### The K-homology class of the Euler characteristic operator is trivial, by Jonathan Rosenberg

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 into *M*).

This paper has been accepted for publication in
*Proceedings of the AMS*.

Jonathan Rosenberg <jmr@math.umd.edu>