A topological splitting theorem for Poincaré duality groups and high-dimensional manifolds
type: Item
instance of: scholarly article
title: A topological splitting theorem for Poincaré duality groups and high-dimensional manifolds
zbMATH Open document ID: 1309.57020
author: A. Kar
author: Graham Niblo
published in: Geometry \& Topology
publication date: 2013-08-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1110.2041
review text: The authors show that, ``for a wide class of manifold pairs, \(N, M\) with \(\dim(M) = \dim (N) + 1\), every \(\pi_1\)--injective map \(f: N \rightarrow M\) factorises up to homotopy as a finite cover of an embedding.'' This main result is reformulated subject to the Borel conjecture. As a corollary the authors obtain a topological rigidity theorem which they compare to the geometric superrigidity theorem by Mok, Siu and Yeung. The proof of the main result proceeds in two stages, the first one group theoretic and the second one topological in nature. The first stage is a splitting theorem for Poincaré duality groups. It involves Kazdhan's property \((T)\), is of independent interest, and provides a splitting of the fundamental group of \(M\). The second stage uses Cappell's surgery methods. These are summarised and require restriction to dimension \(N \geq 5\). The paper under review discusses the case of dimension \(N\) even and the authors intend to treat the odd dimensional case in a forthcoming paper. Finally, the main result yields an obstruction to the existence of \(\pi_1\)--injective continuous maps. The paper is clearly structured and well written.
reviewed by: Beatrice Bleile
Mathematics Subject Classification ID: 57N35
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 57P10
Mathematics Subject Classification ID: 57Q20
Mathematics Subject Classification ID: 57R67
zbMATH DE Number: 6198084
zbMATH Keywords: torus theorem
zbMATH Keywords: Poincaré duality group
zbMATH Keywords: Bass-Serre theory
zbMATH Keywords: Kazhdan's property \((T)\)
zbMATH Keywords: Borel conjecture
zbMATH Keywords: surgery
zbMATH Keywords: Cappell's splitting theorem
zbMATH Keywords: embeddings
zbMATH Keywords: rigidity
zbMATH Keywords: geometric group theory
zbMATH Keywords: quaternionic hyperbolic manifolds
MaRDI profile type: Publication
OpenAlex ID: W3105663029
Wikidata QID: Q125121076
arXiv ID: 1110.2041
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DOI: 10.2140/GT.2013.17.2203
description: scientific article; zbMATH DE number 6198084