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The fundamental crossed module of the complement of a knotted surface


Author: João Faria Martins
Journal: Trans. Amer. Math. Soc. 361 (2009), 4593-4630
MSC (2000): Primary 57M05, 57Q45; Secondary 55Q20
DOI: https://doi.org/10.1090/S0002-9947-09-04576-0
Published electronically: April 3, 2009
MathSciNet review: 2506421
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Abstract: 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 $ \mathcal{G}$ is a finite crossed module and $ M$ is finite, then the number of crossed module morphisms $ \Pi_2(M,M^1) \to \mathcal{G}$ can be re-scaled to a homotopy invariant $ I_{\mathcal{G}}(M)$, depending only on the algebraic 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 $ M$ is the complement of a knotted surface $ \Sigma$ in $ S^4$ and $ M^{(1)}$ is the handlebody of a handle decomposition of $ M$ made from its 0- and $ 1$-handles. Here, $ \Sigma$ is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2-type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant $ I_{\mathcal{G}}$ yields a non-trivial invariant of knotted surfaces in $ S^4$ with good properties with regard to explicit calculations.


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Additional Information

João Faria Martins
Affiliation: Departamentos de Matemática, Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
Email: jnmartins@fc.up.pt

DOI: https://doi.org/10.1090/S0002-9947-09-04576-0
Keywords: Knotted surfaces, crossed modules, homotopy 2-types.
Received by editor(s): June 18, 2007
Published electronically: April 3, 2009
Additional Notes: This work had the financial support of FCT (Portugal), post-doctoral grant number SFRH/BPD/17552/2004, part of the research project POCI/MAT/60352/2004 (“Quantum Topology”), also financed by FCT
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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