Nonvanishing vector fields on orbifolds

Abstract

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 Γ an orbifold called the space of Γ-sectors of Q. The obstruction occurs as the Euler-Satake characteristics of the Γ-sectors for an appropriate choice of Γ; in the case that Q is oriented, this obstruction is expressed as a cohomology class, the Γ-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.

$\Gamma $-extensions of the spectrum of an orbifold

“…If x ∈ M with isotropy group G x and ϕ : Γ → G x ≤ G is a homomorphism with image contained in G x , then a linear orbifold chart {V x , G x , π x } for O at the orbit Gx induces a linear orbifold chart V ϕ x , C Gx (ϕ), π ϕ x for O (ϕ) at the point C G (ϕ)x. Proposition 2.4 ( [14] and [15]). Let Γ be a finitely generated discrete group.…”

“…We define the orbifolds O 1 and O 2 as biquotients of SO (15). To begin, identify SO where B ∈ SO (12).…”

“…To begin, identify SO where B ∈ SO (12). Using this identification, we may think of K ∆2 i < SO(12) as a subgroup of SO (15). Let G i < SO (15), i = 1, 2, be the subgroup isomorphic to Z 5 2 generated by a 12 , a 23 , and K ∆2 i , and note that the K ∆2 i act on coordinates 4 through 15.…”

“…Using this identification, we may think of K ∆2 i < SO(12) as a subgroup of SO (15). Let G i < SO (15), i = 1, 2, be the subgroup isomorphic to Z 5 2 generated by a 12 , a 23 , and K ∆2 i , and note that the K ∆2 i act on coordinates 4 through 15. Furthermore, G 1 and G 2 are almost conjugate but not conjugate in SO (15) for the same reason that K 1 and K 2 are almost conjugate but not conjugate in SO (6).…”

Algebraic Structures Associated to Orbifold Wreath Products

“…The authors study these and other sector decompositions of orbifolds in [13] and [14]. In [15], the authors consider these decompositions for wreath symmetric products of orbifolds presented by the quotient of a smooth, closed manifold by a compact, connected Lie group acting smoothly and almost freely.…”