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Article Exotic $\mathbb{R}^4$ on Wikipedia says that there is at least one maximal smooth structure on $\mathbb{R}^4$, that is such an atlas on $\mathbb{R}^4$ that any other smooth $\mathbb{R}^4$ can be embedded into it. Is the construction of such a maximal exotic $\mathbb{R}^4$ explicit? Can anyone give a reference to the construction? What is a good source (or sources) with examples of exotic smooth structures on $\mathbb{R}^4$? Thanks.

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3 Answers 3

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The paper by Freedman and Taylor mentioned by Carsten Schultz (above or below) is indeed the place to find the explicit construction.

Very roughly, the idea of the construction is as follows. Recall that a Casson handle is, among other things, a smooth 4-manifold which is homeomorphic (but not diffeomorphic) to the standard open 2-handle. It turns out that a countable collection of diffeomorphism classes of Casson handles suffice for solving a certain 5-dimensional h-cobordism problem. The universal $R^4$, call it $U$, is constructed by gluing together countably many copies of each of the Casson handles in the countable collection. Given an arbitrary smooth 4-manifold homeomorphic to $R^4$, we can construct an embedding (but not a proper embedding) into $U$ using the fact that $U$ contains enough Casson handles to solve any link slice problem we might encounter along the way.

Well, that was kind of vague, but hopefully not inaccurate.

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Google + Zentralblatt (Zbl 0586.57007) tell me:

author="Freedman, Michael H. and Taylor, Laurence R.", title="{A universal smoothing of four-space.}", journal="J. Differ. Geom. ", volume="24", pages="69-78", year="1986",

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Another excellent reference is the book "4-Manifolds and Kirby Calculus" by Gompf and Stipsicz. Section 9.4 is devoted to exotic R4s. It describes many constructions of exotic R4s and describes how to find the universal one.

The book gives a beautiful overview of the complexity of smooth structures in dimension 4 and I highly recommend it.

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