# homotopy groups of mapping space

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
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
Ingredients: Fabian's answer, adjunction, and cellular approximation. Yield: Answer. –  Dylan Wilson Dec 12 '12 at 20:48