Poincare Duality Complexes with Highly Connected Universal Covers

TL;DR

This paper characterizes when certain high-dimensional Poincare duality complexes with highly connected universal covers can be realized from algebraic data, extending known results for dimension three.

Contribution

It generalizes Turaev's classification for dimension three to higher dimensions, providing necessary and sufficient conditions for realization of algebraic triples.

Findings

Provides conditions for realization of Poincare duality complexes from algebraic data.

Shows such complexes decompose as connected sums when fundamental groups are free products.

Extends classification results from dimension three to higher dimensions.

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.