Let $X$ be a connected hyperbolic 3-manifold (without boundary), $S^3$ the 3-sphere and $Map(X,S^3)$ the space of continuous maps between $X$ and $S^3$.
Question: Is the space $Map(X,S^3)$ connected ?
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No, homotopic maps have the same degree, but it's an exercise (common to qualifying exams) to construct maps of any degree from a closed, oriented, connected $n$-manifold X to the $n$-sphere. It is less trivial, and I think due to Hopf, that two maps $f,g: X\rightarrow S^n$ are homotopic if and only if they have the same degree. Hence, the components of this space are labelled by the degree of the mapping.
The object you are seeking is the third cohomotopy group $\pi^3(M)$ of a $3$-dimensional manifold $M.$ It is known (H. Hopf, 1953) that the $n$-th cohomotopy group of an $n$-dimensional complex is isomorphic to the $n$-th cohomology group, which is $Z$ for an orientable manifold, so the answer is NO.