The join construction with spheres

Question

I'm trying to prove the following exercise (from http://math.stanford.edu/~ralph/fiber.pdf, page 70 of the pdf). If $*$ denotes the join construction of Milnor, we have that $S^{n} * S^{m} \cong S^{n+m+1}$. I'm not able to find an homeomorphism. Does someone know what could be the homeomorphism or have an intuition about this ? Thank you.

Answer 1

Join is associative. So by induction $S^n=S^0*S^{n-1}=...=S^0*...*S^0$(n+1 times)

Now we have $S^n=S^0*...* S^0$($n+1$ times), $S^m=S^0*...*S^0$($m+1$ times) and so $S^n*S^m=S^0*...*S^0$($n+m+2$ times) $=S^{n+m+1}$