{"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"}