14/06/28 06:05:43.69
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The properity of R4 that has infinite differential structures is related to Yang-Mills field? This post imported from StackExchange Physics at 2014-05-04
1 Answer
I will only address the question to know why R4 admits some non-standard, "exotic", differentiable structure.
The question to know why there are infinitely many (and even uncountably many) examples simply requires an extension of the same kind of techniques.
There are several ways to construct examples of exotic R4.
All use some deep results of topological nature due to Freedman and some deep results of differentiable nature due to Donaldson.
I don't know if the results of Freedman have any physical interpretation.
The Yang-Mills theory appears in Donaldson's results, on the differentiable side
(one needs a differentiable structure to write partial differential equations).
Here is a sketch of one of the standard construction.
（以下略）