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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Nonvanishing vector fields on orbifolds


Authors: Carla Farsi and Christopher Seaton
Journal: Trans. Amer. Math. Soc. 362 (2010), 509-535
MSC (2000): Primary 22A22, 57R25; Secondary 55S91, 58H05
Published electronically: August 7, 2009
MathSciNet review: 2550162
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Abstract: We introduce a complete obstruction to the existence of nonvanishing vector fields on a closed orbifold $ Q$. Motivated by the inertia orbifold, the space of multi-sectors, and the generalized orbifold Euler characteristics, we construct for each finitely generated group $ \Gamma$ an orbifold called the space of $ \Gamma$-sectors of $ Q$. The obstruction occurs as the Euler-Satake characteristics of the $ \Gamma$-sectors for an appropriate choice of $ \Gamma$; in the case that $ Q$ is oriented, this obstruction is expressed as a cohomology class, the $ \Gamma$-Euler-Satake class. We also acquire a complete obstruction in the case that $ Q$ is compact with boundary and in the case that $ Q$ is an open suborbifold of a closed orbifold.


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Additional Information

Carla Farsi
Affiliation: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395
Email: farsi@euclid.colorado.edu

Christopher Seaton
Affiliation: Department of Mathematics and Computer Science, Rhodes College, 2000 N. Parkway, Memphis, Tennessee 38112
Email: seatonc@rhodes.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04938-1
PII: S 0002-9947(09)04938-1
Keywords: Orbifold, orbifold with boundary, vector field, orbifold Euler characteristic, orbifold Euler class, orbifold sector
Received by editor(s): August 12, 2008
Published electronically: August 7, 2009
Additional Notes: The second author was partially supported by a Rhodes College Faculty Development Endowment Grant.
Article copyright: © Copyright 2009 American Mathematical Society