The K-homology class of the Euler characteristic operator is trivial
Operator Algebras
2011-03-10 v1 Differential Geometry
K-Theory and Homology
Abstract
On any manifold M^n, the de Rham operator D=d+d^* (with respect to a complete Riemannian metric), with the grading of forms by parity of degree, gives rise by Kasparov theory to a class [D] in KO_0(M), which when M is closed maps to the Euler characteristic chi(M) in KO_0(point) = Z. The purpose of this note is to give a quick proof of the (perhaps unfortunate) fact that [D] is as trivial as it could be subject to this constraint. More precisely, if M is connected, [D] lies in the image of Z = KO_0(point) in KO_0(M) (induced by the inclusion of a basepoint).
Keywords
Cite
@article{arxiv.math/9806073,
title = {The K-homology class of the Euler characteristic operator is trivial},
author = {Jonathan Rosenberg},
journal= {arXiv preprint arXiv:math/9806073},
year = {2011}
}
Comments
9 pages. This paper has been accepted by Proceedings of the AMS