Exotic Smoothness, Physics and Related Topics

Abstract

In 1854 Riemann, the father of differential geometry, suggested that the geometry of space may be more than just a mathematical tool defining a stage for physical phenomena, and may in fact have profound physical meaning in its own right. Since then various assumptions about the spacetime structure have been put forward. But to what extent the choice of mathematical model for spacetime has important physical significance? With the advent of general relativity physicists began to think of the spacetime in terms a differential manifolds. In this short essay we will discuss to what extent the structure of spacetime can be determined (modelled) and the possible role of differential calculus in the due process. The counterintuitive discovery of exotic four dimensional Euclidean spaces following from the work of Freedman and Donaldson surprised mathematicians. Later, it has been shown that exotic smooth structures are especially abundant in dimension four—the dimension of the physical spacetime. These facts spurred research into possible the physical role of exotic smoothness, an interesting but not an easy task, as we will show.