An infinite set of exotic \({\mathbb{R}}^ 4\)’s. (English) Zbl 0562.57009

Gompf, Robert E.

J. Differ. Geom. 21, 283-300 (1985).

We contruct an infinite family of distinct smooth manifolds homeomorphic to \({\mathbb{R}}^ 4\). In particular, we obtain a family \(\{R_{m,n} | m,n=0,1,2,...,\infty \}\) of exotic \({\mathbb{R}}^ 4\)’s such that \(R_{m,n}\) admits an orientation-preserving smooth embedding in \(R_{m',n'}\) if and only if \(m\leq m'\) and \(n\leq n'\). \((R_{m,n}=R_{n,m}\) with reversed orientation.) This family is constructed inductively. Each \(R_{m,n}\) is constructed so that it contains a compact handlebody which cannot embed in \(R_{m',n'}\) for \(m'<m\) or \(n'<n\). The nonembedding property is proven by the following method: If the handlebody could be embedded in \(R_{m',n'}\), we could complete a certain smooth surgery problem. This would allow us to construct a counterexample to Donaldson’s Theorem on the nonexistence of certain definite manifolds. (This technique was first used by Freedman and Taylor to obtain results about their universal \({\mathbb{R}}^ 4.)\)
A section has been added, subsequent to Taubes’ generalization of Donaldson’s Theorem to open 4-manifolds with ”periodic” ends. This theorem implies (by an observation of Freedman) a one-parameter (\({\mathbb{R}})\) family of distinct exotic \({\mathbb{R}}^ 4\)’s. We describe the requisite ”furling” argument in detail, and generalize the result to obtain a 2-parameter family \(\{R'_{s,t} | 1\leq s,t\leq \infty \}\) with embedding properties analogous to those of \(\{R_{m,n}\}\).

MSC:

57N13	Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R10	Smoothing in differential topology
57R55	Differentiable structures in differential topology
57R65	Surgery and handlebodies
57R20	Characteristic classes and numbers in differential topology

Keywords:

distinct smooth manifolds homeomorphic to \({\mathbb{R}}^ 4\); exotic \({\mathbb{R}}^ 4\); open 4-manifolds with ”periodic” ends; embedding handlebodies