Exotic R4

Donaldson (1983) showed there exists an exotic smooth Differential Structure on $\Bbb{R}^4$. Donaldson's result has been extended to there being precisely a Continuum of nondiffeomorphic Differential Structures on $\Bbb{R}^4$.

See also Exotic Sphere

References

Donaldson, S. K. ``Self-Dual Connections and the Topology of Smooth 4-Manifold.'' Bull. Amer. Math. Soc. 8, 81-83, 1983.

Monastyrsky, M. Modern Mathematics in the Light of the Fields Medals. Wellesley, MA: A. K. Peters, 1997.