On $(n-2)$-connected $2n$-dimensional Poincaré complexes with torsion-free homology

Abstract

Let $X$ be an $(n-2)$-connected $2n$-dimensional Poincaré complex with torsion-free homology, where $n\ge 4$. We prove that $X$ can be decomposed into a connected sum of two Poincaré complexes: one being $(n-1)$-connected, while the other having trivial $n$th homology group. Under the additional assumption that $H_n(X)=0$ and $S{q}^2:H^{n-1}(X;{\mathbb{Z}}_2)\to H^{n+1}(X;{\mathbb{Z}}_2)$ is trivial, we can prove that $X$ can be further decomposed into connected sums of Poincaré complexes whose $(n-1)$th homology is isomorphic to $\mathbb{Z}$. As an application of this result, we classify the homotopy types of such $2$-connected $8$-dimensional Poincaré complexes.

18 pages