An Introduction To Intersection Homology Theory

Author : Frances Kirwan,Jonathan Woolf
Publisher : CRC Press
Page : 250 pages
File Size : 54,5 Mb
Release : 2006-06-07
Category : Mathematics
ISBN : 1584881844

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An Introduction to Intersection Homology Theory, Second Edition by Frances Kirwan,Jonathan Woolf Pdf

Now more that a quarter of a century old, intersection homology theory has proven to be a powerful tool in the study of the topology of singular spaces, with deep links to many other areas of mathematics, including combinatorics, differential equations, group representations, and number theory. Like its predecessor, An Introduction to Intersection Homology Theory, Second Edition introduces the power and beauty of intersection homology, explaining the main ideas and omitting, or merely sketching, the difficult proofs. It treats both the basics of the subject and a wide range of applications, providing lucid overviews of highly technical areas that make the subject accessible and prepare readers for more advanced work in the area. This second edition contains entirely new chapters introducing the theory of Witt spaces, perverse sheaves, and the combinatorial intersection cohomology of fans. Intersection homology is a large and growing subject that touches on many aspects of topology, geometry, and algebra. With its clear explanations of the main ideas, this book builds the confidence needed to tackle more specialist, technical texts and provides a framework within which to place them.

Author : Armand Borel
Publisher : Springer Science & Business Media
Page : 243 pages
File Size : 54,8 Mb
Release : 2009-05-21
Category : Mathematics
ISBN : 9780817647650

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Intersection Cohomology by Armand Borel Pdf

This book is a publication in Swiss Seminars, a subseries of Progress in Mathematics. It is an expanded version of the notes from a seminar on intersection cohomology theory, which met at the University of Bern, Switzerland, in the spring of 1983. This volume supplies an introduction to the piecewise linear and sheaf-theoretic versions of that theory as developed by M. Goresky and R. MacPherson in Topology 19 (1980), and in Inventiones Mathematicae 72 (1983). Some familiarity with algebraic topology and sheaf theory is assumed.

Author : Frances Clare Kirwan
Publisher : Halsted Press
Page : 169 pages
File Size : 46,8 Mb
Release : 1988
Category : Algebra, Homological
ISBN : 0470211989

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An Introduction to Intersection Homology Theory by Frances Clare Kirwan Pdf

Author : Laurenţiu G. Maxim
Publisher : Springer Nature
Page : 270 pages
File Size : 54,8 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 : Greg Friedman
Publisher : Cambridge University Press
Page : 823 pages
File Size : 52,7 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 : James W. Vick
Publisher : Springer Science & Business Media
Page : 258 pages
File Size : 42,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461208815

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Homology Theory by James W. Vick Pdf

This introduction to some basic ideas in algebraic topology is devoted to the foundations and applications of homology theory. After the essentials of singular homology and some important applications are given, successive topics covered include attaching spaces, finite CW complexes, cohomology products, manifolds, Poincare duality, and fixed point theory. This second edition includes a chapter on covering spaces and many new exercises.

Author : Markus Banagl
Publisher : American Mathematical Soc.
Page : 83 pages
File Size : 52,9 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 : Markus Banagl
Publisher : Springer Science & Business Media
Page : 237 pages
File Size : 51,9 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 : David Chataur,Martintxo Saralegi-Aranguren,Daniel Tanré
Publisher : American Mathematical Soc.
Page : 108 pages
File Size : 47,8 Mb
Release : 2018-08-09
Category : Electronic
ISBN : 9781470428877

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Intersection Cohomology, Simplicial Blow-Up and Rational Homotopy by David Chataur,Martintxo Saralegi-Aranguren,Daniel Tanré Pdf

Let X be a pseudomanifold. In this text, the authors use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson. The authors do it simplicially in the setting of a filtered version of face sets, also called simplicial sets without degeneracies, in the sense of C. P. Rourke and B. J. Sanderson. They define perverse local systems over filtered face sets and intersection cohomology with coefficients in a perverse local system. In particular, as announced above when X is a pseudomanifold, the authors get a perverse local system of cochains quasi-isomorphic to the intersection cochains of Goresky and MacPherson, over a field. We show also that these two complexes of cochains are quasi-isomorphic to a filtered version of Sullivan's differential forms over the field Q. In a second step, they use these forms to extend Sullivan's presentation of rational homotopy type to intersection cohomology.

Author : P. J. Hilton,S. Wylie
Publisher : CUP Archive
Page : 504 pages
File Size : 52,8 Mb
Release : 1967
Category : Mathematics
ISBN : 0521094224

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Homology Theory by P. J. Hilton,S. Wylie Pdf

This account of algebraic topology is complete in itself, assuming no previous knowledge of the subject. It is used as a textbook for students in the final year of an undergraduate course or on graduate courses and as a handbook for mathematicians in other branches who want some knowledge of the subject.

Author : Markus Banagl
Publisher : Springer Science & Business Media
Page : 266 pages
File Size : 40,5 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 : William Fulton
Publisher : American Mathematical Soc.
Page : 98 pages
File Size : 55,8 Mb
Release : 1984
Category : Mathematics
ISBN : 9780821807040

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Introduction to Intersection Theory in Algebraic Geometry by William Fulton Pdf

Introduces some of the main ideas of modern intersection theory, traces their origins in classical geometry and sketches a few typical applications. Suitable for graduate students in mathematics, this book describes the construction and computation of intersection products by means of the geometry of normal cones.

Author : David Eisenbud,Joe Harris
Publisher : Cambridge University Press
Page : 633 pages
File Size : 40,5 Mb
Release : 2016-04-14
Category : Mathematics
ISBN : 9781107017085

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3264 and All That by David Eisenbud,Joe Harris Pdf

3264, the mathematical solution to a question concerning geometric figures.

Author : Susan Montgomery
Publisher : American Mathematical Soc.
Page : 258 pages
File Size : 41,5 Mb
Release : 1993-10-28
Category : Mathematics
ISBN : 9780821807385

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Hopf Algebras and Their Actions on Rings by Susan Montgomery Pdf

The last ten years have seen a number of significant advances in Hopf algebras. The best known is the introduction of quantum groups, which are Hopf algebras that arose in mathematical physics and now have connections to many areas of mathematics. In addition, several conjectures of Kaplansky have been solved, the most striking of which is a kind of Lagrange's theorem for Hopf algebras. Work on actions of Hopf algebras has unified earlier results on group actions, actions of Lie algebras, and graded algebras. This book brings together many of these recent developments from the viewpoint of the algebraic structure of Hopf algebras and their actions and coactions. Quantum groups are treated as an important example, rather than as an end in themselves. The two introductory chapters review definitions and basic facts; otherwise, most of the material has not previously appeared in book form. Providing an accessible introduction to Hopf algebras, this book would make an excellent graduate textbook for a course in Hopf algebras or an introduction to quantum groups.

Author : Mark Goresky,Robert MacPherson
Publisher : Springer Science & Business Media
Page : 279 pages
File Size : 43,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642717147

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Stratified Morse Theory by Mark Goresky,Robert MacPherson Pdf

Due to the lack of proper bibliographical sources stratification theory seems to be a "mysterious" subject in contemporary mathematics. This book contains a complete and elementary survey - including an extended bibliography - on stratification theory, including its historical development. Some further important topics in the book are: Morse theory, singularities, transversality theory, complex analytic varieties, Lefschetz theorems, connectivity theorems, intersection homology, complements of affine subspaces and combinatorics. The book is designed for all interested students or professionals in this area.