Poincaré self-duality of A_θ
Date
2020-04-09
Authors
Duwenig, Anna
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Abstract
The irrational rotation algebra A_θ is known to be Poincaré self-dual in the KK-theoretic sense. The spectral triple representing the required K-homology fundamental class was constructed by Connes out of the Dolbeault operator on the 2-torus, but
so far, there has not been an explicit description of the dual element. We geometrically construct, for certain elements g of the modular group, a finitely generated
projective module L_g over A_θ ⊗ A_θ out of a pair of transverse Kronecker flows on
the 2-torus. For upper triangular g, we find an unbounded cycle representing the
dual of said module under Kasparov product with Connes' class, and prove that this
cycle is invertible in KK(A_θ,A_θ), allowing us to 'untwist' L_g to an unbounded cycle
representing the unit for the self-duality of A_θ.
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Keywords
irrational rotation algebra, KK-theory, abstract transversal, groupoid, Kronecker Flow, noncommutative torus, Poincaré duality, Kasparov product, unbounded operator, noncommutative geometry, bivariant K-theory