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Consider the set $Map(T^4,S^2)$ of continuous maps from the 4 dimensonal torus $T^4$ to the 2 dimensional sphere $S^2$, endowed with compact-open topology, can we show it is not connected? How can we calculate its singular homology and $\pi_1$?

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In spite of being more or less standard notations, it'd be better for your question's clearity if you'd define $\;T^4\,,\,S^2\;$ ... –  DonAntonio Nov 5 '13 at 15:24
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1 Answer 1

up vote 5 down vote accepted

For the first part

Hint 1: $$Map(X\times Y,Z)\cong Map(X,Map(Y,Z))$$

Hint 2: $$\pi_i(Map(S^1,X))\cong\pi_{i+1}(X)$$

Hint 3: $$\pi_4(S^2)\cong \mathbb{Z}_2$$

For the second and third parts

Hint 4: $$\pi_5(S^2)\cong\mathbb{Z}_2$$

Hint 5: $$H_1(X)\cong \pi_1(X)^{ab}$$

Hint 6: For higher $H_k$, I think you'll need to iterate the Leray spectral sequence as far as I can tell, which will be messy - there may be an easier way which can be applied to the sphere and its loop-spaces (see this question).

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Feel free to wait to or not answer this since it would provide more than just hints, but is there a good way to give a geometric description of these nontrivial homotopies? –  Zach L. Nov 5 '13 at 16:33
    
@ZachL. Which ones? The first hint is a common homeomorphism/isomorphism colloquially known as currying which is valid in any cartesian closed category - in this case compactly generated Hausdorff spaces. The second is a result of the long exact sequence of the loop-space fibration $\Omega(X)\to P(X)\to X$ where $P(X)$ is the path-space of $X$ and $\Omega(X)\simeq Map(S^1,X)$. –  Daniel Rust Nov 5 '13 at 17:02
    
Sorry that I am not clear with notations about path spaces, What do you mean by $P(X)$? Does it mean the space of all paths that joins any two points in $X$?Is there any reference for that fibration? I didn't find it in wiki.. –  mqx Nov 6 '13 at 7:11
    
@mqx You should be able to find the result in any decent book on algebraic topology - for instance Hatcher proves $\pi_i(Map(S^1,X))\cong\pi_{i+1}(X)$ in several different ways (including the above fibration method). Just search for 'loop space' in his text. –  Daniel Rust Nov 6 '13 at 10:15
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