Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 18, Number 7 (2018), 3749-3788.
Poincaré duality complexes with highly connected universal cover
Turaev conjectured that the classification, realization and splitting results for Poincaré duality complexes of dimension (–complexes) generalize to –complexes with –connected universal cover for . 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 , comprising a group , a cohomology class and a homology class , can be realized by a –complex with –connected universal cover if and only if the Turaev map applied to yields an equivalence. We show that a –complex with –connected universal cover is a nontrivial connected sum of two such complexes if and only if its fundamental group is a nontrivial free product of groups.
We then consider the indecomposable –complexes of this type. When is odd the results are similar to those for the case . The indecomposables are either aspherical or have virtually free fundamental group. When 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 then it has two ends.
Algebr. Geom. Topol., Volume 18, Number 7 (2018), 3749-3788.
Received: 31 October 2016
Revised: 7 July 2018
Accepted: 27 July 2018
First available in Project Euclid: 18 December 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Bleile, Beatrice; Bokor, Imre; Hillman, Jonathan A. Poincaré duality complexes with highly connected universal cover. Algebr. Geom. Topol. 18 (2018), no. 7, 3749--3788. doi:10.2140/agt.2018.18.3749. https://projecteuclid.org/euclid.agt/1545102053