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Loop space decompositions of $(2n-2)$-connected $(4n-1)$-dimensional Poincaré Duality complexes

Ruizhi Huang, Stephen Theriault

Abstract

Beben and Wu showed that if $M$ is a $(2n-2)$-connected $(4n-1)$-dimensional Poincaré Duality complex such that $n\geq 3$ and $H^{2n}(M;\mathbb{Z})$ consists only of odd torsion, then $ΩM$ can be decomposed up to homotopy as a product of simpler, well studied spaces. We use a result from \cite{BT2} to greatly simplify and enhance Beben and Wu's work and to extend it in various directions.

Loop space decompositions of $(2n-2)$-connected $(4n-1)$-dimensional Poincaré Duality complexes

Abstract

Beben and Wu showed that if is a -connected -dimensional Poincaré Duality complex such that and consists only of odd torsion, then can be decomposed up to homotopy as a product of simpler, well studied spaces. We use a result from \cite{BT2} to greatly simplify and enhance Beben and Wu's work and to extend it in various directions.

Paper Structure

This paper contains 6 sections, 27 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. The case when $H^{2n}(M;\mathbb{Z})\cong\mathbb{Z}/p^{r}\mathbb{Z}$
  4. The general case when $H^{2n}(M;\mathbb{Z})$ is odd torsion
  5. An extension to some $2$-torsion cases I
  6. An extension to some $2$-torsion cases II

Key Result

Theorem 1.1

Let $M$ be a $(2n-2)$-connected, $(4n-1)$-dimensional Poincaré Duality complex such that $n\geq 2$. Suppose that where each $p_{k}$ is an odd prime. Then with $V$ and $A$ chosen as above:

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • ...and 42 more