Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.