{"paper":{"title":"M\\\"obius transform, moment-angle complexes and Halperin-Carlsson conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.AT"],"primary_cat":"math.CO","authors_text":"Xiangyu Cao, Zhi L\\\"u","submitted_at":"2009-08-21T17:42:16Z","abstract_excerpt":"In this paper, we give an algebra-combinatorics formula of the M\\\"obius transform for an abstract simplicial complex $K$ on $[m]=\\{1, ..., m\\}$ in terms of the Betti numbers of the Stanley-Reisner face ring of $K$. Furthermore, we employ a way of compressing $K$ to estimate the lower bound of the sum of those Betti numbers by using this formula. As an application, associating with the moment-angle complex $\\mathcal{Z}_K$ (resp. real moment-angle complex ${\\Bbb R}\\mathcal{Z}_K$) of $K$, we show that the Halperin-Carlsson conjecture holds for $\\mathcal{Z}_K$ (resp. ${\\Bbb R}\\mathcal{Z}_K$) under"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.3174","kind":"arxiv","version":2},"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"}