{"paper":{"title":"The mapping index through the lens of the cross-index","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The cross-index of free G-posets obeys the sharp topological union inequality precisely when G is Z2.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hamid Reza Daneshpajouh, Roman Karasev, Vuong Bui","submitted_at":"2026-05-13T02:36:21Z","abstract_excerpt":"We study the cross-index of free \\(G\\)-posets as a combinatorial analogue of the equivariant topological index. We demonstrate that the cross-index exhibits many structural properties closely paralleling those of the topological index, while its behavior with respect to unions displays a pronounced dichotomy depending on the acting group. Specifically, if \\(P = A \\cup B\\) is a union of \\(G\\)-invariant subposets, then for \\(G = \\mathbb{Z}_2\\) we obtain the sharp inequality \\[ \\operatorname{xind} P \\le \\operatorname{xind} A + \\operatorname{xind} B + 1, \\] which is directly analogous to the class"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"if P = A ∪ B is a union of G-invariant subposets, then for G = Z2 we obtain the sharp inequality xind P ≤ xind A + xind B + 1, which is directly analogous to the classical union inequality for the topological index. In contrast, for every group G≠Z2, this phenomenon fails in general, and we establish the best possible weaker estimate xind P ≤ xind A + 2(xind B+1). ... the gap between the cross-index and the topological index can be arbitrarily large.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The cross-index is well-defined for free G-posets and serves as a faithful combinatorial analogue whose union behavior can be compared directly to the topological index without additional topological assumptions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Cross-index of free G-posets satisfies a sharp union bound xind(P) ≤ xind(A) + xind(B) + 1 for Z2 but only xind(P) ≤ xind(A) + 2(xind(B)+1) for other G, with arbitrarily large gap to the topological index.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The cross-index of free G-posets obeys the sharp topological union inequality precisely when G is Z2.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7b8a211b90afd8a282f83c3508c84caa86632443ea34854fbd9d1ada4a72d2dc"},"source":{"id":"2605.12909","kind":"arxiv","version":1},"verdict":{"id":"4b93fa35-29b2-4b86-a0a9-786c750b7a64","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:30:26.704832Z","strongest_claim":"if P = A ∪ B is a union of G-invariant subposets, then for G = Z2 we obtain the sharp inequality xind P ≤ xind A + xind B + 1, which is directly analogous to the classical union inequality for the topological index. In contrast, for every group G≠Z2, this phenomenon fails in general, and we establish the best possible weaker estimate xind P ≤ xind A + 2(xind B+1). ... the gap between the cross-index and the topological index can be arbitrarily large.","one_line_summary":"Cross-index of free G-posets satisfies a sharp union bound xind(P) ≤ xind(A) + xind(B) + 1 for Z2 but only xind(P) ≤ xind(A) + 2(xind(B)+1) for other G, with arbitrarily large gap to the topological index.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The cross-index is well-defined for free G-posets and serves as a faithful combinatorial analogue whose union behavior can be compared directly to the topological index without additional topological assumptions.","pith_extraction_headline":"The cross-index of free G-posets obeys the sharp topological union inequality precisely when G is Z2."},"references":{"count":23,"sample":[{"doi":"","year":2017,"title":"Colorful subhypergraphs in uniform hypergraphs.The Electronic Journal of Combinatorics, pages P1–23, 2017","work_id":"9a75d4eb-7eca-41e6-a99d-3d4d01fa8ce6","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"On the chromatic number of general Kneser hypergraphs.Journal of Combinatorial Theory, Series B, 115:186–209, 2015","work_id":"4ecf3521-6506-4696-947a-cf83d428f3e4","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1986,"title":"The chromatic number of Kneser hypergraphs.Transactions of the American Mathematical Society, 298(1):359–370, 1986","work_id":"6dd826b4-831d-4758-a311-e1ff6e72c6cb","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Systolic inequalities for the number of vertices.Journal of Topology and Analysis, 16(06):955– 977, 2024","work_id":"56c0f1b0-efcb-4f95-9d0f-429f4f0a52c0","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"Envy-free division using mapping degree","work_id":"1bec6c84-836b-47fd-a10b-dd36fff4095c","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":23,"snapshot_sha256":"adfdafed1abd61ce922badc387c45537a93776a07d5ed7508b61dac073f20bed","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"}