Search

3 results

2 - Stratified Spaces
Greg Friedman, Texas Christian University
Book:

Singular Intersection Homology

Published online:

18 September 2020

Print publication:

24 September 2020, pp 16-85
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Summary

Chapter 2 is an introduction to stratified spaces. We begin with filtered spaces and move progressively through more and more constrained classes, including manifold stratified spaces, locally cone-like spaces, the CS sets of Siebenmann, recursive CS sets, and topological and piecewise linear (PL) pseudomanifolds. To facilitate this last definition, we provide some background on PL topology. In the later sections of the chapter, we turn to some more specialized topics, including normalization of pseudomanifolds, pseudomanifolds with boundary, and other more specialized types of spaces, such as Whitney stratified spaces, Thom–Mather stratified spaces, and homotopically stratified spaces. We observe that the class of pseudomanifolds includes many spaces that arise naturally in other mathematical areas, such as singular varieties and orbit spaces of group actions. We also discuss stratified maps between stratified spaces and close with two specialized topics: intrinsic filtrations and products and joins of stratified spaces.

Greg Friedman
Published online:

18 September 2020

Print publication:

24 September 2020
- Book
- - Get access
    
    Buy a print copy
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Intersection homology is a version of homology theory that extends Poincaré duality and its applications to stratified spaces, such as singular varieties. This is the first comprehensive expository book-length introduction to intersection homology from the viewpoint of singular and piecewise-linear chains. Recent breakthroughs have made this approach viable by providing intersection homology and cohomology versions of all the standard tools in the homology tool box, making the subject readily accessible to graduate students and researchers in topology as well as researchers from other fields. This text includes both new research material and new proofs of previously-known results in intersection homology, as well as treatments of many classical topics in algebraic and manifold topology. Written in a detailed but expository style, this book is suitable as an introduction to intersection homology or as a thorough reference.

h-COBORDISMS AND MAPPING CYLINDER OBSTRUCTIONS
Part of
- PL-topology
- Topological manifolds
BOGDAN VAJIAC
Journal:

Journal of the Australian Mathematical Society / Volume 86 / Issue 2 / April 2009

Published online by Cambridge University Press:

01 April 2009, pp. 249-278

Print publication:

April 2009
- Article
- - You have access
- PDF
- Export citation
In this paper we prove a realizability theorem for Quinn’s mapping cylinder obstructions for stratified spaces. We prove a continuously controlled version of the s-cobordism theorem which we further use to prove the relation between the torsion of an h-cobordism and the mapping cylinder obstructions. This states that the image of the torsion of an h-cobordism is the mapping cylinder obstruction of the lower stratum of one end of the h-cobordism in the top filtration. These results are further used to prove a theorem about the realizability of end obstructions.