Riemannian manifolds in noncommutative geometry

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 Poincaré duality. Along the way we clarify the bimodule and first-order conditions for spectral triples.

Examples and details of some topological issues added

References in corpus (5)

• Unbounded bivariant $K$-theory and correspondences in noncommutative geometry
• Reconstruction of manifolds in noncommutative geometry
• The noncommutative geometry of Yang-Mills fields
• Measure Theory in Noncommutative Spaces
• Projective Dirac Operators, Twisted K-Theory and Local Index Formula

Cited by in corpus (22)

• The $K$-theoretic bulk-edge correspondence for topological insulators
• A noncommutative framework for topological insulators
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• Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases
• The Standard Model in Noncommutative Geometry and Morita equivalence
• On stochastic generation of ultrametrics in high-dimension Euclidean spaces
• A spectral geometry for the Standard Model without the fermion doubling
• A reconstruction theorem for Connes-Landi deformations of commutative spectral triples
• Intersecting Quantum Gravity with Noncommutative Geometry - a Review
• C*-algebras of Holonomy-Diffeomorphisms & Quantum Gravity I
• Real structures on almost-commutative spectral triples
• Fermion masses, mass-mixing and the almost commutative geometry of the Standard Model
• Projective Dirac Operators, Twisted K-Theory and Local Index Formula
• A $P$-Adic Spectral Triple
• Application of semifinite index theory to weak topological phases
• Algebraic backgrounds: a framework for noncommutative Kaluza-Klein theory
• Remarks on Morphisms of Spectral Geometries
• Equivariant $KK$-theory for non-Hausdorff groupoids
• Twisted reality and the second-order condition
• Poincare duality for Cuntz-Pimsner algebras of bimodules
• Robustness of topological phases on aperiodic lattices
• Isometric Spectral Subtriples