The $K$-homology class of the Euler characteristic operator is trivial

Abstract

On any manifold Mn, 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] 6 KOo(M), which when M is closed maps to the Euler characteristic x(M) in KOo(pt) = 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 = KO0(pt) KO0(M) (induced by the inclusion of a basepoint into M. © 1999 American Mathematical Society.