The \(RO(G)\)-graded Serre spectral sequence (Q2267990)

In this paper the author extends the Serre spectral sequence of \textit{I. Moerdijk} and \textit{J. A. Svensson} [Proc. Am. Math. Soc. 118, No.1, 263--278 (1993; Zbl 0791.55004)] from Bredon cohomology to the \(RO(G)\)-graded cohomology for finite groups \(G\), in particular to the case where \(G\) is \(\mathbb{Z}_2\). Here \(RO(G)\) is the Grothendieck ring of virtual representations of \(G\). It is noted that an \(RO(G)\)-graded cohomology was constructed by \textit{G. Lewis, J. P. May} and \textit{J. E. McClure} [Bull. Am. Math. Soc., New Ser. 4, 208--212 (1981; Zbl 0477.55009)] extending \textit{G. E. Bredon}'s cohomology [Bull. Am. Math. Soc. 73, 266-268 (1967; Zbl 0162.27301)]. If \(X\) is equivariantly 1-connected and \(f:E \to X\) is a fibration of \(G\)-spaces with fiber \(F\), then for every \(V \in RO(G)\) and every Mackey Functor \(M\), it is proved that there exists a spectral sequence with \[ E^{p,q}_2 = H^{p,0}(X; \underline{H}^{V+q}(F;M)) \Longrightarrow H^{V + p+q}(E;M), \] where \(\underline{H}^{V+q}(F,M)\) is the associated constant coefficient of the local coefficient system \(\mathcal{H}^{V+q}(f;M)\). In the special case where \(G = \mathbb{Z}_2\), \(X\) is equivariantly 1-connected and \(f: E \to X\) is a vector bundle with fiber \(\mathbb{R}^{m,n}\) over the base point, the author considers the spectral sequence above for the associated equivariant projective bundle \(\mathbb{P}(f) : \mathbb{P}(E) \to X\), with constant \(M = \mathbb{Z}_2\) coefficients and proves that it collapses. By using this result, he computes the cohomology of certain projective bundles and loop spaces.