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

Author: Jonathan Rosenberg
Journal: Proc. Amer. Math. Soc. 127 (1999), 3467-3474
MSC (1991): Primary 19K33; Secondary 19K35, 19K56, 58G12
Published electronically: May 13, 1999
MathSciNet review: 1610789
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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 $KO_{0}(\hbox {pt})= \mathbb{Z}$. 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 $\mathbb{Z}=KO_{0}(\hbox {pt})\to KO_{0}(M)$ (induced by the inclusion of a basepoint into $M$).

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Additional Information

Jonathan Rosenberg
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742

Keywords: $K$-homology, de~Rham operator, signature operator, Kasparov theory
Received by editor(s): February 12, 1998
Published electronically: May 13, 1999
Additional Notes: The author was partially supported by NSF Grant # DMS-96-25336 and by the General Research Board of the University of Maryland.
Communicated by: Józef Dodziuk
Article copyright: © Copyright 1999 American Mathematical Society

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