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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 $\\dim Z_f= \\infty$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1302.1494","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-02-06T19:59:20Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"abcbf67d5cadf4b3ff1c3d9302cce1942c3978c2d11485d50d15986fd8695108","abstract_canon_sha256":"007bcba6937ce185617d45c12087955cd88f404b30fc42ae972ab495a2974ea0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:00:54.299795Z","signature_b64":"1pV0soA9mKnLw6Bq1Mqu5d4ICUzctT5BPY+4BRXg9BzEQ7KC4jE9mgKDOJ15MtfAe2cy60X2jeeFDtvBW/EPDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"08940987abe6635fc84e6d37c7dffa960004a91f47412b980918807a8e973971","last_reissued_at":"2026-05-18T01:00:54.299180Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:00:54.299180Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $V^G= W^G=\\{0\\}$. 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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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1302.1494","created_at":"2026-05-18T01:00:54.299261+00:00"},{"alias_kind":"arxiv_version","alias_value":"1302.1494v2","created_at":"2026-05-18T01:00:54.299261+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.1494","created_at":"2026-05-18T01:00:54.299261+00:00"},{"alias_kind":"pith_short_12","alias_value":"BCKATB5L4ZRV","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_16","alias_value":"BCKATB5L4ZRV7SCO","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_8","alias_value":"BCKATB5L","created_at":"2026-05-18T12:27:38.830355+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BCKATB5L4ZRV7SCONU34PX72SY","json":"https://pith.science/pith/BCKATB5L4ZRV7SCONU34PX72SY.json","graph_json":"https://pith.science/api/pith-number/BCKATB5L4ZRV7SCONU34PX72SY/graph.json","events_json":"https://pith.science/api/pith-number/BCKATB5L4ZRV7SCONU34PX72SY/events.json","paper":"https://pith.science/paper/BCKATB5L"},"agent_actions":{"view_html":"https://pith.science/pith/BCKATB5L4ZRV7SCONU34PX72SY","download_json":"https://pith.science/pith/BCKATB5L4ZRV7SCONU34PX72SY.json","view_paper":"https://pith.science/paper/BCKATB5L","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1302.1494&json=true","fetch_graph":"https://pith.science/api/pith-number/BCKATB5L4ZRV7SCONU34PX72SY/graph.json","fetch_events":"https://pith.science/api/pith-number/BCKATB5L4ZRV7SCONU34PX72SY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BCKATB5L4ZRV7SCONU34PX72SY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BCKATB5L4ZRV7SCONU34PX72SY/action/storage_attestation","attest_author":"https://pith.science/pith/BCKATB5L4ZRV7SCONU34PX72SY/action/author_attestation","sign_citation":"https://pith.science/pith/BCKATB5L4ZRV7SCONU34PX72SY/action/citation_signature","submit_replication":"https://pith.science/pith/BCKATB5L4ZRV7SCONU34PX72SY/action/replication_record"}},"created_at":"2026-05-18T01:00:54.299261+00:00","updated_at":"2026-05-18T01:00:54.299261+00:00"}