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

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 $KO_{0}(\mathrm {pt})= \mathbb {Z}$. 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 $\mathbb {Z}=KO_{0}(\mathrm {pt})\to KO_{0}(M)$ (induced by the inclusion of a basepoint into $M$).