Topological Invariants of Stratified Spaces

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Beschreibung

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.

Über den Autor

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.

Zusammenfassung

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.
The book first carefully introduces sheaf theory, derived categories, Verdier duality, stratification theories, intersection homology, t-structures and perverse sheaves. It then explains 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.

Inhaltsverzeichnis

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.

Details

Erscheinungsjahr:	2010
Fachbereich:	Geometrie
Genre:	Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Taschenbuch
Reihe:	Springer Monographs in Mathematics
Inhalt:	xii 264 S. 14 s/w Illustr. 264 p. 14 illus.
ISBN-13:	9783642072482
ISBN-10:	3642072488
Sprache:	Englisch
Ausstattung / Beilage:	Paperback
Einband:	Kartoniert / Broschiert
Autor:	Banagl, Markus
Auflage:	Softcover reprint of hardcover 1st ed. 2007
Hersteller:	Springer-Verlag GmbH Springer Berlin Heidelberg Springer Monographs in Mathematics
Maße:	235 x 155 x 16 mm
Von/Mit:	Markus Banagl
Erscheinungsdatum:	30.11.2010
Gewicht:	0,423 kg