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For a fiber preserving $\\mathbb{Z}_2$-equivariant map $f:E \\to E^{'}$, we estimate the cohomological dimension of the zero set $Z_f = \\{x \\in E | f(x)= 0\\}.$ As an application, we also estimate the cohomological dimension of the $\\mathbb{Z}_2$-coincidence set $A_f=\\{x \\in E "},"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":"0810.4669","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2008-10-26T07:41:53Z","cross_cats_sorted":[],"title_canon_sha256":"037e617a6dd048a4f96eea3612a08bae0f332fb669748366fd3c9e9741ff85dd","abstract_canon_sha256":"041c0858863348e1f83afd3674e182a4238de4692f2e8f14c14b7cbf8b5d547c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:32:22.244120Z","signature_b64":"Tdy1f2HW/FjrJEYiAz7OGrOGS/Bl3kXKlxOROZ53tYR6IQkGnSVU+dNuePl/nkW30LAqaBq1zsCa2O6I602gBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3d45c8a04c0dd05f527352aa8e12f346a6c06de026e88959480fdd4bac2b481b","last_reissued_at":"2026-05-18T04:32:22.243391Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:32:22.243391Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Parametrized Borsuk-Ulam problem for projective space bundles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Mahender Singh","submitted_at":"2008-10-26T07:41:53Z","abstract_excerpt":"Let $\\pi: E \\to B$ be a fiber bundle with fiber having the mod 2 cohomology algebra of a real or a complex projective space and let $\\pi^{'}: E^{'} \\to B$ be vector bundle such that $\\mathbb{Z}_2$ acts fiber preserving and freely on $E$ and $E^{'}-0$, where 0 stands for the zero section of the bundle $\\pi^{'}:E^{'} \\to B$. For a fiber preserving $\\mathbb{Z}_2$-equivariant map $f:E \\to E^{'}$, we estimate the cohomological dimension of the zero set $Z_f = \\{x \\in E | f(x)= 0\\}.$ As an application, we also estimate the cohomological dimension of the $\\mathbb{Z}_2$-coincidence set $A_f=\\{x \\in E "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0810.4669","kind":"arxiv","version":3},"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":"0810.4669","created_at":"2026-05-18T04:32:22.243500+00:00"},{"alias_kind":"arxiv_version","alias_value":"0810.4669v3","created_at":"2026-05-18T04:32:22.243500+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0810.4669","created_at":"2026-05-18T04:32:22.243500+00:00"},{"alias_kind":"pith_short_12","alias_value":"HVC4RICMBXIF","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_16","alias_value":"HVC4RICMBXIF6UTT","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_8","alias_value":"HVC4RICM","created_at":"2026-05-18T12:25:57.157939+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/HVC4RICMBXIF6UTTKKVI4EXTI2","json":"https://pith.science/pith/HVC4RICMBXIF6UTTKKVI4EXTI2.json","graph_json":"https://pith.science/api/pith-number/HVC4RICMBXIF6UTTKKVI4EXTI2/graph.json","events_json":"https://pith.science/api/pith-number/HVC4RICMBXIF6UTTKKVI4EXTI2/events.json","paper":"https://pith.science/paper/HVC4RICM"},"agent_actions":{"view_html":"https://pith.science/pith/HVC4RICMBXIF6UTTKKVI4EXTI2","download_json":"https://pith.science/pith/HVC4RICMBXIF6UTTKKVI4EXTI2.json","view_paper":"https://pith.science/paper/HVC4RICM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0810.4669&json=true","fetch_graph":"https://pith.science/api/pith-number/HVC4RICMBXIF6UTTKKVI4EXTI2/graph.json","fetch_events":"https://pith.science/api/pith-number/HVC4RICMBXIF6UTTKKVI4EXTI2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HVC4RICMBXIF6UTTKKVI4EXTI2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HVC4RICMBXIF6UTTKKVI4EXTI2/action/storage_attestation","attest_author":"https://pith.science/pith/HVC4RICMBXIF6UTTKKVI4EXTI2/action/author_attestation","sign_citation":"https://pith.science/pith/HVC4RICMBXIF6UTTKKVI4EXTI2/action/citation_signature","submit_replication":"https://pith.science/pith/HVC4RICMBXIF6UTTKKVI4EXTI2/action/replication_record"}},"created_at":"2026-05-18T04:32:22.243500+00:00","updated_at":"2026-05-18T04:32:22.243500+00:00"}