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The cohomology of \(C_2\)-surfaces with \({\underline{{\mathbb {Z}}}}\)-coefficients

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Abstract

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.

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Acknowledgements

Much of this work was done while the author was a graduate student at University of Oregon. She thanks her doctoral advisor, Daniel Dugger, for all of his advice and guidance on this project. She also thanks Mike Hill and Clover May for many helpful conversations. Lastly, many thanks to the anonymous referee for their careful reading and helpful feedback.

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  1. University of California Los Angeles, Los Angeles, USA

    Christy Hazel

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  1. Christy Hazel
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Correspondence to Christy Hazel.

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Communicated by Anna Marie Bohmann.

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Hazel, C. The cohomology of \(C_2\)-surfaces with \({\underline{{\mathbb {Z}}}}\)-coefficients. J. Homotopy Relat. Struct. 18, 71–114 (2023). https://doi.org/10.1007/s40062-022-00321-y

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