Christopher Seaton

Two Gauss–Bonnet and Poincaré–Hopf theorems for orbifolds with boundary

Abstract We generalize the Gauss–Bonnet and Poincare–Hopf theorems to the case of orbifolds with boundary. We present two such generalizations, the first in the spirit of Satake, in which the local data (i.e. integral of the curvature in the case of the Gauss–Bonnet theorem and the index of the vector field in the case of the Poincare–Hopf theorem) is related to Satakes orbifold Euler–Satake characteristic, a rational number which depends on the orbifold structure. For the second pair of generalizations, we use the Chen–Ruan orbifold cohomology to express the local data in a way which can be related to the Euler characteristic of the underlying space of the orbifold.

Generalized orbifold Euler characteristics for general orbifolds and wreath products

We introduce the A ‐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 A ‐sectors of the orbifold where A is a finitely generated discrete group. We study the behavior of these Euler characteristics under product operations applied to the group A 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

GENERALIZED TWISTED SECTORS OF ORBIFOLDS

For a finitely generated discrete group Γ, the Γ-sectors of an orbifold Q are a disjoint union of orbifolds corresponding to homomorphisms from Γ into a groupoid presenting Q. Here, we show that the inertia orbifold and k-multisectors are special cases of the Γ-sectors, and that the Γ-sectors are orbifold covers of Leidas fixed-point sectors. In the case of a global quotient, we show that the Γ-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 Γ-sectors corresponding to a generalized loop space.

NONVANISHING VECTOR FIELDS ON ORBIFOLDS

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.

The Hilbert Series of a Linear Symplectic Circle Quotient

We compute the Hilbert series of the graded algebra of regular functions on a symplectic quotient of a unitary circle representation. Additionally, we elaborate explicit formulas for the lowest coefficients of the Laurent expansion of such a Hilbert series in terms of rational symmetric functions of the weights. Considerable effort is devoted to including the cases in which the weights are degenerate. We find that these Laurent expansions formally resemble Laurent expansions of Hilbert series of graded rings of real invariants of finite subgroups of . Moreover, we prove that certain Laurent coefficients are strictly positive. Experimental observations are presented concerning the behavior of these coefficients as well as relations among higher coefficients, providing empirical evidence that these relations hold in general.

On Orbifold Criteria for Symplectic Toric Quotients

We introduce the notion of regular symplectomorphism and graded regular sym- plectomorphism between singular phase spaces. Our main concern is to exhibit examples of unitary torus representations whose symplectic quotients cannot be graded regularly sym- plectomorphic to the quotient of a symplectic representation of a finite group, while the corresponding GIT quotients are smooth. Additionally, we relate the question of simplicial- ness of a torus representation to Gaussian elimination.

CLASSIFYING CLOSED 2-ORBIFOLDS WITH EULER CHARACTERISTICS

We determine the extent to which the collection of

Stratifications of Inertia Spaces of Compact Lie Group Actions

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An Impossibility Theorem for Linear Symplectic Circle Quotients

-Euler-Satake characteristics classify closed 2-orbifolds. In particular, we show that the closed, connected, effective, orientable 2-orbifolds are classified by the collection of

WHEN IS A SYMPLECTIC QUOTIENT AN ORBIFOLD

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