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- Title
TRANSVERSALS, DUALITY, AND IRRATIONAL ROTATION.
- Authors
DUWENIG, ANNA; EMERSON, HEATH
- Abstract
An early result of Noncommutative Geometry was Connes' observation in the 1980's that the Dirac-Dolbeault cycle for the 2-torus 핋², which induces a Poincaré self-duality for 핋², can be 'quantized' to give a spectral triple and a K-homology class in KK0(Aθ ⊗ Aθ, ℂ) providing the co-unit for a Poincar´e self-duality for the irrational rotation algebra Aθ for any θ ∈ ℝ \ ℚ. Connes' proof, however, relied on a K-theory computation and does not supply a representative cycle for the unit of this duality. Since such representatives are vital in applications of duality, we supply such a cycle in unbounded form in this article. Our approach is to construct, for any non-zero integer b, a finitely generated projective module Lb over Aθ ⊗ Aθ by using a reduction-to-a-transversal argument of Muhly, Renault, and Williams, applied to a pair of Kronecker foliations along lines of slope θ and θ + b, using the fact that these flows are transverse to each other. We then compute Connes' dual of [Lb] and prove that we obtain an invertible τb ∈ KK0(Aθ, Aθ), represented by an equivariant bundle of Dirac-Schr¨odinger operators. An application of equivariant Bott Periodicity gives a form of higher index theorem describing functoriality of such 'b-twists' and this allows us to describe the unit of Connes' duality in terms of a combination of two constructions in KK-theory. This results in an explicit spectral representative of the unit -- a kind of 'quantized Thom class' for the diagonal embedding of the noncommutative torus.
- Subjects
RENAULT SA; TORIC varieties; TRANSVERSAL lines; FOLIATIONS (Mathematics); ROTATIONAL motion; K-theory; ALGEBRA
- Publication
Transactions of the American Mathematical Society, Series B, 2020, Vol 7, p254
- ISSN
2330-0000
- Publication type
Article
- DOI
10.1090/btran/54