Topological Invariants of Stratified Spaces

ISBN-13: 9783642072482 / Angielski / Miękka / 2010 / 264 str.

Markus Banagl

Opis

The homology of manifolds enjoys a remarkable symmetry: Poincare 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. Poincare 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 Poincare 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.

Recenzje

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)

Spis treści

Elementary Sheaf Theory.- Homological Algebra.- Verdier Duality.- Intersection Homology.- Characteristic Classes and Smooth Manifolds.- Invariants of Witt Spaces.- T-Structures.- Methods of Computation.- Invariants of Non-Witt Spaces.- L2 Cohomology.

O autorze

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.

Blurb

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.

After carefully introducing sheaf theory, derived categories,
Verdier duality, stratification theories, intersection homology,
t-structures and perverse sheaves, the ultimate objective is to explain
the construction as well as algebraic and geometric properties of
invariants such as the signature and characteristic classes effectuated
by self-dual sheaves.

Highlights never before presented in book form include complete and
very 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.