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Topological Invariants of Stratified Spaces

  • Book
  • © 2007

Overview

Authors:
  1. M. Banagl

    1. Mathematisches Institut, Universität Heidelberg, Heidelberg, Germany

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Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-viii
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  2. Elementary Sheaf Theory

    Pages 1-25

  3. Homological Algebra

    Pages 27-57

  4. Verdier Duality

    Pages 59-70

  5. Intersection Homology

    Pages 71-98

  6. Characteristic Classes and Smooth Manifolds

    Pages 99-122

  7. Invariants of Witt Spaces

    Pages 123-140

  8. T-Structures

    Pages 141-159

  9. Methods of Computation

    Pages 161-216

  10. Invariants of Non-Witt Spaces

    Pages 217-241

  11. L2 Cohomology

    Pages 243-249

  12. Back Matter

    Pages 251-259
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Keywords

About this book

The homology of manifolds enjoys a remarkable symmetry: Poincaré duality. If the manifold is triangulated, then this duality can be established by associating to a s- plex its dual block in the barycentric subdivision. In a manifold, the dual block is a cell, so the chain complex based on the dual blocks computes the homology of the manifold. Poincaré duality then serves as a cornerstone of manifold classi cation theory. One reason is that it enables the de nition of a fundamental bordism inva- ant, the signature. Classifying manifolds via the surgery program relies on modifying a manifold by executing geometric surgeries. The trace of the surgery is a bordism between the original manifold and the result of surgery. Since the signature is a b- dism invariant, it does not change under surgery and is thus a basic obstruction to performing surgery. Inspired by Hirzebruch’s signature theorem, a method of Thom constructs characteristic homology classes using the bordism invariance of the s- nature. These classes are not in general homotopy invariants and consequently are ne enough to distinguish manifolds within the same homotopy type. Singular spaces do not enjoy Poincaré duality in ordinary homology. After all, the dual blocks are not cells anymore, but cones on spaces that may not be spheres. This book discusses when, and how, the invariants for manifolds described above can be established for singular spaces.

Reviews

From the reviews:

"This is an excellent book, highly recommended to anyone interested in studying the topology of singular spaces. With modest prerequisites, the author defines intersection homology (both chain- and sheaf-theoretic), gives a self-contained treatment of t-structures and perverse sheaves, and explains the construction as well as algebraic and geometric properties of invariants such as the signature and L-classes associated to self-dual sheaves." (Laurentiu G. Maxim, Mathematical Reviews, Issue 2007 j)

"In the book, the construction of these invariants for stratified singular spaces is presented, as well as some methods for their computation. Well written and with modest prerequisites concerning (co)homology theory, simplicial complexes and some basic notions of differential topology, the book is accessible to graduate students. Also, it is useful for the research mathematician wishing to learn about intersection homology and the invariants of singular spaces." (Gheorghe Pitis, Zentralblatt MATH, Vol. 1108 (10), 2007)

Authors and Affiliations

  • Mathematisches Institut, Universität Heidelberg, Heidelberg, Germany

    M. Banagl

About the author

EMPLOYMENT: Since 2004: Professor at the Ruprecht-Karls-Universität Heidelberg, Germany
2002 - 2004: Assistant Professor (tenure track) at the University of Cincinnati, USA
1999 - 2002: Van Vleck Assistant Professor at the University of Wisconsin - Madison, USA

EDUCATION: Ph.D. Mathematics, Courant Institute (New York University), May 1999.
Field: Topology.
Dissertation Title: Extending Intersection Homology Type Invariants to non-Witt Spaces.

RESEARCH AREA: Algebraic and Geometric Topology, Stratified Spaces.

Bibliographic Information

  • Book Title: Topological Invariants of Stratified Spaces

  • Authors: M. Banagl

  • Series Title: Springer Monographs in Mathematics

  • DOI: https://doi.org/10.1007/3-540-38587-8

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2007

  • Hardcover ISBN: 978-3-540-38585-1Published: 09 January 2007

  • Softcover ISBN: 978-3-642-07248-2Published: 30 November 2010

  • eBook ISBN: 978-3-540-38587-5Published: 16 February 2007

  • Series ISSN: 1439-7382

  • Series E-ISSN: 2196-9922

  • Edition Number: 1

  • Number of Pages: XII, 264

  • Number of Illustrations: 14 b/w illustrations

  • Topics: Topology, Differential Geometry, Algebraic Topology

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