Exotic R4

Remarks

For $n \ne 4$, every smooth manifold homeomorphic to $\R^n$ is diffeomorphic to $\R^n$, so dimension $4$ is the only exception. Clifford Taubes showed that there are uncountably many pairwise non-diffeomorphic exotic $\R^4$'s. Some of them, the small exotic $\R^4$'s, embed smoothly as open subsets of the standard $\R^4$; others, the large ones, do not. The constructions use handle decompositions and Casson handles rather than explicit coordinates.