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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

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    You might want to construct a map from $X\star Y$ into $I\times X\times Y$ and make sure it passes through all the appropriate quotients.2011-10-02
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    More details, please? That's what I've been trying to do.2011-10-02
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    Maybe even start one higher: Make a map from $X \times Y \times I$ to $X \times Y \times I$, then pass through all the quotients. Hint: Don't think too hard.2011-10-02

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