The fundamental crossed module of the complement of a knotted surface

Faria Martins, João

doi:10.1090/S0002-9947-09-04576-0

The fundamental crossed module of the complement of a knotted surface
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Trans. Amer. Math. Soc. 361 (2009), 4593-4630 Request permission

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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