Obstructions for constructing equivariant fibrations
If \(G\) is a finite group and \({\mathcal H}\) a collection of subgroups of \(G\) closed under conjugation and taking subgroups, then a \(G\)-fibration over a \(G\)-\(CW\)-complex \(B\) is a \(G\)-equivariant fibration over \(B\) with fibers equivariantly homotopy equivalent to a space in a compatible family of \(H\)-spaces, \(H\in{\mathcal H}\). If the isotropy groups are in \({\mathcal H}\) and \(B^H\) is simply connected for every isotropy group \(H\), then there is an obstruction cocycle defined for a \(G\)-fibration over \(B^n\). The cocylce is an element of the Bredon cochain group at level \(n+ 1\) with coefficients in a local system of homotopy groups at level \(n\).NEWLINENEWLINE The main result of this paper is that this Bredon cocycle behaves like a classical obstruction cocycle, that is the cocycle vanishes if and only if the \(G\)-fibration extends to a \(G\)-fibration over \(B^{n+1}\) and the cohomology class of the cocycle vanishes if and only if the restriction of the \(G\)-fibration to \(B^{n-1}\) extends to a \(G\)-fibration over \(B^{n+1}\). In both cases, the total space of the extension is \(G\)-homotopy equivalent to a \(G\)-\(CW\)-complex.
scientific article; zbMATH DE number 6068599
scientific article