Topological Invariants Of Stratified Spaces

Author : Markus Banagl
Publisher : Springer Science & Business Media
Page : 266 pages
File Size : 52,5 Mb
Release : 2007-02-16
Category : Mathematics
ISBN : 9783540385875

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Topological Invariants of Stratified Spaces by Markus Banagl Pdf

The central theme of this book is the restoration of Poincaré duality on stratified singular spaces by using Verdier-self-dual sheaves such as the prototypical intersection chain sheaf on a complex variety. Highlights include complete and detailed proofs of decomposition theorems for self-dual sheaves, explanation of methods for computing twisted characteristic classes and an introduction to the author's theory of non-Witt spaces and Lagrangian structures.

Author : Greg Friedman
Publisher : Cambridge University Press
Page : 491 pages
File Size : 51,5 Mb
Release : 2011-03-28
Category : Mathematics
ISBN : 9780521191678

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Topology of Stratified Spaces by Greg Friedman Pdf

This book explores the study of singular spaces using techniques from areas within geometry and topology and the interactions among them.

Author : Shmuel Weinberger
Publisher : University of Chicago Press
Page : 308 pages
File Size : 51,7 Mb
Release : 1994
Category : Mathematics
ISBN : 0226885674

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The Topological Classification of Stratified Spaces by Shmuel Weinberger Pdf

This book provides the theory for stratified spaces, along with important examples and applications, that is analogous to the surgery theory for manifolds. In the first expository account of this field, Weinberger provides topologists with a new way of looking at the classification theory of singular spaces with his original results. Divided into three parts, the book begins with an overview of modern high-dimensional manifold theory. Rather than including complete proofs of all theorems, Weinberger demonstrates key constructions, gives convenient formulations, and shows the usefulness of the technology. Part II offers the parallel theory for stratified spaces. Here, the topological category is most completely developed using the methods of "controlled topology." Many examples illustrating the topological invariance and noninvariance of obstructions and characteristic classes are provided. Applications for embeddings and immersions of manifolds, for the geometry of group actions, for algebraic varieties, and for rigidity theorems are found in Part III. This volume will be of interest to topologists, as well as mathematicians in other fields such as differential geometry, operator theory, and algebraic geometry.

Author : Shmuel Weinberger
Publisher : University of Chicago Press
Page : 314 pages
File Size : 43,6 Mb
Release : 1994
Category : Mathematics
ISBN : 0226885666

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The Topological Classification of Stratified Spaces by Shmuel Weinberger Pdf

This book provides the theory for stratified spaces, along with important examples and applications, that is analogous to the surgery theory for manifolds. In the first expository account of this field, Weinberger provides topologists with a new way of looking at the classification theory of singular spaces with his original results. Divided into three parts, the book begins with an overview of modern high-dimensional manifold theory. Rather than including complete proofs of all theorems, Weinberger demonstrates key constructions, gives convenient formulations, and shows the usefulness of the technology. Part II offers the parallel theory for stratified spaces. Here, the topological category is most completely developed using the methods of "controlled topology." Many examples illustrating the topological invariance and noninvariance of obstructions and characteristic classes are provided. Applications for embeddings and immersions of manifolds, for the geometry of group actions, for algebraic varieties, and for rigidity theorems are found in Part III. This volume will be of interest to topologists, as well as mathematicians in other fields such as differential geometry, operator theory, and algebraic geometry.

Author : Jerome Kaminker,American Mathematical Society
Publisher : American Mathematical Soc.
Page : 312 pages
File Size : 46,5 Mb
Release : 1990
Category : Mathematics
ISBN : 9780821851128

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Geometric and Topological Invariants of Elliptic Operators by Jerome Kaminker,American Mathematical Society Pdf

This volume contains the proceedings of the AMS-IMS-SIAM Summer Research Conference on ``Geometric and Topological Invariants of Elliptic Operators,'' held in August 1988 at Bowdoin College. Some of the themes covered at the conference and appearing in the articles are: the use of more sophisticated asymptotic methods to obtain index theorems, the study of the $\eta$ invariant and analytic torsion, and index theory on open manifolds and foliated manifolds. The current state of noncommutative differential geometry, as well as operator algebraic and $K$-theoretic methods, are also presented in several the articles. This book will be useful to researchers in index theory, operator algebras, foliations, and mathematical physics. Topologists and geometers are also likely to find useful the view the book provides of recent work in this area. In addition, because of the expository nature of several of the articles, it will be useful to graduate students interested in working in these areas.

