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 $XY$? 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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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 Seifertvan Kampen Theorem. So for the case in question this paper uses the fundamental crossed module and a $2$dimensional Seifertvan Kampen Theorem. 

