{"paper":{"title":"Homological Filling Functions with Coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Fedor Manin, Xingzhe Li","submitted_at":"2020-09-28T17:27:49Z","abstract_excerpt":"How hard is it to fill a loop in a Cayley graph with an unoriented surface? Following a comment of Gromov in \"Asymptotic invariants of infinite groups\", we define homological filling functions of groups with coefficients in a group $R$. Our main theorem is that the coefficients make a difference. That is, for every $n \\geq 1$ and every pair of coefficient groups $A, B \\in \\{\\mathbb{Z},\\mathbb{Q}\\} \\cup \\{\\mathbb{Z}/p\\mathbb{Z} : p\\text{ prime}\\}$, there is a group whose filling functions for $n$-cycles with coefficients in $A$ and $B$ have different asymptotic behavior."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.13489","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2009.13489/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}