{"paper":{"title":"Cobordism Categories and Parametrized Morse Theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.GT"],"primary_cat":"math.AT","authors_text":"Nathan Perlmutter","submitted_at":"2017-03-03T06:17:14Z","abstract_excerpt":"Fix a tangential structure $\\theta: B \\longrightarrow BO(d+1)$ and an integer $k < d/2$. In this paper we determine the homotopy type of a cobordism category $\\mathbf{Cob}^{\\text{mf}, k}_{\\theta}$, where morphisms are given by $\\theta$-cobordisms $W: P \\rightsquigarrow Q$ equipped with a choice of proper Morse function $h_{W}: W \\longrightarrow [0, 1]$, with the property that all critical points $c \\in W$ of $h_{W}$ satisfy the condition: $k < \\text{index}(c) < d-k+1$. In particular, we prove that there is a weak homotopy equivalence $B\\mathbf{Cob}^{\\text{mf}, k}_{\\theta} \\simeq\\Omega^{\\infty}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.01047","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"}