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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
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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.

Free and Properly Discontinuous Actions of Groups on Homotopy 2n-spheres
Part of
Marek Golasiński, Daciberg Lima Gonçalves, Rolando Jimenez
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 61 / Issue 2 / May 2018

Published online by Cambridge University Press:

08 February 2018, pp. 305-327
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Let G be a group acting freely, properly discontinuously and cellularly on some finite dimensional CW-complex Σ(2n) which has the homotopy type of the 2n-sphere 𝕊2n. Then, that action induces a homomorphism G → Aut(H2n(Σ(2n))). We classify all pairs (G, φ), where G is a virtually cyclic group and φ: G → Aut(ℤ) is a homomorphism, which are realizable in the way above and the homotopy types of all possible orbit spaces as well. Next, we consider the family of all groups which have virtual cohomological dimension one and which act on some Σ(2n). Those groups consist of free groups and semi-direct products F ⋊ ℤ2 with F a free group. For a group G from the family above and a homomorphism φ: G → Aut(ℤ), we present an algebraic criterion equivalent to the realizability of the pair (G, φ). It turns out that any realizable pair can be realized on some Σ(2n) with dim Σ(2n) ≤ 2n + 1.

THE COHOMOLOGY RING OF ORBIT SPACES OF FREE $\mathbb{Z}_{2}$-ACTIONS ON SOME DOLD MANIFOLDS
Part of
- Fiber spaces and bundles
ANA MARIA M. MORITA, DENISE DE MATTOS, PEDRO L. Q. PERGHER
Journal:

Bulletin of the Australian Mathematical Society / Volume 97 / Issue 2 / April 2018

Published online by Cambridge University Press:

02 February 2018, pp. 340-348

Print publication:

April 2018
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We determine the possible $\mathbb{Z}_{2}$-cohomology rings of orbit spaces of free actions of $\mathbb{Z}_{2}$ (or fixed point free involutions) on the Dold manifold $P(1,n)$, where $n$ is an odd natural number.