Poincare Duality and Spin^c Structures for Noncommutative Manifolds

TL;DR

This paper establishes a noncompact Serre-Swan theorem, characterizes modules over noncommutative manifolds, and demonstrates that Poincare Duality leads to Morita equivalence between the algebra of functions and Clifford algebra in noncommutative spin manifolds.

Contribution

It extends classical geometric theorems to noncommutative, noncompact settings, linking Poincare Duality with Morita equivalence in noncommutative geometry.

Findings

Proved a noncompact Serre-Swan theorem for vector bundle modules.

Identified endomorphism algebras of modules over noncommutative algebras.

Showed Poincare Duality implies Morita equivalence between function algebra and Clifford algebra.

Abstract

We prove a noncompact Serre-Swan theorem characterising modules which are sections of vector bundles not necessarily trivial at infinity. We then identify the endomorphism algebras of the resulting modules. The endomorphism results continue to hold for the generalisation of these modules to noncommutative, nonunital algebras. Finally, we apply these results to not necessarily compact noncommutative spin manifolds, proving that Poincare Duality implies the Morita equivalence of the `algebra of functions' and the `Clifford algebra.'