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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2012.08237 |
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Table of Contents:
- 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 $\mathbb{E}_{\mathscr{M}}$.