Poincar\'e duality complexes with highly connected universal cover

TL;DR

This paper proves Turaev's conjectures on the classification, realization, and splitting of high-dimensional Poincaré duality complexes with highly connected universal covers, extending known results from the 3-dimensional case.

Contribution

It establishes conditions for realizing and decomposing PD$_n$-complexes with highly connected universal covers, confirming Turaev's conjectures on their classification and structure.

Findings

Classification up to homotopy by fundamental group, orientation, and fundamental class image.

Connected sums correspond to free product decompositions of fundamental groups.

Indecomposable complexes are either aspherical or have virtually free fundamental groups.

Abstract

Turaev conjectured that the classification, realization and splitting results for Poincar\'e duality complexes of dimension $3$ (PD$_{3}$-complexes) generalize to PD$_{n}$-complexes with $(n-2)$-connected universal cover for $n \ge 3$. Baues and Bleile showed that such complexes are classified, up to oriented homotopy equivalence, by the triple consisting of their fundamental group, orientation class and the image of their fundamental class in the homology of the fundamental group, verifying Turaev's conjecture on classification. We prove Turaev's conjectures on realization and splitting. We show that a triple $(G, \omega, \mu)$, comprising a group, $G$, a cohomology class $\omega \in H^{1}\left(G, \mathbb{Z}/2\mathbb{Z}\right)$ and a homology class $\mu \in H_{n}(G, \mathbb{Z}^{\omega})$, can be realized by a PD$_{n}$-complex with $(n-2)$-connected universal cover if and only if the Turaev map applied to $\mu$ yields is an equivalence. We show that such a PD$_{n}$-complex is a connected sum of two such complexes if and only if its fundamental group is a free product of groups. We then consider the indecomposable $PD_n$-complexes of this type. When $n$ is odd the results are similar to those for the case $n=3$. The indecomposables are either aspherical or have virtually free fundamental group. When $n$ is even the indecomposables include manifolds which are neither aspherical nor have virtually free fundamental group, but if the group is virtually free and has no dihedral subgroup of order $>2$ then it has two ends.