Exotic Smoothness on Spacetime

Abstract

In this paper we review some of the recent mathematical discoveries in the field of differential topology as they might possibly influence our understanding of physical theories. In particular, we refer to the discovery of so-called “exotic” differentiate structures on topologicals simple spaces, such as S7 and R4 and discuss their impact on the construction of physical models for spacetime and related bundles. Relevant basic features of differential topology are surveyed, with special attention to the distinction between merely “different” and the physically significant “nondiffeomorphic” differentiate structures on a given topological manifold. Some early exotic structures including nonstandard complex structures on R2, Whitehead manifolds, and Milnor spheres are reviewed as easy-to-understand analogous examples. Gauge theory and resultant moduli spaces were important contributors to the discovery of R4Θs, SO brief mention is made of them. Next, we provide a very sketchy outline of the way in which Freedman and Donaldson put together the various pieces from different mathematical areas to show the existence of at least one R4Θ. This is followed by a review of some properties of these spaces, some more recent discoveries about them, and finally statements and conjectures of relevance to general relativity and other physical theories.