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Suppose we have two (smooth) functions $f,g:X\to Y$, where $X,Y$ are smooth (second-countable, Hausdorff) manifolds which are locally homotopic (that is, any point in $X$ has a neighbourhood $U$ such that for any $V$ contained in $U$ with its closure, there is a homotopy which turns $f$ into $g$ in $V$, but keeps them fixed outside $U$).

Are $f,g$ necessarily homotopic? If the answer is yes, how much can we weaken the assumptions?

This seems rather obvious if $X$ is compact, for example, but I can't think of an easy way to show it in general, although it seems to be intuitively true.

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    @lee: I'm not sure. I think the same example should work for any connected manifold with nontrivial homotopy group. Though I may be missing something, considering it's been some time since I've considered the problem, or even done any differential topology at all...2013-05-25

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Take spheres for example. The identity and constant map from $S^n$ to itself satisfy the condition. Because one can always choose $V$ relatively small compared to $U$, so that we can use geodesics to construct the local homotopy.