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] E 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 = KOo(pt) -KOo(M) (induced by the inclusion of a basepoint into M). Let Mn be a complete Riemannian manifold without boundary (possibly compact, possibly non-compact). Recall that the de Rham operator D = d+ d*, acting on differential forms on M (of all possible degrees) is a formally self-adjoint elliptic operator, and that on the Hilbert space of L2 forms, it is essentially self-adjoint [Ga]. With a certain grading on the form bundle (coming from the Hodge *-operator), D becomes the signature operator; with the more obvious grading of forms by parity of the degree, D becomes the Euler characteristic operator. When M is compact, the kernel of D, the space of harmonic forms, is naturally identified with the real or complex1 cohomology of M by the Hodge Theorem, and in this way one observes that the index of D (with respect to the parity grading) is simply the Euler characteristic of M, whereas the index with respect to the other grading is the signature [AS3]. Now by Kasparov theory (good general references are [B1] and [Higl]), an elliptic operator such as D gives rise to a K-homology class. In the case of a compact manifold, the index of the operator is recovered by looking at the image of this class under the map collapsing M to a point. However, the K-homology class usually carries far more information than the index alone; for example, it determines the index of the operator with coefficients in any vector bundle, and even determines the families index in K* (X) of a family of twists of the operator, as determined by a vector bundle on M x X. (X here is a parameter space.) When M is non-compact, things are similar, except that usually there is no index, and the class lives in an Received by the editors February 12, 1998. 1991 Mathematics Subject Classification. Primary 19K33; Secondary 19K35, 19K56, 58G12.