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Homotopy truncations of homotopically stratified spaces

David Chataur, Martintxo Saralegi-Aranguren, Daniel Tanré

Abstract

Intersection homology of Goresky and MacPherson can be defined from the Deligne sheaf, obtained from truncations of complexes of sheaves. As intersection homology is not the homology of a particular space, the search for a family of spaces whose homologies have properties analogous to intersection homology has developed. For some stratified spaces, M. Banagl has introduced such a family by using a topological truncation: the original link is replaced by a truncation of its homological Moore resolution. In this work, we study the dual approach in the Eckmann-Hilton sense : we consider the stratified space obtained by replacing the original link by a Postnikov approximation. The main result is that our construction restores the space constructed by Gajer to establish an intersection Dold-Thom theorem. We are conducting this study within the general framework of Quinn's homotopically stratified spaces.

Homotopy truncations of homotopically stratified spaces

Abstract

Intersection homology of Goresky and MacPherson can be defined from the Deligne sheaf, obtained from truncations of complexes of sheaves. As intersection homology is not the homology of a particular space, the search for a family of spaces whose homologies have properties analogous to intersection homology has developed. For some stratified spaces, M. Banagl has introduced such a family by using a topological truncation: the original link is replaced by a truncation of its homological Moore resolution. In this work, we study the dual approach in the Eckmann-Hilton sense : we consider the stratified space obtained by replacing the original link by a Postnikov approximation. The main result is that our construction restores the space constructed by Gajer to establish an intersection Dold-Thom theorem. We are conducting this study within the general framework of Quinn's homotopically stratified spaces.
Paper Structure (8 sections, 10 theorems, 13 equations)

This paper contains 8 sections, 10 theorems, 13 equations.

Table of Contents

  1. Outline of the paper
  2. Notation and convention
  3. Perversity on stratified spaces
  4. Quinn's homotopically stratified spaces
  5. Gajer simplicial set
  6. Postnikov truncation of links
  7. Linkwise localization
  8. Postnikov truncation

Key Result

Theorem A

Let $(X,\overline{p})$ be a perverse manifold homotopically stratified space with connected links and two strata, $S$ and $X\backslash S$. Then there is a homotopy equivalence between the realisation of ${\mathscr G}_{\overline{p}}X$ and the homotopy pushout of ${\tt eval}_{1}$ with the fibrewise Po

Theorems & Definitions (30)

  • Theorem A
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 2.1
  • ...and 20 more