A product theorem for \(\Omega B \Gamma_ G\)
type: Item
instance of: scholarly article
title: A product theorem for \(\Omega B \Gamma_ G\)
zbMATH Open document ID: 0786.57012
DOI: 10.1016/0166-8641(93)90075-O
published in: Topology and its Applications
publication date: 1993-09-22T00:00:00Z
review text: Let \(G\subset \text{GL}(q,\mathbb{R})\) and let \(B\Gamma_ G\) be the classifying space of \(G\)-foliations, i.e. of codimension \(q\) foliations whose normal bundle have a \(G\) structure invariant under the parallelism along the leaves. Let \(F\Gamma_ G\) be the homotopy fibre of the classifying map \(B\Gamma_ G\to BG\) of the universal normal bundle. The main result of the paper is: If \(F\Gamma_ G\) is \(N\)-connected and if there are subgroups \(K_ 1,\dots,K_ r\) of \(G\) such that \(\dim K_ j\leq N\) for all \(j\) and such that the map \(K_ 1\times\dots\times K_ r\to G\) induced by multiplication in \(G\) is a homotopy equivalence, then the loop space \(\Omega B\Gamma_ G\) is homotopy equivalent (as space, not as an \(H\)- space) to \(G\times\Omega F\Gamma_ G\). As an application it is proved that for \(B\Gamma_ q^ +\) (i.e. \(G= \text{GL}^ + (q,\mathbb{R})\)) and \(q\leq 4\) one has \(\Omega B\Gamma_ q^ + \simeq \text{SO}(q)\times \Omega F\Gamma_ q\). It is conjectured that this homotopy equivalence holds for all \(q\).
Mathematics Subject Classification ID: 57R32
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 55R40
zbMATH DE Number: 416870
zbMATH Keywords: classifying space of \(G\)-foliations
zbMATH Keywords: \(G\) structure invariant under the parallelism along the leaves
zbMATH Keywords: universal normal bundle
zbMATH Keywords: loop space
author: Steven Hurder
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 416870