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] in KO0(M), which when
M is closed maps
to the Euler characteristic chi(M) in KO0(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 = KO0(point) in KO0(M)
(induced by the inclusion of a basepoint into M).
This paper has been accepted for publication in
Proceedings of the AMS.
This paper has been accepted for publication in Proceedings of the AMS.