The \(C_2\)-equivariant cohomology of complex projective spaces (Q2078879)

This paper contributes to a small but growing body of computations of equivariant cohomology graded on the representations of the fundamental groupoid of a \(G\)-space, \(RO(\Pi_G X)\). This ordinary equivariant cohomology theory, developed by \textit{S. R. Costenoble} et al. [Homology Homotopy Appl. 3, No. 2, 265--339 (2001; Zbl 0994.55009)], and further explored by \textit{S. R. Costenoble} and \textit{S. Waner} [Equivariant ordinary homology and cohomology. Cham: Springer (2016; Zbl 1362.55001)], is an extension of \(RO(G)\)-graded Bredon cohomology with coefficients in a Mackey functor. In the extended grading, Costenoble previously computed the cohomology of complex projective spaces of complete \(C_p\)-universes, where \(C_p\) is the cyclic group of prime order \(p\) (see [\textit{S. R. Costenoble}, ``The \(\mathbb{Z}/2\) ordinary cohomology of \(B_GU(1)\) for \(p=2\)'', Preprint, \url{arXiv:1312.0926}; \textit{S. R. Costenoble}, ``The \(\mathbb{Z}/p\) ordinary cohomology of \(B_GU(1)\)'', Preprint, \url{arXiv:1708.06009}] for odd primes). The present work concentrates on \(C_2\)-spaces. The authors use the computation of the cohomology of the infinite projective space \(B = \mathbb{P}(\mathbb{C}^{\infty} \oplus \mathbb{C}^{\infty \sigma})\) from [loc. cit.] to compute the \(RO(\Pi B)\)-graded cohomology of the finite complex projective spaces \(\mathbb{P}(\mathbb{C}^{n} \oplus \mathbb{C}^{m\sigma})\). They give a complete description of the cohomology as an algebra over the \(RO(C_2)\)-graded cohomology of a point with coefficients in the Burnside ring Mackey functor. As an application, they formulate and prove an equivariant version of Bézout's theorem.