Relative K-homology of higher-order differential operators

TL;DR

This paper generalizes spectral triples to higher-order relative spectral triples to include hypoelliptic operators on manifolds with boundary, leading to new index theorems in relative K-homology.

Contribution

It introduces higher-order relative spectral triples and computes their boundary maps, extending the Baum-Douglas-Taylor index theorem to a broader class of differential operators.

Findings

Defined higher-order relative spectral triples for hypoelliptic operators

Calculated the K-homology boundary map for these triples

Generalized the Baum-Douglas-Taylor index theorem

Abstract

We extend the notion of a spectral triple to that of a higher-order relative spectral triple, which accommodates several types of hypoelliptic differential operators on manifolds with boundary. The bounded transform of a higher-order relative spectral triple gives rise to a relative K-homology cycle. In the case of an elliptic differential operator on a compact smooth manifold with boundary, we calculate the K-homology boundary map of the constructed relative K-homology cycle to obtain a generalization of the Baum-Douglas-Taylor index theorem.