Take $S^n$ and consider the union $Z$ of $k\_1$ circles, $k\_2$ 2-dimensional spheres, ..., $k\_{n-2}$ $(n−2)$-dimensional spheres, embedded in $S^n$ in an unknotted way, with no mutual intersection and no mutual linking.

I want to understand the complement $S^n \setminus Z$.

To that end, I calculated its homology via Mayer-Vietoris and I think that complement is homotopy equivalent to $(S^{n-1})^{k\_1+k\_2+\dots+k\_{n-2}-1} \vee (S^{n-2})^{k\_1} \vee \dots \vee (S^{1})^{k\_{n-2}}$.

Is it true? Where can I find a reference? The problem is that we cannot deduce this isomorphism from knowing the homology... Is there any way to see this explicitly (if it's true)?

Originally asked as [Math Stack Exchange question 3991178](https://math.stackexchange.com/questions/3991178).