K-theory of Etesi C*-algebras

TL;DR

This paper investigates the K-theory of Etesi's $C^*$-algebra associated with smooth 4-manifolds, revealing its structure as a stationary AF-algebra and linking it to manifold invariants and the Brauer group of a number field.

Contribution

It demonstrates that the $C^*$-algebra is a stationary AF-algebra and connects the K-theory to topological invariants and the Brauer group, providing new insights into 4-manifold smoothings.

Findings

$ ext{E}_{ ext{M}}$ is a stationary AF-algebra.

Manifold invariants are expressed via K-theory of $ ext{E}_{ ext{M}}$.

Smoothings form a torsion abelian group isomorphic to a Brauer group.

Abstract

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