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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 $f\circ \alpha$.2012-12-12
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    I didn't see how that could help. I'm think of using $\sigma':(S^k\times X\to Y)$2012-12-12
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    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 $i\geq 1$.2012-12-12
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    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 $$\pi_i(map(X,Y),f)\simeq \begin{cases} C(\pi;f), &i=1;\\ 0,& i>1.\end{cases}$$2012-12-12
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    Obviously I can't assume Y is an Eilenberg-Maclane space;and the result is to show the case when k$\le$n-m2012-12-12
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    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?2012-12-12
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    Ingredients: Fabian's answer, adjunction, and cellular approximation. Yield: Answer.2012-12-12
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    Would do please be a little more specific? That will help a lot! Thanks!2012-12-13
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    I can't be more specific without doing your homework for you. I might be a little more inclined to help if you showed us all you'd put in some effort on the problem and explained what you have tried, what you know, and what you don't understand. It's the courteous thing to do.2012-12-13

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