Riemannian manifolds in noncommutative geometry
K-Theory and Homology
2015-05-30 v2 Differential Geometry
Operator Algebras
Abstract
We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spin^c manifolds; and conversely, in the presence of a spin^c structure. We also show how to obtain an analogue of Kasparov's fundamental class for a Riemannian manifold, and the associated notion of Poincar\'e duality. Along the way we clarify the bimodule and first-order conditions for spectral triples.
Cite
@article{arxiv.1109.2196,
title = {Riemannian manifolds in noncommutative geometry},
author = {Steven Lord and Adam Rennie and Joseph C. Varilly},
journal= {arXiv preprint arXiv:1109.2196},
year = {2015}
}
Comments
Examples and details of some topological issues added