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.

中文翻译：

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$$\underline{{\mathbb {Z}}/2}$$-coefficients 中 $$C_2$$-surfaces 的 $$RO(C_2)$$-graded cohomology

具有对合的曲面可以被视为 $$C_2$$ C 2 -空间，其中 $$C_2$$ C 2 是二阶循环群。直到等变同构，表面上的所有对合都在 Bujalance 等人中分类。(Math Z 211:461–478, 1992) 和最近在 Dugger 中使用等变手术进行分类 (J Homotopy Relat Struct 14(4):919–992, 2019)。我们使用 Dugger (2019) 中给出的分类来计算常量 $${\mathbb {Z 中所有 $$C_2$$ C 2 表面的 $$RO(C_2)$$RO ( C 2 ) 分级 Bredon 上同调}}/2$$ Z / 2 系数作为一个点上同调的模块。我们证明上同调在非自由情况下仅取决于三个数值不变量，而在自由情况下仅取决于两个数值不变量。