K-theory of Etesi C*-algebras
Igor V. Nikolaev
TL;DR
This work links smooth structures on 4-manifolds to operator algebra invariants by constructing the Etesi C*-algebra $\mathbb{E}_{\mathscr{M}}=C^*(G)$ with $G=\mathrm{Diff}(\mathscr{M})/\mathrm{Diff}_0(\mathscr{M})$ and showing it is a stationary AF-algebra. The K-theory of $\mathbb{E}_{\mathscr{M}}$ is encoded by the Handelman triple $(\Lambda,[\mathfrak{m}],K)$, where $K=\mathbf{Q}(\lambda_A)$ comes from the Perron–Frobenius eigenvalue of the stationary matrix, and the pair $(\Lambda,[\mathfrak{m}])$ captures Morita equivalence data. Gompf's Stable Diffeomorphism Theorem yields that all smoothings of a topological 4-manifold form a torsion abelian group isomorphic to the Brauer group $Br(K)$, providing a global invariant that classifies smooth structures within a homeomorphism class. The paper also establishes Morita invariance under connected sums, computes explicit examples (e.g., $\mathbb{E}_{S^2\times S^2}\cong M_4(\mathbf{C})$), and situates the smoothing classification in the framework of stationary AF-algebras and their K-theory, linking to a functor into division algebras via the Brauer group.
Abstract
We study the $C^*$-algebra $\mathbb{E}_{\mathscr{M}}$ of a smooth 4-dimensional manifold $\mathscr{M}$ introduced by Gábor 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 $\mathbb{E}_{\mathscr{M}}$.