direction of macpherson.
Stratified morse theory by mark goresky, 9783642717161, available at book depository with free delivery worldwide. We extend the theory of euler integration from the class of constructible functions to that of tame r valued functions definable with respect to an ominimal structure. Hereby the contribution of a singular point of to the morse inequalities can be expressed in terms of the intersection cohomology of the local morse data of at. While the standard referenceforthispartis15, thereadermayalsoconsult16,25,26forabriefintroductionto thetheoryanditsapplications. The transition from topology to constructible sheaves in full generality is indicated there in an. A standard reference is the book of goresky macpherson gm2. Ausersguide kevin knudson departmentofmathematics,universityofflorida. We study the behavior of this stratification under deformations of the. We conclude the essay by considering the special case of complex algebraic curves. In chapter 1 we introduce the basic notions of the theory and we describe the main properties of. Many of the results in his thesis were published in 1981 by the american mathematical society. There are essentially no previouslyknown results which show how milnor fibers, real links, and complex links detect the dimension of the singular locus of a local complete intersection.
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