Uniform K-theory, and Poincare duality for uniform K-homology

TL;DR

This paper develops uniform K-theory and K-homology, establishing their properties and duality on manifolds of bounded geometry, extending the framework of noncommutative geometry.

Contribution

It constructs the external product for uniform K-homology, proves homotopy invariance, and establishes Poincare duality with uniform K-theory on spin-c manifolds.

Findings

Constructed the external product for uniform K-homology.

Proved homotopy invariance of uniform K-homology.

Established Poincare duality between uniform K-theory and K-homology.

Abstract

We revisit Spakula's uniform K-homology, construct the external product for it and use this to deduce homotopy invariance of uniform K-homology. We define uniform K-theory and on manifolds of bounded geometry we give an interpretation of it via vector bundles of bounded geometry. We further construct a cap product with uniform K-homology and prove Poincare duality between uniform K-theory and uniform K-homology on spin-c manifolds of bounded geometry.