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Boundaries, spectral triples and K-homology

Journal article
Authors I. Forsyth
Magnus Goffeng
B. Mesland
A. Rennie
Published in Journal of Noncommutative Geometry
Volume 13
Issue 2
Pages 407-472
ISSN 1661-6952
Publication year 2019
Published at Department of Mathematical Sciences
Pages 407-472
Language en
Keywords Spectral triple, manifold-with-boundary, K-homology, positive scalar curvature, higher rho-invariants, index theory, operators, manifolds, signature, algebras, Mathematics, Physics, eeger j, 1983, journal of differential geometry, v18, p575
Subject categories Mathematics


This paper extends the notion of a spectral triple to a relative spectral triple, an unbounded analogue of a relative Fredholm module for an ideal J (sic) A. Examples include manifolds with boundary, manifolds with conical singularities, dimension drop algebras, theta-deformations and Cuntz-Pimsner algebras of vector bundles. The bounded transform of a relative spectral triple is a relative Fredholm module, making the image of a relative spectral triple under the boundary mapping in K-homology easy to compute. We introduce an additional operator called a Clifford normal with which a relative spectral triple can be doubled into a spectral triple. The Clifford normal also provides a boundary Hilbert space, a representation of the quotient algebra, a boundary Dirac operator and an analogue of the Calderon projection. In the examples this data does assemble to give a boundary spectral triple, though we can not prove this in general. When we do obtain a boundary spectral triple, we provide sufficient conditions for the boundary triple to represent the K-homological boundary. Thus we abstract the proof of Baum- Douglas-Taylor's "boundary of Dirac is Dirac on the boundary" theorem into the realm of non-commutative geometry.

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