The Homotopy Type of a Poincar\'e Duality Complex after Looping

TL;DR

This paper investigates the homotopy types of certain high-dimensional manifolds, showing that their loop space homotopy type is determined by higher Bockstein operations under specific conditions.

Contribution

It establishes that for a class of high-dimensional Poincaré complexes, the loop space homotopy type is uniquely determined by algebraic operations on cohomology groups.

Findings

Loop space homotopy type determined by Bockstein actions

Results hold for n-2 connected, orientable Poincaré complexes with specific cohomology conditions

Stronger localization results at odd primes

Abstract

We answer a weaker version of the classification problem for the homotopy types of $(n-2)$-connected closed orientable $(2n-1)$-manifolds. Let $n\geq 6$ be an even integer, and $X$ be a $(n-2)$-connected finite orientable Poincar\'e $(2n-1)$-complex such that $H^{n-1}(X;\mathbb{Q})=0$ and $H^{n-1}(X;\mathbb{Z}_2)=0$. Then its loop space homotopy type is uniquely determined by the action of higher Bockstein operations on $H^{n-1}(X;\mathbb{Z}_p)$ for each odd prime $p$. A stronger result is obtained when localized at odd primes.