{"paper":{"title":"An $A_{\\infty}$-coalgebra Structure on a Closed Compact Surface","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Quinn Minnich, Ronald Umble","submitted_at":"2018-01-24T16:53:18Z","abstract_excerpt":"Let $P$ be an $n$-gon with $n\\geq3.$ There is a formal combinatorial $A_\\infty$-coalgebra structure on cellular chains $C_*(P)$ with non-vanishing higher order structure when $n\\geq5$. If $X_g$ is a closed compact surface of genus $g\\geq2$ and $P_g$ is a polygonal decomposition, the quotient map $q:P_g\\to X_g$ projects the formal $A_\\infty$-coalgebra structure on $C_*(P_g)$ to a quotient structure on $C_*(X_g)$, which persists to homology $H_{\\ast}\\left( X_g;\\mathbb{Z}_{2}\\right) $, whose operations are determined by the quotient map $q$, and whose higher order structure is non-trivial if and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.08071","kind":"arxiv","version":5},"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"}