Homology of complement of a $m$-sphere in $\mathbb R^n$, with $mTrying to solve the problem in the title, I found this post where a particular case is described and the answer gives a generalization that I think could help me.
Here is my first doubt: How to see that $S^{n-1}\backslash(\{*\}\cup S^{k-1})$ is homotopic equivalent to $S^{n-k-1}\vee S^{n-2}$?
Assuming this result, I define the set $A$ as one of these spheres together with a small open neighborhood of the point shared with the other sphere. And $B$ the same, but with the other sphere and the open neighborhood corresponding to $A$.
This election is intended to make $A$ and $B$ open sets that can be retracted to the spheres. The union is the space I want to solve ($X$) and the intersection is the gluing point.
This way, Mayer-Vietoris says that we have
$\ldots \rightarrow H_n(\{p\}) \overset{}{\rightarrow} H_n(S^{n-k-1})\oplus H_n(S^{n-2}) \overset{}{\rightarrow} H_n(X) \overset{}{\rightarrow} H_{n-1}(\{p\}) \overset{}{\rightarrow} \ldots$,
And since these components are known, I think I could be able to finish the problem.
Does it seem ok to you?
Do you have other suggestions/approaches to solve the original problem?
Thank you in advance.

Use the Mayer-Vietoris sequence with $\mathbb{R}^n-S^m$ and a small neighborhood $N\cong S^m\times D^{n-m}$ of $S^m$. We have $X\cup N=\mathbb{R}^n$, $N\simeq S^m$, and $X\cap N\simeq \partial N=S^m\times S^{n-m-1}$. Then the reduced M-V sequence gives isomorphisms
$$\tilde{H}_i(X)\oplus\tilde{H}_i(S^m)\cong\tilde{H}_i(S^m\times S^{n-m-1})$$
for all $i$. You can then use either the Kunneth Theorem or induction on $k$ in $S^m\times S^k$ by representing $S^k$ as a union of two hemispheres and using the M-V sequence to derive the Betti numbers
$$b_i(S^m\times S^{n-m-1})=\delta_{0i}+\delta_{mi}+\delta_{(n-m-1)i}+\delta_{(n-1)i}$$
Since $b(S^m)=\delta_{mi}$, we find
$$b_i(X)=\delta_{0i}+\delta_{(n-m-1)i}+\delta_{(n-1)i}$$
There is no torsion involved here, so this is sufficient.
