Is there any general theory about existence of smooth structures of a topological manifold?

Asked Jan 28 '19 at 19:57

Active Jan 28 '19 at 19:57

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I've read that there exists some topological manifold (of dimension $>3$) either with no smooth structure or having some multiple non-diffeomorphic smooth structures.

How to show that the topological manifold $\mathbb{R}^4$ has no smooth structure? And also what is about higher dimensions?

Is there any general theory about the existence of smooth structures of a topological manifold?

Thank You in advance.

asked Jan 28 '19 at 19:57

MAS

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4

$\mathbb{R}^4$ has a smooth structure – Maxime Ramzi Jan 28 '19 at 20:00
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See this question on Simon Donaldson and Michael Freedman in 1984 and the links given there. Try to search bit on this site here, there are many posts about this topic. – Dietrich Burde Jan 28 '19 at 20:02
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Yes, there is such a general theory. As a starting point you can look at Milnor's paper "Differential topology, 46 years later". – Jan 28 '19 at 22:17

Is there any general theory about existence of smooth structures of a topological manifold?

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