{"paper":{"title":"On 2-Dimensional Homotopy Invariants of Complements of Knotted Surfaces","license":"","headline":"","cross_cats":["math.QA"],"primary_cat":"math.GT","authors_text":"Jo\\~ao Faria Martins","submitted_at":"2005-07-12T17:34:40Z","abstract_excerpt":"We prove that if $M$ is a CW-complex and $*$ is a 0-cell of $M$, then the crossed module $\\Pi_2(M,M^1,*)$ does not depend on the cellular decomposition of $M$ up to free products with $\\Pi_2(D^2,S^1,*)$, where $M^1$ is the 1-skeleton of $M$. From this it follows that if $G$ is a finite crossed module and $M$ is finite, then the number of crossed module morphisms $\\Pi_2(M,M^1,*) \\to G$ (which is finite) can be re-scaled to a homotopy invariant $I_G(M)$ (i. e. not dependent on the cellular decomposition of $M$). We describe an algorithm to calculate $\\pi_2(M,M^{(1)},*)$ as a crossed module over "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0507239","kind":"arxiv","version":3},"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"}