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

TL;DR

This paper studies the structure of high-dimensional Poincaré complexes with torsion-free homology, showing they can be decomposed into simpler complexes and classifying certain 8-dimensional cases.

Contribution

It proves a decomposition theorem for $(n-2)$-connected $2n$-dimensional Poincaré complexes with torsion-free homology, and classifies homotopy types of 2-connected 8-dimensional cases.

Findings

Decomposition into connected sums of Poincaré complexes with specific connectivity properties.

Further decomposition under additional homology and Steenrod square conditions.

Complete classification of homotopy types for 2-connected 8-dimensional Poincaré complexes.

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.