Approximately fibering a manifold over an aspherical one
type: Item
instance of: scholarly article
title: Approximately fibering a manifold over an aspherical one
zbMATH Open document ID: 1385.57032
author: Wolfgang Lück
author: Wolfgang Steimle
author: F. Thomas Farrell
published in: Mathematische Annalen
publication date: 2018-02-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1603.07934
review text: A map \(p: M \;\rightarrow \;B\) between closed manifolds is a manifold approximate fibration (MAF) iff and only if the preimage of any open ball in \(B\) has the weak homotopy type of the homotopy fiber of \(p\). A key intermediate result of this paper is that given a continuous map \(p: M \rightarrow B\) that satisfies certain conditions there exists a homotopy equivalence \(f: M \rightarrow E\) with \(E\) a compact ENR and an approximate fibration \(q : E \rightarrow B\) with \(p = q\circ f\). Given such a factorization denote by \(\tau(f) \in Wh(\pi_{1}(E))\) the Whitehead torsion of the homotopy equivalence \(f\). The image of this in the cokernel \(NWh(p)\) of the assembly map \[ H_{1}^{\pi}(\tilde{B}, Wh(p)) \rightarrow \pi_{1}(Wh(M)) \] is called the tight torsion. A naive conjecture is that the map \(p\) is homotopic to a MAF if and only if \(N\tau(p) = 0\). The authors give an example to show that this conjecture is false. However they also give several conditions where they show that the conjecture is true. A stabilization of \(p\) is a map \(M \times T^{s} \rightarrow B \times T^{s}\), where \(T^{s}\) is the \(s\)-dimensional torus. The authors show that in case \(B\) is PL and the map \(p\) is surjective on \(\pi_{1}\), the homotopy fibre of \(p\) is finitely dominated and the \(L\)-theoretic Farrrell-Jones conjecture is true for \(\pi_{1}(M)\) then there exists an \(s\) such that the stabilized map is a MAF. The main results then follow from theorems on splitting such a stabilization.
Mathematics Subject Classification ID: 57R65
Mathematics Subject Classification ID: 19J10
zbMATH DE Number: 6837440
zbMATH Keywords: manifold approximate fibration
zbMATH Keywords: tight torsion
zbMATH Keywords: aspherical manifold
zbMATH Keywords: Farrell-Jones Conjecture
zbMATH Keywords: aspherical closed manifold
reviewed by: Jonathan Hodgson
MaRDI profile type: Publication
OpenAlex ID: W2963920007
arXiv ID: 1603.07934
Wikidata QID: Q122860379
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Approximately fibering a manifold over an aspherical one (DOI)
DOI: 10.1007/S00208-017-1535-1

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