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A surface with an involution can be viewed as a $$C_2$$ C 2 -space where $$C_2$$ C 2 is the cyclic group of order two. Up to equivariant isomorphism, all involutions on surfaces were classified in Bujalance et al. (Math Z 211:461–478, 1992) and recently classified using equivariant surgery in Dugger (J Homotopy Relat Struct 14(4):919–992, 2019). We use the classification given in Dugger (2019) to compute the $$RO(C_2)$$ R O ( C 2 ) -graded Bredon cohomology of all $$C_2$$ C 2 -surfaces in constant $${\mathbb {Z}}/2$$ Z / 2 coefficients as modules over the cohomology of a point. We show the cohomology depends only on three numerical invariants in the nonfree case, and only on two numerical invariants in the free case.

The $$RO(C_2)$$-graded cohomology of $$C_2$$-surfaces in $$\underline{{\mathbb {Z}}/2}$$-coefficients