The Homotopy Type of a Poincaré Duality Complex after Looping

Abstract

We answer a weaker version of the classification problem for the homotopy types of $(n-2)$-connected closed orientable $(2n-1)$-manifolds. Let $n\ge 6$ be an even integer, and $X$ be a $(n-2)$-connected finite orientable Poincaré $(2n-1)$-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.

To be published in PEMS

Cited by in corpus (1)

• The Free Loop Space Homology of $(n-1)$-connected $2n$-manifolds