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Abstract Let $$M$$ be a compact 3-manifold and $$\Gamma =\pi _1(M)$$. Work by Thurston and Culler–Shalen established the $${\operatorname{\textrm{SL}}}_2({\mathbb{C}})$$ character variety $$X(\Gamma )$$ as fundamental tool in the study of the geometry and topology of $$M$$. This is particularly the case when $$M$$ is the exterior of a hyperbolic knot $$K$$ in $S^3$. The main goals of this paper are to bring to bear tools from algebraic and arithmetic geometry to understand algebraic and number theoretic properties of the so-called canonical component of $$X(\Gamma )$$, as well as distinguished points on the canonical component, when $$\Gamma $$ is a knot group. In particular, we study how the theory of quaternion Azumaya algebras can be used to obtain algebraic and arithmetic information about Dehn surgeries, and perhaps of most interest, to construct new knot invariants that lie in the Brauer groups of curves over number fields.
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Brauer groups and Galois cohomology of commutative ring spectra
In this paper we develop methods for classifying Baker, Richter, and Szymik's Azumaya algebras over a commutative ring spectrum, especially in the largely inaccessible case where the ring is nonconnective. We give obstruction-theoretic tools, constructing and classifying these algebras and their automorphisms with Goerss–Hopkins obstruction theory, and give descent-theoretic tools, applying Lurie's work on $$\infty$$ -categories to show that a finite Galois extension of rings in the sense of Rognes becomes a homotopy fixed-point equivalence on Brauer spaces. For even-periodic ring spectra $$E$$ , we find that the ‘algebraic’ Azumaya algebras whose coefficient ring is projective are governed by the Brauer–Wall group of $$\pi _0(E)$$ , recovering a result of Baker, Richter, and Szymik. This allows us to calculate many examples. For example, we find that the algebraic Azumaya algebras over Lubin–Tate spectra have either four or two Morita equivalence classes, depending on whether the prime is odd or even, that all algebraic Azumaya algebras over the complex K-theory spectrum $KU$ are Morita trivial, and that the group of the Morita classes of algebraic Azumaya algebras over the localization $KU[1/2]$ is $$\mathbb {Z}/8\times \mathbb {Z}/2$$ . Using our descent results and an obstruction theory spectral sequence, we also study Azumaya algebras over the real K-theory spectrum $KO$ which become Morita-trivial $KU$ -algebras. We show that there exist exactly two Morita equivalence classes of these. The nontrivial Morita equivalence class is realized by an ‘exotic’ $KO$ -algebra with the same coefficient ring as $$\mathrm {End}_{KO}(KU)$$ . This requires a careful analysis of what happens in the homotopy fixed-point spectral sequence for the Picard space of $KU$ , previously studied by Mathew and Stojanoska.
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- Award ID(s):
- 1714273
- PAR ID:
- 10287492
- Date Published:
- Journal Name:
- Compositio Mathematica
- Volume:
- 157
- Issue:
- 6
- ISSN:
- 0010-437X
- Page Range / eLocation ID:
- 1211 to 1264
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/S0010437X21007065
