$RO(C_2)$-graded equivariant cohomology and classical Steenrod squares

We investigate the restriction to fixed-points and change of coefficient functors in $RO(C_2)$-graded equivariant cohomology, with applications to the equivariant cohomology of spaces with a trivial $C_2$-action for $\underline{\mathbb{Z}}$ and $\underline{\mathbb{F}_2}$ coefficients. To this end, we study the nonequivariant spectra representing these theories and the corresponding functors. In particular, we show that the $RO(C2)$-graded homology class determined by a Real submanifold $Y$ (in the sense of Atiyah) of a Real compact manifold $X$ encodes the total Steenrod square of the dual to $Y^{C_2}$ in $X^{C_2}$.