Let \(C_2\) denote the cyclic group of order 2. We compute the \(RO(C_2)\)-graded cohomology of all \(C_2\)-surfaces with constant integral coefficients. We show when the action is nonfree, the answer depends only on the genus, the orientability of the underlying surface, the number of isolated fixed points, the number of fixed circles with trivial normal bundles, and the number of fixed circles with nontrivial normal bundles. When the action on the surface is free, we show the answer depends only on the genus, the orientability of the underlying surface, whether or not the action preserves the orientation, and one other invariant.

中文翻译：

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具有 $${\underline{{\mathbb {Z}}}}$$ -系数的 $$C_2$$ -曲面的上同调

让\(C_2\)表示 2 阶循环群。我们计算\(RO(C_2)\) -所有具有常积分系数的\(C_2\)表面的渐变上同调。我们表明当动作是非自由的时，答案仅取决于属、下表面的可定向性、孤立不动点的数量、具有平凡法束的固定圆的数量以及具有非平凡法束的固定圆的数量. 当表面上的动作是自由的时，我们表明答案仅取决于属、底层表面的可定向性、动作是否保持方向，以及另一个不变量。