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.
Categories: math.AT math.GT