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: H^n-1 (X;Z₂) H^n+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.