## Second homotopy of a torus complement in the 4-sphere

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Let $T$ be the boundary of a solid torus in $S^4$. Are there any theorems or methods which would help one to compute $\pi_2(S^4 -T)$? Or to say if, e.g., it had finite rank and no torsion?
More generally, suppose we have a closed manifold $X$ and we remove a submanifold $Y$ of at least codimension 2. Is there a general method for computing the homotopy groups of $X-Y$? The above case was the original example that got me to think about this but a more general result would also perhaps be helpful.

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Is your $T$ topologically equal to $S^1\times S^1$? – Wlodzimierz Holsztynski Feb 14 at 4:46

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If $X$ is the universal covering space of $S^4 - T$, $H_2 X$ is isomorphic to $\pi_2(S^4 - T)$, this is just the fact that $X$ is simply connected, together with the Hurewicz theorem that says the first non-trivial homotopy and homology groups agree above dimension $2$, and that covering spaces preserve higher homotopy groups. This is a standard technique to compute homotopy groups, for example, Serre made great mileage out of this in his dissertation. – Ryan Budney Feb 14 at 5:35
See also Sam Lomonaco's Pacific Math Journal paper. Here he is dealing with spheres, but the crossed module structure of $\pi_2$ should be interesting to you as well. – Scott Carter Feb 15 at 1:40

## 1 Answer

The paper

Martins, João Faria The fundamental crossed module of the complement of a knotted surface. Trans. Amer. Math. Soc. 361 (2009), no. 9, 4593–4630.

looks relevant.

Note that for links in $\mathbb R^3$ it is standard to use the fundamental group and a Seifert-van Kampen Theorem. So for the case in question this paper uses the fundamental crossed module and a $2$-dimensional Seifert-van Kampen Theorem.

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