Poincare Duality Complexes with Highly Connected Universal Covers

Abstract

Baues and Bleille showed that, up to oriented homotopy equivalence, a Poincare duality complex of dimension $n \ge 3$ with $(n-2)$-connected universal cover, is classified by its fundamental group, orientation class and the image of its fundamental class in the homology of the fundamental group. We generalise Turaev's results for the case $n=3$, by providing necessary and sufficient conditions for a triple $(G, \omega, \mu)$, comprising a group, $G$, $\omega \in H^{1}(G;\mathbb{Z}/2\mathbb{Z})$ and $\mu \in H_{3}(G, \mathbb{Z}^{\omega})$ to be realised by a Poincar\'e duality complex of dimension $n$ with $(n-2)$-connected universal cover, and by showing that such a complex is a connected sum of two such complexes if and only if its fundamental group is a free product of groups.