The $RO(\Pi B)$-graded $C_2$-equivariant ordinary cohomology of $B_{C_2}U(1)$

Abstract

We calculate the ordinary $C_2$-cohomology, with Burnside ring coefficients, of $C{P}_{C_2}^{\infty }=B_{C_2}U(1)$, the complex projective space, a model for the classifying space for $C_2$-equivariant complex line bundles. The $RO(C_2)$-graded Bredon ordinary cohomology was calculated by Gaunce Lewis, but here we extend to a larger grading in order to capture a more natural set of generators. These generators include the Euler class of the tautological bundle, which lies outside of the $RO(C_2)$-graded theory.

arXiv comment52 pages. Accepted manuscript; main difference from previous versions is the removal of Part 4 with general results about equivariant cohomology and the calculation of the cohomology of a point and EC_2

References in corpus (3)

• Bigraded cohomology of Z/2-equivariant Grassmannians
• The $C_2$-equivariant cohomology of complex projective spaces
• Equivariant fundamental classes in $RO(C_2)$-graded cohomology in $\underline{\mathbb{Z}/2}$-coefficients

Cited by in corpus (1)

• An algebraic $C_2$-equivariant Bézout's theorem