I am having some trouble with this. Any help would be very appreciated. Thanks.
Exercise 24. Let $X$ and $Y$ be $CW$ complexes with $0$ cells $x_{0}$ and $y_{0}$. Show that the quotient spaces $X * Y / (X*\left\{y_{0}\right\} \cup \left\{x_{0}\right\} * Y)$ and $S(X \wedge Y)/S(\left\{y_{0}\right\} \wedge \left\{x_{0}\right\})$ are homeomorphic, and deduce that $X * Y$ and $S(X \wedge Y)$ are homotopy equivalent.
Best, Anna