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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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 $M$).
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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