Prove that $S^m * S^n$ is homotopically equivalent to $S^{m+n+1}$

Question

I feel like the proof is very similar to the one in this link:

$S^n \backslash S^m $ homotopy equivalent to $ S^{n-m-1} $

But I could not spot out where will be the difference? could anyone help me with this, please?

Thanks!

Answer 1

Here are some facts that you can assemble:

1. For any CW complexes $A,B$ the join $A*B$ is homotopy equivalent to $\Sigma(A\wedge B)$
2. For any $n,m$ we have a homeomorphism $S^n\wedge S^m\simeq S^{n+m}$
3. For any $n$ we have a homeomorphism $\Sigma S^n\simeq S^{n+1}$
4. If $A\simeq B$ then $\Sigma A\simeq \Sigma B$
5. Spheres are CW complexes (the simpliest there are)

Can you combine them together to get the result?

Answer 2

In fact $S^m *S^n$ and $S^{m+n+1}$ are homeomorphic. This is proved as in https://math.stackexchange.com/q/3354659:

$S^m \approx S^0 * \ldots * S^0$ witk $m+1$ factors, $S^n \approx S^0 * \ldots * S^0$ witk $n+1$ factors, thus $S^m * S^n \approx S^0 * \ldots * S^0$ witk $m+n+2$ factors, the latter being homeomorphic to $S^{m+n+1}$.