On $(n-2)$-connected $2n$-dimensional Poincar\'e complexes with torsion-free homology

Let X be an $(n-2)$-connected $2n$-dimensional Poincar\'e complex with torsion-free homology, where n≥ 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 nth homology group. Under the additional assumption that Hₙ(X)=0 and Sq²:Hⁿ⁻¹(X;Z₂)→ Hⁿ⁺¹(X;Z₂) 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 Z. As an application of this result, we classify the homotopy types of such $2$-connected $8$-dimensional Poincar\'e complexes.