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  Intuitive notion of multiple differential structures

+ 2 like - 0 dislike

I was reading Nakahara Chapter 5 and came across this paragraph:

"Clearly a diffeomorphism is a homeomorphism. What about the converse? Is a homeomorphism a diffeomorphism? In the previous section, we defined the differentiable structure as an equivalence class of atlases. Is it possible for a topological space to carry many differentiable structures?"

He then goes on to state that \(S^7\) and \(\mathbb{R}^4\)have multiple differentiable structures. I understand this has something to do with Donaldson invariants and Seiberg-Witten theory, but for now I'm just trying to understand physically/pictorially what a differentiable structure is and how it isn't unique. Is there an intuitive notion of how a single topological space can admit multiple differentiable structures?

asked Aug 1, 2018 in Mathematics by Sam Makhoul (25 points) [ no revision ]

It's not my affair, but take a Laplasian $\Delta$ and consider different separated variables like Descartes ones, spherical, and elliptical, for example.

Look for "Instantons and the Topology of 4-Manifolds" by Ronald J. Stern ( see the "abstract" page 5 ) with another proof of the Donaldson theorem.

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