# Are exotic spheres still exotic in generalised smooth spaces?

This is really more of a philosophical question, and the title is somewhat rhetorical:

Exotic spheres are a feature of smooth manifold theory, where certain spheres can have more than one differentiable structure. In a generalised smooth space, say for definiteness a diffeological space, its trivial to construct different diffelogies on the space.

Are then exotic spheres really just an artifact of using the wrong smooth technology?

I guess what I'm getting is that the surprise was that there were manifolds with more than one smooth structure, one could argue that this was a first pointer that we should work in a category where the manifolds could be given different smooth structures, since the natural & usual one couldn't maintain uniqueness of smooth structure for certain manifolds.

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I'm also having some trouble understanding the question. David C is correct about there being a full embedding of ordinary manifolds into (say) diffeological spaces. But let's understand first of all that exotic structures are not just about a category $Diff$ of smooth structures; they are about a forgetful functor $U: Diff \to Top$ and the structure of fibers of this functor (where we could change $Top$ to something else like $PL$). So that's where the focus should be: on a forgetful functor of some type. –  Todd Trimble Dec 2 '12 at 23:11
Your question appears to be circular. –  Ryan Budney Dec 2 '12 at 23:15
I strongly suspect that your question and concerns are largely linguistic and have little to do with mathematics. You talk about "wrong" technology. But you do not say what your mathematical purpose is, so there is no right or wrong available for discussion. Bringing in an undefined notion of "right" or "wrong" is an easy route to purposeless mathematical speculation. –  Ryan Budney Dec 2 '12 at 23:20
Is there a forgetful functor $U:DiffEsp\to Top$ from diffeological spaces to topological spaces? If yes, an interesting question could perhaps be: « Given a differentiable manifold $X$ (thought of as a diffeological space), does the fiber $U ^{−1}(U(X))$ contain genuine diffeological spaces (i.e. spaces that do not lie in the essential image of the full embedding $DiffMan\to DiffEsp$)? ». That is: can a manifold have even more exotic structures? (Perhaps this question is trivial for "diffeologists") –  Qfwfq Dec 2 '12 at 23:40
@Qfwfq: there is a canonical topology associated with a diffeological space, the D-topology = finest topology for which plots are continuous. This is mentioned briefly at Wikipedia: en.wikipedia.org/wiki/Diffeology Perhaps if Andrew Stacey sees this, he will have more to say. –  Todd Trimble Dec 3 '12 at 2:01
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As the category of smooth manifolds is a full subcategory of the category of diffeological spaces the answer is yes they are still exotic.

Edit: let me add that the forgetful functor $U$ from diffeological spaces to topological spaces given by the $D$-topology has an adjoint let call it $Sm$. If you take a topological space $X$ it has a diffeology where the plots are given by all continuous maps from numerical domains to $X$. If you begin with a non-empty smooth manifold $M$ of $dim>0$ viewed as a diffeological space then you can notice that $Sm(U(M))$ are not isomorphic. This is not surprising of course.

Let me also add that even if you can put plenty of diffeological structures on a topological space, actually we have no classification results nor surgery techniques available in the diffeological world. People are developping homotopy theory for diffeological spaces (Enxin Wu's Phd thesis is a very good place to look at) maybe one day these techniques will be helpful in order to understand our "classical smooth manifolds". Thus actually exotic spheres are not easier to build nor to recognize in the diffeological world.

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agreed. but thats not the real intent of the question. –  Mozibur Ullah Dec 2 '12 at 21:23
I am very sorry, I did not understand your question. It will be nice if you can reformulate it or made it a little bit more precise. Thanks in advance. –  David C Dec 2 '12 at 21:26
I guess what I'm getting as that the surprise was that there were manifolds with more than one smooth structure, one could argue that this was a first pointer that we should work in a category where the manifolds could be given different smooth structures, since the natural & usual one couldn't maintain uniqueness of smooth structure for certain manifolds. –  Mozibur Ullah Dec 2 '12 at 21:47

This is a bit long for a comment, so I am posting it as an answer.

With the "clarification" you made, you seem to be saying two (apparently contradictory) things: (1) Since it is easier to construct exotic smooth structures working diffeologically, one should replace the classical viewpoint on smooth manifolds with the categorical one. (2) Since there are smooth manifolds which are homeomorphic but not diffeomorphic, one should replace the classical concept of a smooth manifold with a different (unspecified) categorical one.

If you consider history of smooth manifolds, the classical notion of a smooth manifold was the result of a long chain of events, roughly, from Riemann (whose notion of a "manifold" included rainbow) to Weyl (who gave the first precise definition, in the context of Riemann surfaces). This development was driven by needs of analysis and geometry. Yes, existence of exotic spheres was a rather shocking discovery, followed by a series of exciting developments in geometric topology. However, none of this has diminished the needs coming from analysis and geometry. Incidentally (even though this is not a part of your question), it was proven by Sullivan (in 1977) that in all dimensions but 4, every topological manifold has a Lipschitz structure and this structure is unique. This result could be (potentially) used for development of analytical tools for the study of topological manifolds. In spite of efforts to develop such tools, not much has happend in this direction. On the geometric side, while some of the differential geometry could be done in the category of Lipschitz manifolds, one still does not have a complete replacement for the curvature tensor, which makes Lipschitz setting less appealing than the smooth one.

The bottom line is: Until categorical viewpoint on smoothness proves itself superior to the classical one in the context of analysis on manifolds, there does not seem to be any need to replace the classical notion of smoothness for manifolds.

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Agreed, the generalised smooth spaces have to prove its worth (I'm guessing that it will), but that doesn't diminish the importance of the usual smooth manifolds. I guess an analogy would be the discovery of the complex plane and the realisation of its importance (it had to prove its worth - it wasn't enough to know that it was there), but that doesn't diminish the importance of the real line. –  Mozibur Ullah Dec 2 '12 at 23:14
@Misha: « Since there are smooth manifolds which are homeomorphic but not diffeomorphic, one should replace the classical concept of a smooth manifold with a different (unspecified) categorical one. » - If the purpose is to "kill exoticity", wouldn't this "unspecified categorical concept" just concide with that of topological manifold? –  Qfwfq Dec 2 '12 at 23:47
@Qfwfq: This is what I was trying to get to in my discussion of the Lipschitz category and history. The problem (as I tried to explain it) is lack of analytical tools in the topological setting, since Lipschitz does not quite cut it. (Also, in dimension 4 Lipschitz is not equivalent to TOP, as proven by Donaldson and Sullivan.) Lastly, TOP does not completely kill exoticity in the sense that there are homotopy-equivalences which are not homotopic to homeomorphisms. One can also argue for Poincar\'e complexes, and so on. My viewpoint is that there are different manifold-like categories ... –  Misha Dec 3 '12 at 0:18
...which are all different and that's OK: Different categories address different needs, even though, DIFF, so far, is the most (but not exclusively) useful one for geometry and analysis. –  Misha Dec 3 '12 at 0:22
I see; thank you for the explanations. –  Qfwfq Dec 3 '12 at 0:37