Celebratio Mathematica — Freedman

Topology in dImension 4

Works connected to Stefano DeMichelis

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article S. De Michelis and M. H. Freedman: “Uncountably many exotic \( \mathbf{R}^ 4 \)’s in standard 4-space,” J. Differential Geom. 35 : 1 (1992), pp. 219–254. MR 1152230 Zbl 0780.57012

It is known that the standard (Euclidean) smooth structure on 4-space when restricted to certain open subsets homeomorphic to \( \mathbb{R}^4 \) gives a smooth structure which is not diffeomorphic to the standard one. This behavior is a consequence of Donaldson’s counterexample [1987] to the smooth 5-dimensional \( h \)-cobordism theorem, and was noticed (in anticipation of Donaldson’s result) by A. Casson and the second named author (see [Kirby 1989]). Taubes [1987] developed a technically demanding theory of the Yang–Mills equation on “asymptotically end periodic” 4-manifolds in part to verify that a known family of exotic \( \mathbb{R}^4 \)’s were mutually distinct. That family lays smoothly in \( \mathbb{S}^2\times\mathbb{S}^2 \) but not \( \mathbb{R}^4 \). We combine ideas from the above-mentioned papers to address a nested family of \( \mathbb{R}^4 \) homeomorphs called “ribbon \( \mathbb{R}^4 \)’s” lying in \( \mathbb{R}^4 \) standard. There are continuum many pairwise distinct smooth structures represented within this family.