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17
The Euler Characteristic of a Category
 DOCUMENTA MATH.
, 2008
"... The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula is proved for the cardinality of a colimit of sets, generalizing the classical inclusionexclusion formula. Both rest on a generali ..."
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Cited by 34 (4 self)
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The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula is proved for the cardinality of a colimit of sets, generalizing the classical inclusionexclusion formula. Both rest on a generalization of Rota’s Möbius inversion from posets to categories.
GROUPOIDS, BRANCHED MANIFOLDS AND MULTISECTIONS
, 2006
"... Abstract. Cieliebak, Mundet i Riera and Salamon recently formulated a definition of branched submanifold of Euclidean space in connection with their discussion of multivalued sections and the Euler class. This note proposes an intrinsic definition of a weighted branched manifold Z that is obtained f ..."
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Cited by 11 (1 self)
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Abstract. Cieliebak, Mundet i Riera and Salamon recently formulated a definition of branched submanifold of Euclidean space in connection with their discussion of multivalued sections and the Euler class. This note proposes an intrinsic definition of a weighted branched manifold Z that is obtained from the usual definition of oriented orbifold groupoid by relaxing the properness condition and adding a weighting. We show that if Z is compact, finite dimensional and oriented, then it carries a fundamental class [Z]. Adapting a construction of Liu and Tian, we also show that the fundamental class [X] of any oriented orbifold X may be represented by a map Z → X, where the branched manifold Z is unique up to a natural equivalence relation. This gives further insight into the structure of the virtual moduli cycle in the new polyfold theory recently constructed by Hofer, Wysocki and Zehnder. 1.
GENERALIZED ORBIFOLD EULER CHARACTERISTICS FOR GENERAL ORBIFOLDS AND WREATH PRODUCTS
, 2009
"... We introduce the ΓEulerSatake characteristics of a general orbifold Q presented by an orbifold groupoid G, generalizing to orbifolds that are not necessarily global quotients the generalized orbifold Euler characteristics of BryanFulman and Tamanoi. Each of these Euler characteristics is define ..."
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Cited by 5 (2 self)
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We introduce the ΓEulerSatake characteristics of a general orbifold Q presented by an orbifold groupoid G, generalizing to orbifolds that are not necessarily global quotients the generalized orbifold Euler characteristics of BryanFulman and Tamanoi. Each of these Euler characteristics is defined as the EulerSatake characteristic of the space of Γsectors of the orbifold where Γ is a finitely generated discrete group. We study the behavior of these characteristics under product operations applied to the group Γ 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 almost freely.
Holonomy for Gerbes over Orbifolds
"... In this paper we compute explicit formulas for the holonomy map for a gerbe with connection over an orbifold. We show that the holonomy descends to a transgression map in Deligne cohomology. We prove that this recovers both the inner local systems in Ruan’s theory of twisted orbifold cohomology [18] ..."
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Cited by 4 (1 self)
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In this paper we compute explicit formulas for the holonomy map for a gerbe with connection over an orbifold. We show that the holonomy descends to a transgression map in Deligne cohomology. We prove that this recovers both the inner local systems in Ruan’s theory of twisted orbifold cohomology [18] and the local system of FreedHopkinsTeleman in their work in twisted Ktheory [7]. In the case in which the orbifold is simply a manifold we recover previous results of Gaw¸edzki[8] and Brylinski[3].
A complete obstruction to the existence of nonvanishing vector fields on almostcomplex, closed, cyclic orbifolds
, 408
"... We determine several necessary and sufficient conditions for a closed almostcomplex orbifold Q to admit a nonvanishing vector field. These conditions are stated separately in terms of the orbifold EulerSatake characteristics of Q and the connected components of its twisted sectors, the Euler chara ..."
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Cited by 2 (2 self)
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We determine several necessary and sufficient conditions for a closed almostcomplex orbifold Q to admit a nonvanishing vector field. These conditions are stated separately in terms of the orbifold EulerSatake characteristics of Q and the connected components of its twisted sectors, the Euler characteristics of the underlying topological spaces of Q and the components of its twisted sectors, and in terms of the orbifold Euler class eorb(Q) in ChenRuan orbifold cohomology H ∗ orb(Q; R). 1
The index of a vector field on an orbifold with boundary, preprint. arXiv:0806.2113v1 [math.DG
, 2008
"... double A PoincaréHopf theorem in the spirit of Pugh is proven for compact orbifolds with boundary. The theorem relates the index sum of a smooth vector field in generic contact with the boundary orbifold to the EulerSatake characteristic of the orbifold and a boundary term. The boundary term is ex ..."
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Cited by 1 (1 self)
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double A PoincaréHopf theorem in the spirit of Pugh is proven for compact orbifolds with boundary. The theorem relates the index sum of a smooth vector field in generic contact with the boundary orbifold to the EulerSatake characteristic of the orbifold and a boundary term. The boundary term is expressed as a sum of Euler characteristics of tangency and exitregion orbifolds. As a corollary, we express the index sum of the vector field induced on the inertia orbifold to the Euler characteristics of the associated underlying topological spaces. 1