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I got this homework problem: $X,Y$ finite CW-complexes with $\dim X=m$ and $Y$ is $n$-connected.

Prove that $\pi_k(map(X,Y))=0$ for all $k \le n-m$.

Thanks for the help!

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Is this fact helpful? If $f:X\to Y$ then any $\alpha\in \pi_k(X)$ gives us an element $\beta\in\pi_k(Y)$ representing $f\circ \alpha$. –  Sigur Dec 12 '12 at 0:44
I didn't see how that could help. I'm think of using $\sigma':(S^k\times X\to Y)$ –  John0417 Dec 12 '12 at 1:18
Maybe this theorem could help you: Suppose that $Y$ is an Eilenberg--MacLane space of type $(\pi,n)$ for $n\geq 1$with $\pi$ abelian. Then $$\pi_i(map(X,Y),f)\simeq H^{n-i}(X;\pi)$$ for all $i\geq 1$. –  Sigur Dec 12 '12 at 1:32
Or this: Suppose that $Y$ is an Eilenberg--MacLane space of type $(\pi,1)$ with $\pi$ nonabelian. For any based map $f: X\to Y$, consider the induced homomorphism $f_*: \pi_1(X)\to \pi_1(Y)$ and denote by $C(\pi;f)$ the centralizer for $f_*(\pi_1(X))$ in $\pi_1(Y)\simeq \pi$. For any finite dimensional CW-complex $X$ we then get $$\pi_i(map(X,Y),f)\simeq \begin{cases} C(\pi;f), &i=1;\\ 0,& i>1.\end{cases}$$ –  Sigur Dec 12 '12 at 1:38
Replace $Y$ by a homotopy equivalent complex whose n-skeleton is trivial. Note that $\pi_k(map(X,Y)) = [S^k, map(X,Y)]$ where $[]$ denotes pointed homotopy classes. What do you know about mapping spaces? –  Fabian Lenhardt Dec 12 '12 at 11:27

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