Nonvanishing vector fields on orbifolds

Details Nonvanishing vector fields on orbifolds Nonvanishing vector fields on orbifolds

2008-07-17

arxiv.org/abs/0807.2738v2

Trans. Amer. Math. Soc. 362 (2010), 509--535

Mathematics

Differential Geometry

28 pages; edited for clearer exposition, fixed Example 2.12, added Example 4.6

Scientific paper

10.1090/S0002-9947-09-04938-1

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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