A complement to a set of spheres in a sphere

Question

Take $S^n$ and consider a set $Z$ of $k_1$ circles, $k_2$ 2-dimensional spheres, .., $k_{n-2}$ $n-2$-dimenional spheres in $S^n$ (there spheres do not intersect inside $S^n$). I want to understand what is $S^n \setminus Z$. To do this, I calculated homology of this space via Mayer-Vietoris and I think that the complement is homotopy equivalent to $(S^{n-1})^{k_1+k_2+..+k_{n-2}-1} \vee (S^{n-2})^{k_1} \vee .. \vee (S^{1})^{k_{n-2}}$. Is it true? Where can I find a reference? The problem is that we cannot deduce this isomorphism from knowing homology, is there any way to see this explicitly?