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

We describe the main properties of the $RO(C_2\times {\mathrm{\Sigma }}_2)$-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 $RO(C_2\times {\mathrm{\Sigma }}_2)$-graded cohomology ring of $E_{{\mathrm{\Sigma }}_2}{C}_2$. This generalizes Voevodsky’s identification of motivic cohomology of real numbers with the positive cone of $RO(C_2)$ graded cohomology of a point.