Maximal exotic \$\mathbb{R}^4\$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T22:59:37Z http://mathoverflow.net/feeds/question/6941 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/6941/maximal-exotic-mathbbr4 Maximal exotic \$\mathbb{R}^4\$ Konstantin Slutsky 2009-11-27T07:12:02Z 2009-11-27T17:11:26Z <p>Article <a href="http://en.wikipedia.org/wiki/Exotic%5FR4" rel="nofollow">Exotic \$\mathbb{R}^4\$</a> 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.</p> http://mathoverflow.net/questions/6941/maximal-exotic-mathbbr4/6954#6954 Answer by Joel Fine for Maximal exotic \$\mathbb{R}^4\$ Joel Fine 2009-11-27T14:48:55Z 2009-11-27T14:48:55Z <p>Another excellent reference is the book "4-Manifolds and Kirby Calculus" by Gompf and Stipsicz. Section 9.4 is devoted to exotic R<sup>4</sup>s. It describes many constructions of exotic R<sup>4</sup>s and describes how to find the universal one. </p> <p>The book gives a beautiful overview of the complexity of smooth structures in dimension 4 and I highly recommend it.</p> http://mathoverflow.net/questions/6941/maximal-exotic-mathbbr4/6961#6961 Answer by Carsten Schultz for Maximal exotic \$\mathbb{R}^4\$ Carsten Schultz 2009-11-27T16:29:20Z 2009-11-27T16:29:20Z <p>Google + Zentralblatt (Zbl 0586.57007) tell me:</p> <p>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",</p> http://mathoverflow.net/questions/6941/maximal-exotic-mathbbr4/6965#6965 Answer by Kevin Walker for Maximal exotic \$\mathbb{R}^4\$ Kevin Walker 2009-11-27T17:11:26Z 2009-11-27T17:11:26Z <p>The paper by Freedman and Taylor mentioned by Carsten Schultz (above or below) is indeed the place to find the explicit construction.</p> <p>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.</p> <p>Well, that was kind of vague, but hopefully not inaccurate.</p>