exotic R4’s

If $n\neq 4$ then the smooth manifolds homeomorphic to a given topological $n$ - manifold, $M$ , are parameterized by some discrete algebraic invariant of $M$ . In particular there is a unique smooth manifold homeomorphic to $\mathbb{R}^{n}$ .

By contrast one may choose uncountably many open sets in $\mathbb{R}^{4}$ , which are all homeomorphic to $\mathbb{R}^{4}$ , but which are pairwise non-diffeomorphic.

A smooth manifold homeomorphic to $\mathbb{R}^{4}$ , but not diffeomorphic to it is called an exotic $\mathbb{R}^{4}$ .

Given an exotic $\mathbb{R}^{4}$ , $E$ , we have a diffeomorphism $E\times\mathbb{R}\to\mathbb{R}^{5}$ . (As there is only one smooth manifold homeomorphic to $\mathbb{R}^{5}$ ). Hence exotic $\mathbb{R}^{4}$ ’s may be identified with closed submanifolds of $\mathbb{R}^{5}$ . In particular this means the cardinality of the set of exotic $\mathbb{R}^{4}$ ’s is precisely continuum.

Historically, Donaldson’s theorem led to the discovery of the Donaldson Freedman exotic $\mathbb{R}^{4}$ .

Title	exotic R4’s
Canonical name	ExoticR4s
Date of creation	2013-03-22 15:37:33
Last modified on	2013-03-22 15:37:33
Owner	whm22 (2009)
Last modified by	whm22 (2009)
Numerical id	21
Author	whm22 (2009)
Entry type	Definition
Classification	msc 57R12
Classification	msc 14J80
Related topic	DonaldsonsTheorem
Related topic	DonaldsonFreedmanExoticR4