How does homotopy groups of wedge sums relate to the individual spheres?

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Thanks to all MHB members who have participated this year in the Graduate POTWs. (Happy)
Here is the last problem of the year, concerning homotopy groups:

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Explain why $\pi_r(S^m \lor S^n) \approx \pi_r(S^m) \oplus \pi_r(S^n)$ for $2 \le r < m + n - 1$.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

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No one answered this week's problem. You can read my solution below.

Spoiler

For $r \ge 2$, there is an isomorphism $\pi_r(S^m\lor S^n) \approx \pi_r(S^m) \oplus \pi_r(S^n) \oplus \pi_r(S^m\times S^n, S^m\lor S^n)$. The product $S^m \times S^n$ has an $(m + n)$-dimensional CW-complex structure with $1$ $0$-cell, $1$ $m$-cell, $1$ $n$-cell, and $1$ $(m+n)$-cell. The $m + n - 1$ skeleton is $S^m \lor S^n$, and hence $\pi_r(S^m \times S^n, S^m\lor S^n) = 0$ for $r < m + n - 1$. Therefore, $\pi_r(S^m \lor S^n) \approx \pi_r(S^m) \lor \pi_r(S^n)$ for $2\le r < m + n - 1$.