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

Abstract: Let $X$ be an $(n-2)$-connected $2n$-dimensional Poincar\'e complex with torsion-free homology, where $n\geq 4$. We prove that $X$ can be decomposed into a connected sum of two Poincar\'e 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 $Sq^2:H^{n-1}(X;\mathbb{Z}_2)\to H^{n+1}(X;\mathbb{Z}_2)$ is trivial, we can prove that $X$ can be further decomposed into connected sums of Poincar\'e 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 Poincar\'e complexes.