K-theory of Etesi C*-algebras

Abstract

We study the $C^{\ast }$-algebra ${\mathbb{E}}_{\mathcal{M}}$ of a smooth 4-dimensional manifold $\mathcal{M}$ introduced by Gábor Etesi. It is proved that the ${\mathbb{E}}_{\mathcal{M}}$ is a stationary AF-algebra. We calculate the topological and smooth invariants of $\mathcal{M}$ in terms of the K-theory of the $C^{\ast }$-algebra ${\mathbb{E}}_{\mathcal{M}}$. Using Gompf's Stable Diffeomorphism Theorem, it is shown that all smoothings of $\mathcal{M}$ form a torsion abelian group. The latter is isomorphic to the Brauer group of a number field associated to the K-theory of ${\mathbb{E}}_{\mathcal{M}}$.

11 pages; an update