Transversals, duality, and irrational rotation




Duwenig, Anna
Emerson, Heath

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Transactions of the American Mathematical Society Series B


An early result of Noncommutative Geometry was Connes’ observation in the 1980’s that the Dirac-Dolbeault cycle for the -torus , which induces a Poincaré self-duality for , can be ‘quantized’ to give a spectral triple and a K-homology class in providing the co-unit for a Poincaré self-duality for the irrational rotation algebra for any ⧵. 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 , a finitely generated projective module over by using a reduction-to-a-transversal argument of Muhly, Renault, and Williams, applied to a pair of Kronecker foliations along lines of slope and , using the fact that these flows are transverse to each other. We then compute Connes’ dual of and prove that we obtain an invertible , represented by an equivariant bundle of Dirac-Schrödinger operators. An application of equivariant Bott Periodicity gives a form of higher index theorem describing functoriality of such ‘-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.


This research was carried out in the course of the first-named author’s Ph.D. at the University of Victoria, and forms part of her thesis [5]. We would like to thank Marcelo Laca and Ian Putnam for their sage remarks and advice during the production of this paper, and the referees for their careful reading and helpful corrections.



Duwenig, A., & Emerson, H. (2020). Transversals, duality, and irrational rotation. Transactions of the American Mathematical Society Series B, 7, 254-289.