Showing that $X * Y$ is homotopically equivalent to $\sum (X \wedge Y)$ .

Question

Showing that $X * Y$ is homotopically equivalent to $\sum (X \wedge Y)$ . I am really stuck in this problem and any help will be greatly appreciated.

An Edit:

I was looking at the properties of the join from Wikipedia and I found the following property :

However,I do not understand why if $A * {b_{0}} \cup {a_{0}}*B $ is contractible then we are sure that there is a homotopy equivalence between $X * Y$ and $\sum (X \wedge Y)$, could anyone explain for me this, is there is a theorem relating contractibility to homotopy equivalence?

Answer 1

This is Allen Hatcher, "Algebraic Topology" Proposition 0.17:

Proposition 0.17. If the pair $(X, A)$ satisfies the homotopy extension property and $A$ is contractible, then the quotient map $q:X\to X/A$ is a homotopy equivalence.

Note that for CW complexes it is enough if $A$ is a closed subcomplex (which happens here).