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

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

1998-06-12

arxiv.org/abs/math/9806073v1

Proc. Amer. Math. Soc. 127 (1999), no. 12, 3467-3474

Mathematics

Operator Algebras

9 pages. This paper has been accepted by Proceedings of the AMS

Scientific paper

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).

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