TITLE: Is Map($T^4$,$S^2$) connected?
QUESTION [5 upvotes]: 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 $\pi_1$?

REPLY [3 votes]: The accepted answer is incorrect. The problem is in Hint 2, which conflates based maps with unbased maps, and in particular which conflates the based loop space $\Omega X$ of a pointed space $(X, x)$ (the space of maps $S^1 \to X$ sending a fixed basepoint in $S^1$ to $x$) with the unbased or free loop space $LX$ of a space $X$ (the space of maps $S^1 \to X$, with no further hypotheses). Hint 1 and Hint 2 together were supposed to convince you that the space you're looknig at is the 4-fold based loop space of $S^2$, which satisfies
$$\pi_0(\Omega^4 S^2) \cong \pi_4(S^2) \cong \mathbb{Z}_2$$
but that's not true; the 4-fold based loop space of $S^2$ is the space of maps $S^4 \to S^2$ sending a fixed basepoint of $S^4$ to a fixed basepoint of $S^2$, and has nothing to do with $T^4$. The space you're looking at is in fact the 4-fold free loop space $L^4 S^2$.