Poincare duality for Cuntz-Pimsner algebras of bimodules

TL;DR

This paper develops a new method to establish Poincare duality in Cuntz-Pimsner algebras, enabling the construction of fundamental classes in K-theory and K-homology for various examples, including non-commutative cases.

Contribution

It introduces sufficient conditions for lifting Poincare self-duality from coefficient algebras to Cuntz-Pimsner algebras, broadening duality applications.

Findings

Constructive methods for fundamental classes in K-theory.

Application to Cuntz-Krieger algebras and crossed products.

Extension to non-commutative analogues.

Abstract

We present a new approach to Poincare duality for Cuntz-Pimsner algebras. We provide sufficient conditions under which Poincare self-duality for the coefficient algebra of a Hilbert bimodule lifts to Poincare self-duality for the associated Cuntz-Pimsner algebra. With these conditions in hand, we can constructively produce fundamental classes in K-theory for a wide range of examples. We can also produce K-homology fundamental classes for the important examples of Cuntz-Krieger algebras (following Kaminker-Putnam) and crossed products of manifolds by isometries, and their non-commutative analogues.