Author : Markus Pflaum
Publisher : Springer Science & Business Media
Page : 233 pages
File Size : 54,5 Mb
Release : 2001-10-23
Category : Mathematics
ISBN : 9783540426264

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Analytic and Geometric Study of Stratified Spaces by Markus Pflaum Pdf

The book provides an introduction to stratification theory leading the reader up to modern research topics in the field. The first part presents the basics of stratification theory, in particular the Whitney conditions and Mather's control theory, and introduces the notion of a smooth structure. Moreover, it explains how one can use smooth structures to transfer differential geometric and analytic methods from the arena of manifolds to stratified spaces. In the second part the methods established in the first part are applied to particular classes of stratified spaces like for example orbit spaces. Then a new de Rham theory for stratified spaces is established and finally the Hochschild (co)homology theory of smooth functions on certain classes of stratified spaces is studied. The book should be accessible to readers acquainted with the basics of topology, analysis and differential geometry.

Author : David N Yetter
Publisher : World Scientific
Page : 238 pages
File Size : 48,7 Mb
Release : 2001-04-16
Category : Mathematics
ISBN : 9789814492249

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Functorial Knot Theory: Categories Of Tangles, Coherence, Categorical Deformations And Topological Invariants by David N Yetter Pdf

Almost since the advent of skein-theoretic invariants of knots and links (the Jones, HOMFLY, and Kauffman polynomials), the important role of categories of tangles in the connection between low-dimensional topology and quantum-group theory has been recognized. The rich categorical structures naturally arising from the considerations of cobordisms have suggested functorial views of topological field theory.This book begins with a detailed exposition of the key ideas in the discovery of monoidal categories of tangles as central objects of study in low-dimensional topology. The focus then turns to the deformation theory of monoidal categories and the related deformation theory of monoidal functors, which is a proper generalization of Gerstenhaber's deformation theory of associative algebras. These serve as the building blocks for a deformation theory of braided monoidal categories which gives rise to sequences of Vassiliev invariants of framed links, and clarify their interrelations.

Author : Dan Burghelea
Publisher : World Scientific
Page : 260 pages
File Size : 51,5 Mb
Release : 2017-08-16
Category : Mathematics
ISBN : 9789814618267

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New Topological Invariants For Real- And Angle-valued Maps: An Alternative To Morse-novikov Theory by Dan Burghelea Pdf

This book is about new topological invariants of real- and angle-valued maps inspired by Morse-Novikov theory, a chapter of topology, which has recently raised interest outside of mathematics; for example, in data analysis, shape recognition, computer science and physics. They are the backbone of what the author proposes as a computational alternative to Morse-Novikov theory, referred to in this book as AMN-theory.These invariants are on one side analogues of rest points, instantons and closed trajectories of vector fields and on the other side, refine basic topological invariants like homology and monodromy. They are associated to tame maps, considerably more general than Morse maps, that are defined on spaces which are considerably more general than manifolds. They are computable by computer implementable algorithms and have strong robustness properties. They relate the dynamics of flows that admit the map as 'Lyapunov map' to the topology of the underlying space, in a similar manner as Morse-Novikov theory does.

Author : Markus Banagl
Publisher : American Mathematical Soc.
Page : 83 pages
File Size : 41,9 Mb
Release : 2002
Category : Mathematics
ISBN : 9780821829882

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Extending Intersection Homology Type Invariants to Non-Witt Spaces by Markus Banagl Pdf

Intersection homology theory provides a way to obtain generalized Poincare duality, as well as a signature and characteristic classes, for singular spaces. For this to work, one has had to assume however that the space satisfies the so-called Witt condition. We extend this approach to constructing invariants to spaces more general than Witt spaces. We present an algebraic framework for extending generalized Poincare duality and intersection homology to singular spaces $X$ not necessarily Witt. The initial step in this program is to define the category $SD(X)$ of complexes of sheaves suitable for studying intersection homology type invariants on non-Witt spaces. The objects in this category can be shown to be the closest possible self-dual 'approximation' to intersection homology sheaves.It is therefore desirable to understand the structure of such self-dual sheaves and to isolate the minimal data necessary to construct them. As the main tool in this analysis we introduce the notion of a Lagrangian structure (related to the familiar notion of Lagrangian submodules for $(-1)^k$-Hermitian forms, as in surgery theory). We demonstrate that every complex in $SD(X)$ has naturally associated Lagrangian structures and conversely, that Lagrangian structures serve as the natural building blocks for objects in $SD(X).Our main result asserts that there is in fact an equivalence of categories between $SD(X)$ and a twisted product of categories of Lagrangian structures. This may be viewed as a Postnikov system for $SD(X)$ whose fibers are categories of Lagrangian structures. The question arises as to which varieties possess Lagrangian structures. To begin to answer that, we define the model-class of varieties with an ordered resolution and use block bundles to describe the geometry of such spaces. Our main result concerning these is that they have associated preferred Lagrangian structures, and hence self-dual generalized intersection homology sheaves.

