| Topological Invariants of Stratified Spaces 2007 Edition Contributor(s): Banagl, Markus (Author) |
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ISBN: 3540385851 ISBN-13: 9783540385851 Publisher: Springer OUR PRICE: $104.49 Product Type: Hardcover - Other Formats Published: January 2007 Annotation: The central theme of this book is the restoration of Poincar?? 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. |
| Additional Information |
| BISAC Categories: - Mathematics | Geometry - Differential - Mathematics | Topology - General |
| Dewey: 514.34 |
| LCCN: 2006935017 |
| Series: Springer Monographs in Mathematics |
| Physical Information: 0.82" H x 6.37" W x 9.38" (1.20 lbs) 264 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 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. |