{"paper":{"title":"Parametrized $\\underline{\\mathbb{F}}_2$-Cohomology of $B_{C_2}O(1)$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Agn\\`es Beaudry, Chloe Lewis, Clover May, Elizabeth Tatum, Sabrina Pauli","submitted_at":"2026-07-29T19:10:52Z","abstract_excerpt":"We compute the parametrized (or twisted) ordinary cohomology of the classifying space for real $C_2$-line bundles, $B_{C_2}O(1)$ with coefficients in the constant Mackey functor $\\underline{\\mathbb{F}}_2$. Parametrized cohomology refines $RO(G)$-graded Bredon cohomology by assembling equivariant cohomology for all local coefficients into a single graded ring. For this reason, our work also encodes a computation of the $RO(C_2)$-graded cohomology of all Thom spaces of real $C_2$-vector bundles over $B_{C_2}O(1)$.\n  Along the way, we prove many general results that can be applied to computations"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.27398","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.27398/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}