We now have enough mathematical tools to form some intuition about what exotic structures on manifolds are and now we proceed to build a model of an exotic cosmos and the resulting exotic spacetime. We will roughly outline the arguments and procedures in Asselmeyer-Maluga et al. “Does Our Universe Prefer Exotic Smoothness?”. The main mathematical tool in constructing an exotic 4D spacetime is that of a cobordism. A cobordism between two manifolds M and N of the same dimension is a manifold W such that “at one end” W is M and “at the other end” W is N. We think of W as representing a topological or smooth change from M to N, as seen below. It’s a way of transitioning from M to N in some suitably controlled fashion, it’s a topological “tube” where one edge is M and the other edge is N.

It’s important to note there that W arises as a manifold of one dimension greater than M (or N). So, if M and N represent the cosmos at some fixed times, then they are each 3D and W is a 4D manifold.

Asselmeyer-Maluga et al. begin with M=S^3, the Einstein Cosmos. This manifold is the set of all points in 4D flat space that lie a fixed distance from the origin, i.e. S^3 is the standard 3D sphere representing the compact vacuum solutions to Einstein’s equations. The exotic spacetime will result from a choice of N, a 3-manifold that is a so-called homological sphere satisfying certain topological conditions inferred from physical characteristics of the cosmos. Indeed, the authors choose N from a certain class of 3-manifolds called Brieskorn spheres which, in some sense, can be thought of as three-spheres that have been “wildly knotted up inside”. (These ideas are similar to the Witten Construction of exotic spheres.)

With M and N at hand, the authors then construct a so-called Akbulut cork (which Akbulut refers to simply as a ‘cork’). The cork is a 4-manifold C that is topologically trivial (it’s continuously contractible to a point) that is bounded by M at one end and N at the other and is intricately interwoven in between.

Once the cork C is constructed, one picks a smooth cobordism W between M and N, “drills” out a contractible center of the cobordism and “glues” the cork in using a map that extends to a topological equivalence but not a smooth equivalence. The idea is, then, that the “exotic-ness” of the resulting spacetime W is concentrated near the cork, but is only detectable globally on W. (The technical details roughly are that one starts with the K3 surface and removes a four-ball, then one performs surgery on the cork with a suitable guiling that yields the exotic structure desired.)

Properties of this construction include the fact that a neighborhood of the cork in W is an exotic R^4, hence we expect that physics in this region behaves different from that in a standard R^4. Indeed, with some reasonable physical assumptions, with this cork and a certain choice of Brieskorn sphere N, the manifold W will have non-flat Ricci curvature. What Ricci curvature has to do with physics will have to be explained; we begin with this next time. Thanks for reading!