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

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- by Jonathan Rosenberg PDF
- Proc. Amer. Math. Soc.
**127**(1999), 3467-3474 Request permission

## 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 $KO_{0}(\mathrm {pt})= \mathbb {Z}$. 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 $\mathbb {Z}=KO_{0}(\mathrm {pt})\to KO_{0}(M)$ (induced by the inclusion of a basepoint into $M$).## References

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## Additional Information

**Jonathan Rosenberg**- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- MR Author ID: 298722
- ORCID: 0000-0002-1531-6572
- Email: jmr@math.umd.edu
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**127**(1999), 3467-3474 - MSC (1991): Primary 19K33; Secondary 19K35, 19K56, 58G12
- DOI: https://doi.org/10.1090/S0002-9939-99-04943-6
- MathSciNet review: 1610789