An infinite set of exotic $\mathbf{R}^4$'s

Abstract

In 1982, Michael Freedman startled the topological community by pointing out the existence of an "exotic R 4 ", a smooth manifold homeomorphic to R 4 , but not diffeomorphic to it. This result follows easily from Donaldson's Theorem [2] on the nonexistence of certain smooth 4-manifolds, together with Freedman's powerful techniques This exotic R 4 was shocking to topologists, because in dimensions n 4, it is a fundamental result of smoothing theory that there are no exotic R"'s. (Since R" is contractible, there is no place for any bundle-theoretic obstruction to live.) Thus, this exotic R 4 implies a catastrophic failure in dimension 4 of the basic philosophy of smoothing theory, as well as other high-dimensional techniques.