{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:IE3ISIBPSZU56BOW4W3BGHM6EX","short_pith_number":"pith:IE3ISIBP","schema_version":"1.0","canonical_sha256":"413689202f9669df05d6e5b6131d9e25cbddb10f47743aa3684d1194ad09c4c1","source":{"kind":"arxiv","id":"1207.5229","version":1},"attestation_state":"computed","paper":{"title":"The graph cohomology ring of the GKM graph of a flag manifold of type $G_2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AT","authors_text":"Yukiko Fukukawa","submitted_at":"2012-07-22T13:21:11Z","abstract_excerpt":"Suppose a compact torus $T$ acts on a closed smooth manifold $M$. Under certain conditions, Guillemin and Zara associate to $(M, T)$ a labeled graph $\\mG_M$ where the labels lie in $H^2(BT)$. They also define the subring $H_T^*(\\mG_M)$ of $\\bigoplus_{v\\in V(\\mG_M)}H^*(BT)$, where $V(\\mG_M)$ is the set of vertices of $\\mG_M$ and we call $H_T^*(\\mG_M)$ the \"graph cohomology\" ring of $\\mG_M$. It is known that the equivariant cohomology ring of $M$ can be described by using combinatorial data of the labeled graph. The main result of this paper is to determine the ring structure of equivariant coho"},"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":"1207.5229","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2012-07-22T13:21:11Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"355eedd0390158aaab308691046587e4db825b3dcf941ff697949ba2e0107a9d","abstract_canon_sha256":"2547d5e3dcdc3c5c0c076ce7a1577d999b450ebc2b4a2ad908113a6827af1b57"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:24.122416Z","signature_b64":"S+Z2+m36JdWMQQ2BM5IBCuCZ+qlGip2STGmTchV2TtEzrzOhgOU0PG6UdeM+70mqRaWSkncp1h8MUcMoEUH9DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"413689202f9669df05d6e5b6131d9e25cbddb10f47743aa3684d1194ad09c4c1","last_reissued_at":"2026-05-18T03:50:24.121564Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:24.121564Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The graph cohomology ring of the GKM graph of a flag manifold of type $G_2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AT","authors_text":"Yukiko Fukukawa","submitted_at":"2012-07-22T13:21:11Z","abstract_excerpt":"Suppose a compact torus $T$ acts on a closed smooth manifold $M$. Under certain conditions, Guillemin and Zara associate to $(M, T)$ a labeled graph $\\mG_M$ where the labels lie in $H^2(BT)$. They also define the subring $H_T^*(\\mG_M)$ of $\\bigoplus_{v\\in V(\\mG_M)}H^*(BT)$, where $V(\\mG_M)$ is the set of vertices of $\\mG_M$ and we call $H_T^*(\\mG_M)$ the \"graph cohomology\" ring of $\\mG_M$. It is known that the equivariant cohomology ring of $M$ can be described by using combinatorial data of the labeled graph. The main result of this paper is to determine the ring structure of equivariant coho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.5229","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":""},"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":"1207.5229","created_at":"2026-05-18T03:50:24.121699+00:00"},{"alias_kind":"arxiv_version","alias_value":"1207.5229v1","created_at":"2026-05-18T03:50:24.121699+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.5229","created_at":"2026-05-18T03:50:24.121699+00:00"},{"alias_kind":"pith_short_12","alias_value":"IE3ISIBPSZU5","created_at":"2026-05-18T12:27:09.501522+00:00"},{"alias_kind":"pith_short_16","alias_value":"IE3ISIBPSZU56BOW","created_at":"2026-05-18T12:27:09.501522+00:00"},{"alias_kind":"pith_short_8","alias_value":"IE3ISIBP","created_at":"2026-05-18T12:27:09.501522+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/IE3ISIBPSZU56BOW4W3BGHM6EX","json":"https://pith.science/pith/IE3ISIBPSZU56BOW4W3BGHM6EX.json","graph_json":"https://pith.science/api/pith-number/IE3ISIBPSZU56BOW4W3BGHM6EX/graph.json","events_json":"https://pith.science/api/pith-number/IE3ISIBPSZU56BOW4W3BGHM6EX/events.json","paper":"https://pith.science/paper/IE3ISIBP"},"agent_actions":{"view_html":"https://pith.science/pith/IE3ISIBPSZU56BOW4W3BGHM6EX","download_json":"https://pith.science/pith/IE3ISIBPSZU56BOW4W3BGHM6EX.json","view_paper":"https://pith.science/paper/IE3ISIBP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1207.5229&json=true","fetch_graph":"https://pith.science/api/pith-number/IE3ISIBPSZU56BOW4W3BGHM6EX/graph.json","fetch_events":"https://pith.science/api/pith-number/IE3ISIBPSZU56BOW4W3BGHM6EX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IE3ISIBPSZU56BOW4W3BGHM6EX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IE3ISIBPSZU56BOW4W3BGHM6EX/action/storage_attestation","attest_author":"https://pith.science/pith/IE3ISIBPSZU56BOW4W3BGHM6EX/action/author_attestation","sign_citation":"https://pith.science/pith/IE3ISIBPSZU56BOW4W3BGHM6EX/action/citation_signature","submit_replication":"https://pith.science/pith/IE3ISIBPSZU56BOW4W3BGHM6EX/action/replication_record"}},"created_at":"2026-05-18T03:50:24.121699+00:00","updated_at":"2026-05-18T03:50:24.121699+00:00"}