Author : Greg Friedman
Publisher : Cambridge University Press
Page : 823 pages
File Size : 41,7 Mb
Release : 2020-09-24
Category : Mathematics
ISBN : 9781107150744

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Singular Intersection Homology by Greg Friedman Pdf

The first expository book-length introduction to intersection homology from the viewpoint of singular and piecewise linear chains.

Author : Markus Banagl
Publisher : Springer Science & Business Media
Page : 237 pages
File Size : 46,8 Mb
Release : 2010-07-08
Category : Mathematics
ISBN : 9783642125881

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Intersection Spaces, Spatial Homology Truncation, and String Theory by Markus Banagl Pdf

The present monograph introduces a method that assigns to certain classes of stratified spaces cell complexes, called intersection spaces, whose ordinary rational homology satisfies generalized Poincaré duality.

Author : Jean-Paul Brasselet
Publisher : American Mathematical Soc.
Page : 370 pages
File Size : 47,6 Mb
Release : 2008
Category : Singularities (Mathematics)
ISBN : 9780821844588

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Singularities I by Jean-Paul Brasselet Pdf

Author : Markus Banagl,Denis Vogel
Publisher : Springer Science & Business Media
Page : 363 pages
File Size : 48,8 Mb
Release : 2010-11-25
Category : Mathematics
ISBN : 9783642156373

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The Mathematics of Knots by Markus Banagl,Denis Vogel Pdf

The present volume grew out of the Heidelberg Knot Theory Semester, organized by the editors in winter 2008/09 at Heidelberg University. The contributed papers bring the reader up to date on the currently most actively pursued areas of mathematical knot theory and its applications in mathematical physics and cell biology. Both original research and survey articles are presented; numerous illustrations support the text. The book will be of great interest to researchers in topology, geometry, and mathematical physics, graduate students specializing in knot theory, and cell biologists interested in the topology of DNA strands.

Author : Laurenţiu G. Maxim
Publisher : Springer Nature
Page : 270 pages
File Size : 51,8 Mb
Release : 2019-11-30
Category : Mathematics
ISBN : 9783030276447

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Intersection Homology & Perverse Sheaves by Laurenţiu G. Maxim Pdf

This textbook provides a gentle introduction to intersection homology and perverse sheaves, where concrete examples and geometric applications motivate concepts throughout. By giving a taste of the main ideas in the field, the author welcomes new readers to this exciting area at the crossroads of topology, algebraic geometry, analysis, and differential equations. Those looking to delve further into the abstract theory will find ample references to facilitate navigation of both classic and recent literature. Beginning with an introduction to intersection homology from a geometric and topological viewpoint, the text goes on to develop the sheaf-theoretical perspective. Then algebraic geometry comes to the fore: a brief discussion of constructibility opens onto an in-depth exploration of perverse sheaves. Highlights from the following chapters include a detailed account of the proof of the Beilinson–Bernstein–Deligne–Gabber (BBDG) decomposition theorem, applications of perverse sheaves to hypersurface singularities, and a discussion of Hodge-theoretic aspects of intersection homology via Saito’s deep theory of mixed Hodge modules. An epilogue offers a succinct summary of the literature surrounding some recent applications. Intersection Homology & Perverse Sheaves is suitable for graduate students with a basic background in topology and algebraic geometry. By building context and familiarity with examples, the text offers an ideal starting point for those entering the field. This classroom-tested approach opens the door to further study and to current research.

Author : Akihiro Kanamori
Publisher : Springer Science & Business Media
Page : 555 pages
File Size : 47,6 Mb
Release : 2008-11-23
Category : Mathematics
ISBN : 9783540888673

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The Higher Infinite by Akihiro Kanamori Pdf

Over the years, this book has become a standard reference and guide in the set theory community. It provides a comprehensive account of the theory of large cardinals from its beginnings and some of the direct outgrowths leading to the frontiers of contemporary research, with open questions and speculations throughout.