{"paper":{"title":"An upper bound for topological complexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Gregory Lupton, John Oprea, Mark Grant, Michael Farber","submitted_at":"2018-07-11T08:31:54Z","abstract_excerpt":"In arXiv:1711.10132 a new approximating invariant ${\\mathsf{TC}}^{\\mathcal{D}}$ for topological complexity was introduced called $\\mathcal{D}$-topological complexity. In this paper, we explore more fully the properties of ${\\mathsf{TC}}^{\\mathcal{D}}$ and the connections between ${\\mathsf{TC}}^{\\mathcal{D}}$ and invariants of Lusternik-Schnirelmann type. We also introduce a new $\\mathsf{TC}$-type invariant $\\widetilde{\\mathsf{TC}}$ that can be used to give an upper bound for $\\mathsf{TC}$, $$\\mathsf{TC}(X)\\le {\\mathsf{TC}}^{\\mathcal{D}}(X) + \\left\\lceil \\frac{2\\dim X -k}{k+1}\\right\\rceil,$$ wh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.03994","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"}