2015/09/07 by Beatrice Bleile, Imre Bokor, Jonathan A. Hillman +1
Computer Science · Mathematics · #Cohomology #Conjecture #Cover (algebra) #Covering space #Duality (order theory) #Fundamental group #Geometric and Algebraic Topology #Homology (biology) #Homotopy #Homotopy and Cohomology in Algebraic Topology #Indecomposable module #Mapping class group #Topological and Geometric Data Analysis #math.AT #math.GT #msc:57N65 #msc:57P10
paper · pdf · doi:10.2140/agt.2018.18.3749
published as Algebraic and Geometric Topology 18 (2018), 3749--3788 · This paper incorporates and supersedes arXiv:1509.01928
openalex created_date 2016/06/24 · arxiv created 2017/03/20 · openalex publication_date 2018/12/11 · arxiv updated 2021/02/24 · openalex updated_date 2026/08/05
Turaev conjectured that the classification, realization and splitting results for Poincaré duality complexes of dimension [math] ( [math] –complexes) generalize to [math] –complexes with [math] –connected universal cover for [math] . 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.\n¶ We prove Turaev’s conjectures on realization and splitting. We show that a triple [math] , comprising a group [math] , a cohomology class [math] and a homology class [math] , can be realized by a [math] –complex with [math] –connected universal cover if and only if the Turaev map applied to [math] yields an equivalence. We show that a [math] –complex with [math] –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.\n¶ We then consider the indecomposable [math] –complexes of this type. When [math] is odd the results are similar to those for the case [math] . The indecomposables are either aspherical or have virtually free fundamental group. When [math] 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 [math] then it has two ends.