We introduce the -Euler-Satake characteristics of a general orbifold Q presented by an orbifold groupoid G , extending to orbifolds that are not global quotients the generalized orbifold Euler characteristics of Bryan-Fulman and Tamanoi. Each of these Euler characteristics is defined as the Euler-Satake characteristic of the space of -sectors of the orbifold where is a finitely generated discrete group. We study the behavior of these Euler characteristics under product operations applied to the group as well as the orbifold and establish their relationships to existing Euler characteristics for orbifolds. As applications, we generalize formulas of Tamanoi, Wang and Zhou for the Euler characteristics and Hodge numbers of wreath symmetric products of global quotient orbifolds to the case of quotients by compact, connected Lie groups acting locally freely, in particular including all closed, effective orbifolds. 22A22, 55S15; 58E40, 55N91
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.
DateThe final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline.iii Seaton, Christopher W. (Ph.D., Mathematics)Two Gauss-Bonnet and Poincaré-Hopf Theorems for Orbifolds with Boundary Thesis directed by Assoc. Prof. Carla FarsiThe goal of this work is to generalize the Gauss-Bonnet and Poincaré-Hopf Theorems to the case of orbifolds with boundary. We present two such generalizations, the first in the spirit of [20], which uses an argument parallel to that contained in [22]. In this case, the local data (i.e. integral of the curvature in the case of the Gauss-BonnetTheorem and the index of the vector field in the case of the Poincaré-Hopf Theorem)is related to Satake's orbifold Euler characteristic, a rational number which depends on the orbifold structure.For the second pair of generalizations, we use a more recent orbifold cohomology[3] to express the local data in a way which can be related to the Euler characteristic of the underlying space of the orbifold. This case applies only to orbifolds which admit almost-complex structures.
For a finitely generated discrete group , the -sectors of an orbifold Q are a disjoint union of orbifolds corresponding to homomorphisms from into a groupoid presenting Q. Here, we show that the inertia orbifold and k-multisectors are special cases of the -sectors, and that the -sectors are orbifold covers of Leida's fixed-point sectors. In the case of a global quotient, we show that the -sectors correspond to orbifolds considered by other authors for global quotient orbifolds, as well as their direct generalization to the case of an orbifold given by a quotient by a Lie group. Furthermore, we develop a model for the -sectors corresponding to a generalized loop space.
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