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ORBIFOLDS ARE NOT COMMUTATIVE GEOMETRIES

Part of: Infinite-dimensional manifolds Generalized manifolds

Published online by Cambridge University Press:  01 February 2008

ADAM RENNIE  and
JOSEPH C. VÁRILLY
ADAM RENNIE*
Affiliation:
Institute for Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark Department of Mathematics, Australian National University, Canberra, ACT, 0200, Australia (email: adam.rennie@maths.anu.edu.au)
JOSEPH C. VÁRILLY
Affiliation:
Departamento de Física Teórica I, Universidad Complutense, Madrid 28040, Spain Departamento de Matemáticas, Universidad de Costa Rica, 2060 San José, Costa Rica (email: varilly@cariari.ucr.ac.cr)
*
✉For correspondence; e-mail: adam.rennie@maths.anu.edu.au
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Abstract

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In this note we show that the crucial orientation condition for commutative geometries fails for the natural commutative spectral triple of an orbifold M/G.

Keywords

noncommutative geometryspectral tripleorbifold

MSC classification

Secondary: 57P05: Local properties of generalized manifolds 58B34: Noncommutative geometry (à la Connes)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 84 , Issue 1 , February 2008 , pp. 109 - 116
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

This work was supported by an ARC grant, DP0211367, and by the Statens Naturvidenskabelige Forskningsråd, Denmark. Support from the University Complutense de Madrid and the Vicerrectoría de Investigación of the Universidad de Costa Rica is acknowledged.

References

[1]Carey, A. L., Phillips, J., Rennie, A. and Sukochev, F. A., ‘The Hochschild class of the Chern character for semifinite spectral triples’, J. Funct. Anal. 213 (2004), 111–153.CrossRefGoogle Scholar
[2]Connes, A., ‘Gravity coupled with matter and foundation of noncommutative geometry’, Commun. Math. Phys. 182 (1996), 155–176.CrossRefGoogle Scholar
[3]Moerdijk, I. and Mrčun, J., Introduction to foliations and Lie groupoids (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
[4]Rennie, A. and Várilly, J. C., ‘Reconstruction of manifolds in noncommutative geometry’, Preprint arXiv:math/0610418 [math.OA], Copenhagen, 2006.Google Scholar