# 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 ExoticR4s 2013-03-22 15:37:33 2013-03-22 15:37:33 whm22 (2009) whm22 (2009) 21 whm22 (2009) Definition msc 57R12 msc 14J80 DonaldsonsTheorem DonaldsonFreedmanExoticR4