$RO(C_2\times C_2)$-graded cohomology ring of a point and applications

Abstract: We describe the main properties of the $RO(C_2\times \Sigma_2)$-graded cohomology ring of a point and apply the results to compute the subring of motivic classes given by the Bredon motivic cohomology of real numbers and to compute $RO(C_2\times \Sigma_2)$-graded cohomology ring of $E_{\Sigma_2}C_2$. This generalizes Voevodsky's identification of motivic cohomology of real numbers with the positive cone of $RO(C_2)$ graded cohomology of a point.