A product theorem for \(\Omega B \Gamma_ G\)
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 \(q\).
\(G\) structure invariant under the parallelism along the leaves
scientific article; zbMATH DE number 416870