Under what circumstances can it be concluded that if two items from the smooth category are related by a topological relationship, then they are also smoothly related in the corresponding way? For instance, it is true that two smooth maps between manifolds without boundary $M$ and $N$ if $N$ is compact always satisfy that if $f, g$ map $M$ into $N$ and $f, g$ are homotopic, then they are also smoothly homotopic. Here the items are f and g, while the relation is homotopy which upgrades to smooth homotopy. (Although a different homotopy may be required to mediate this.) I'm not sure if this claim persists when boundaries are allowed, or when N is allowed to be not compact. Please let me know if boundaries are allowed in your claims/proofs, in addition to whatever other conditions are required.

If $M$ and $N$ are homotopy equivalent, what can one say? What if one is a deformation retract of the other? What if they are homeomorphic?

More esoterically, what if one is a (continuous) covering space for the other? Feel free to add to this if anything hits you, and please let me know if there is a general technique that catches all questions of this type. Thanks.

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    $\begingroup$ For homotopic maps being automatically smoothly homotopic, I believe the answer is yes, and IRRC you can find a proof of that in Lee's Introduction to Smooth Manifolds. Another possible source would be Hirsch's book Differentiable Topology. What with homeomorphy, you can't always conclude diffeomorphy as I'm sure somebody will explain better than I can, as there may be essentially different smooth structures on one given topological manifold. $\endgroup$ – Olivier Bégassat Jun 18 '12 at 2:33
  • $\begingroup$ Well I've heard of how R^4 with the usual topology has uncountably many diffeomorphically distinct structures, but I want something that I can actually prove. It is from Lee that I got the claim I stated in my original post. For some reason in my notes stronger hypotheses were required, but you are right that they are not. However, boundaryless is still required, at least from my understanding of what I read. $\endgroup$ – Jeff Jun 18 '12 at 2:49
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    $\begingroup$ If $M\rightarrow N$ is a covering map and $N$ has a smooth structure, this smooth structure can be lifted to a smooth structure on $M$ (note that the charts in a smooth atlas are simply connected). If $M$ admits multiple smooth structures, then this lifted structure may or may not agree with a given smooth structure. I think these statements work just fine for manifolds with boundary, and (non-)compactness is not an issue either. $\endgroup$ – Dan Ramras Jun 18 '12 at 5:44
  • $\begingroup$ You are right I suppose this question together with the homeomorphism version of the question can be solved by just looking for nice examples of when a locally Euclidean topological space admits two manifold structures that are nondiffeomorphic. $\endgroup$ – Jeff Jun 19 '12 at 1:08

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