The RO(ΠP)-graded F2-cohomology of P=B_{C2}O(1) is the explicit ring M2[u10,u11,a10,a11,e,ν] with u10u11=ue, a10u11+a11u10=ae, e2=ν2=1.
The $C_2$-equivariant ordinary cohomology of $BU(2)$
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abstract
We calculate the ordinary $C_2$-cohomology, with Burnside ring coefficients, of $BU(2)$, the classifying space for $C_2$-equivariant complex 2-plane bundles, using an extended grading that allows us to capture a more natural set of generators. This allows us to define characteristic classes for such bundles. Combined with earlier calculations, it also allows us to define characteristic numbers for equivariant complex lines and surfaces and we give some sample computations.
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Parametrized $\underline{\mathbb{F}}_2$-Cohomology of $B_{C_2}O(1)$
The RO(ΠP)-graded F2-cohomology of P=B_{C2}O(1) is the explicit ring M2[u10,u11,a10,a11,e,ν] with u10u11=ue, a10u11+a11u10=ae, e2=ν2=1.