{"paper":{"title":"The Fundamental Crossed Module of the Complement of a Knotted Surface","license":"","headline":"","cross_cats":["hep-th","math.QA"],"primary_cat":"math.GT","authors_text":"Jo\\~ao Faria Martins","submitted_at":"2008-01-25T14:21:10Z","abstract_excerpt":"We prove that if $M$ is a CW-complex and $M^1$ is its 1-skeleton then the crossed module $\\Pi_2(M,M^1)$ depends only on the homotopy type of $M$ as a space, up to free products, in the category of crossed modules, with $\\Pi_2(D^2,S^1)$. From this it follows that, if $G$ is a finite crossed module and $M$ is finite, then the number of crossed module morphisms $\\Pi_2(M,M^1) \\to G$ can be re-scaled to a homotopy invariant $I_G(M)$, depending only on the homotopy 2-type of $M$. We describe an algorithm for calculating $\\pi_2(M,M^{(1)})$ as a crossed module over $\\pi_1(M^{(1)})$, in the case when $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0801.3921","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"}