{"paper":{"title":"Bourgin-Yang versions of the Borsuk-Ulam theorem for $p$-toral groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.AT","authors_text":"Denise De Mattos, Edivaldo L. dos Santos, Wac{\\l}aw Marzantowicz","submitted_at":"2013-02-06T19:59:20Z","abstract_excerpt":"Let $V$ and $W$ be orthogonal representations of $G$ with $V^G= W^G=\\{0\\}$. Let $S(V )$ be the sphere of $V$ and $f : S(V ) \\to W$ be a $G$-equivariant mapping. We give an estimate for the dimension of the set $Z_f=f^{-1}\\{0\\}$ in terms of $ \\dim V$ and $\\dim W$, if $G$ is the torus $\\mathbb T^k$, or the $p$-torus $\\mathbb Z_p^k$. This extends the classical Bourgin-Yang theorem onto this class of groups. Finally, we show that for any $p$-toral group $G$ and a $G$-map $f:S(V) \\to W$, with $\\dim V=\\infty$ and $\\dim W<\\infty$, we have $\\dim Z_f= \\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.1494","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}