{"paper":{"title":"The complexity of prime 3-manifolds and the first $\\mathbb{Z}_{/2\\mathbb{Z}}$-cohomology of small rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Kei Nakamura","submitted_at":"2017-12-07T13:22:13Z","abstract_excerpt":"For a closed orientable connected 3-manifold $M$, its complexity $\\boldsymbol{T}(M)$ is defined to be the minimal number of tetrahedra in its triangulations. Under the assumption that $M$ is prime (but not necessarily atoroidal), we establish a lower bound for the complexity $\\boldsymbol{T}(M)$ in terms of the $\\mathbb{Z}_{/2\\mathbb{Z}}$-coefficient Thurston norm for $H^1(M;\\mathbb{Z}_{/2\\mathbb{Z}})$: (1) for any rank-1 subgroup $\\{0,\\varphi\\} \\leqslant H^1(M;\\mathbb{Z}_{/2\\mathbb{Z}})$, we have $\\boldsymbol{T}(M) \\geqslant 2+2||\\varphi||$ unless $M$ is a lens space with $\\boldsymbol{T}(M)=1+"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.02607","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"}