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Top cell attachment for a Poincare Duality complex

Stephen Theriault

Abstract

Let M be a simply-connected closed Poincare Duality complex of dimension n. Then M is obtained by attaching a cell of highest dimension to its (n-1)-skeleton M'. Conditions are given for when the skeletal inclusion i:M' --> M has the property that the based loops on i has a right homotopy inverse. This is an integral version of the rational statement that such a right homotopy inverse always exists provided the rational cohomology of M is not generated by a single element. New methods are developed in order to do the integral case. These lead to p-local versions and recover the full rational statement. Families for which the integral statement holds include moment-angle manifolds and quasi-toric manifolds.

Top cell attachment for a Poincare Duality complex

Abstract

Let M be a simply-connected closed Poincare Duality complex of dimension n. Then M is obtained by attaching a cell of highest dimension to its (n-1)-skeleton M'. Conditions are given for when the skeletal inclusion i:M' --> M has the property that the based loops on i has a right homotopy inverse. This is an integral version of the rational statement that such a right homotopy inverse always exists provided the rational cohomology of M is not generated by a single element. New methods are developed in order to do the integral case. These lead to p-local versions and recover the full rational statement. Families for which the integral statement holds include moment-angle manifolds and quasi-toric manifolds.
Paper Structure (12 sections, 29 theorems, 104 equations)

This paper contains 12 sections, 29 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. A decomposition theorem
  3. Homotopy fibrations and homotopy co-actions
  4. A splitting criterion
  5. Inert attaching maps I
  6. A localized version of Theorem \ref{['inert']} when $m$ is odd
  7. A localized version of Theorem \ref{['inert']} when $m$ is even
  8. Inert attaching maps II
  9. The rational case
  10. Two applications
  11. The homotopy fibre of the inclusion $\overline{M}\stackrel{i} {\longrightarrow}M$
  12. Connected sums

Key Result

Theorem 1.1

Let $M$ be an $(m-1)$-connected, closed Poincaré Duality complex of dimension $n$, where $2\leq m<n$. If there is a map $M\stackrel{} {\longrightarrow}S^{m}$ having a right homotopy inverse, then the attaching map for the top cell of $M$ is inert.

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Definition 3.4
  • Lemma 3.5
  • proof
  • ...and 49 more