4

I apologize in advance for the vagueness of this question.

It is known that there exist differential manifolds that are homeomorphic but not diffeomorphic to spheres (Milnor), and likewise there are manifolds homeomorphic but not diffeomorphic to $\mathbb R^4$ (due to Donaldson I think).

I'm having a hard time developing a mental picture of these objects. Does anyone have a good idea of how to think (more or less intuitively) about such spaces?

doetoe
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  • For example the Bolyai hyperbolic plane is homeomorphic to $\Bbb R^2$. – Berci Sep 17 '18 at 00:55
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    But it's diffeomorphic as well, right? Only not isometric. – doetoe Sep 17 '18 at 00:56
  • Well, yes, you're right, but probably it's already something along the way. – Berci Sep 17 '18 at 00:59
  • https://mathoverflow.net/questions/24970/exotic-differentiable-structures-on-r4 – Anubhav Mukherjee Sep 17 '18 at 19:41
  • In dimension 4 most of these sorts of results are due to Freedman and Donaldson (Two fields medal). Everything is basically depends on intersection form and how to operate surgery without changing intersection form. – Anubhav Mukherjee Sep 17 '18 at 19:44
  • The main observation of Freedman was the failure of smooth h-cobordism and how to topologically we(Freedman) can modify it. May be another good idea to get an intuitive idea is to read the proof of h-cobordism theorem. I won't recomend to read Freedman's proof. Just get the idea of Freedman of using Casson handle. [Gompf and Stipschiz has a small paragraph] – Anubhav Mukherjee Sep 17 '18 at 19:48
  • @AnubhavMukherjee Thanks! I'll try to follow your suggestions – doetoe Sep 18 '18 at 09:21
  • https://math.stackexchange.com/questions/29123/explicit-exotic-charts?noredirect=1&lq=1 – Moishe Kohan Sep 28 '18 at 17:21
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    One way to think about exotic smooth 4-manifolds is via their (infinite) Kirby diagram. See Gompf & Stipsicz - 4-manifolds and Kirby calculus – Henry Nov 29 '18 at 19:25

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