Dorette Pronk, Laura Scull
We construct a 2-category version of tom Dieck's equivariant fundamental groupoid for representable orbifolds and show that the discrete fundamental groupoid is Morita invariant; hence an orbifold invariant for representable orbifolds.
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This is a survey paper based on my talk at the Workshop on Orbifolds and String Theory, the goal of which was to explain the role of groupoids and their classifying spaces as a foundation for the theory of orbifolds.
We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps. We use this result to define an orbifold version of Bredon cohomology.
Torus orbifolds are topological generalization of symplectic toric orbifolds. We give a construction of smooth orbifolds with torus actions whose boundary is a disjoint union of torus orbifolds using toric topological method. As a result, we show that any orientable locally standard torus orbifold is equivariantly cobordant to some copies of orbifold complex projective spaces. We also discuss some further equivariant cobordism results including the cases when torus orbifolds are actually torus manifolds.
Lie groupoids generalize transformation groups, and so provide a natural language for studying orbifolds and other noncommutative geometries. In this paper, we investigate a connection between orbifolds and equivariant stable homotopy theory using such groupoids. A different sort of twisted sector, along with a classical theorem of tom Dieck, allows for a natural definition of stable orbifold homotopy groups, and motivates defining extended unstable orbifold homotopy groups generalizing previous definitions.
For a finitely presented discrete group $Γ$, we introduce two generalizations of the orbifold Euler characteristic and $Γ$-orbifold Euler characteristic to a class of proper topological groupoids large enough to include all cocompact proper Lie groupoids. The $Γ$-Euler characteristic is defined as an integral with respect to the Euler characteristic over the orbit space of the groupoid, and the $Γ$-inertia Euler characteristic is the usual Euler characteristic of the $Γ$-inertia space associated to the groupoid. A key ingredient is the application of o-minimal structures to study orbit spaces …
We calculate the integral equivariant cohomology, in terms of generators and relations, of locally standard torus orbifolds whose odd degree ordinary cohomology vanishes. We begin by studying GKM-orbifolds, which are more general, before specialising to half-dimensional torus actions.
Equivariant elliptic cohomology and the 2-loop groupoid
本文遵循 Rezk 的提纲，为任意紧 Lie 群构造复解析等变椭圆上同调，证明其基本性质，并以二维环面主丛群胚的轨道范畴参数化，推广 Grojnowski 的构造。
We define a 2-category structure (Pre-Orb) on the category of reduced complex orbifold atlases. We construct a 2-functor F from (Pre-Orb) to the 2-category (Grp) of proper étale effective groupoid objects over the complex manifolds. Both on (Pre-Orb) and (Grp) there are natural equivalence relations on objects: (a natural extension of) equivalence of orbifold atlases in (Pre-Orb) and Morita equivalences in (Grp). We prove that F induces a bijection between the equivalence classes of its source and target.
Integral equivariant $K$-theory and cobordism ring of simplicial GKM orbifold complexes
本文定义了单纯GKM轨形复形，引入了可构建与可分性质，并计算了可分单纯GKM轨形复形的整等变上同调环、等变K-理论环及等变配边环。
Toric orbifolds are a generalization of simplicial projective toric varieties. In this paper, we show that there is a resolution of singularities of a toric orbifold. In a different category, the class of quasi-contact toric manifolds contains the class of good contact toric manifolds. We prove that a quasi-contact toric manifold is equivariantly a boundary. Moreover, we conclude that good contact toric manifolds and generalized lens spaces are equivariantly boundaries.
We define the orbit category for transitive topological groupoids and their equivariant CW-complexes. By using these constructions we define equivariant Bredon homology and cohomology for actions of transitive topological groupoids. We show how these theories can be obtained by looking at the action of a single isotropy group on a fiber of the anchor map, extending equivariant results for compact group actions. We also show how this extension from a single isotropy group to the entire groupoid action can be applied to the structure of principal bundles and classifying spaces.
In this article we give a characterisation of the Baum-Connes assembly map with coefficients. The technical tools needed are the K-theory of C*-categories, and equivariant KK-theory in the world of groupoids.
We propose a notion of 1-homotopy for generalized maps. This notion generalizes those of natural transformation and ordinary homotopy for functors. The 1-homotopy type of a Lie groupoid is shown to be invariant under Morita equivalence. As an application we consider orbifolds as groupoids and study the notion of orbifold 1-homotopy type induced by a 1-homotopy between presentations of the orbifold maps.
The purpose of this paper is to introduce the notion of loop groupoid associated to a groupoid. After studying the general properties of the loop groupoid, we show how this notion provides a very natural geometric interpretation for the twisted sectors of an orbifold, and for the inner local systems introduced by Ruan by means of a natural generalization of the concept holonomy of a gerbe.
In this paper we define twisted equivariant K-theory for actions of Lie groupoids. For a Bredon-compatible Lie groupoid, this defines a periodic cohomology theory on the category of finite CW-complexes with equivariant stable projective bundles. A classification of these bundles is shown. We also obtain a completion theorem and apply these results to proper actions of groups.
The first purpose of this paper is to examine the relationship between equivariant elliptic genera and orbifold elliptic genera. We apply the character theory of Hopkins et. al. to the Borel-equivariant genus associated to the sigma orientation of Ando-Hopkins-Strickland to define an orbifold genus for certain total quotient orbifolds and supersingular elliptic curves. We show that our orbifold genus is given by the same sort of formula as the orbifold ``two-variable'' genus of Dijkgraaf et al. In the case of a finite cyclic orbifold group, we use the characteristic series for the two-variable…
We propose a new homotopy invariant for Lie groupoids which generalizes the classical Lusternik-Schnirelmann category for topological spaces. We use a bicategorical approach to develop a notion of contraction in this context. We propose a notion of homotopy between generalized maps given by the 2-arrows in a certain bicategory of fractions. This notion is invariant under Morita equivalence. Thus, when the groupoid defines an orbifold, we have a well defined LS-category for orbifolds. We prove an orbifold version of the classical Lusternik-Schnirelmann theorem for critical points.
In this paper we define complex equivariant K-theory for actions of Lie groupoids using finite-dimensional vector bundles. For a Bredon-compatible Lie groupoid, this defines a periodic cohomology theory on the category of finite equivariant CW-complexes. We also establish an analogue of the completion theorem of Atiyah and Segal. Some examples are discussed.
We develop equivariant KK-theory for locally compact groupoid actions by Morita equivalences on real and complex graded C*-algebras. Functoriality with respect to generalised morphisms and Bott periodicity are discussed. We introduce Stiefel-Whitney classes for real or complex equivariant vector bundles over locally compact groupoids to establish the Thom isomorphism theorem in twisted groupoid K-theory.
On aspherical configuration Lie groupoids
本文研究了2维orbifold超平面补空间的非球面性，证明了一类2维非球面orbifold的情形，并在李群胚范畴中推广了该问题。
Configuration Lie groupoids and orbifold braid groups
本文提出配置李群胚的两种定义，证明一类李群胚的Fadell-Neuwirth型纤维化定理，并利用纯轨道折叠辫群的多重几乎自由结构，推导出相应Artin群及轨道折叠辫群的Farrell-Jones猜想。
We introduce a new definition of weighted Grassmann orbifolds. We study their several invariant $q$-cell structures and the orbifold singularities on these $q$-cells. We discuss when the integral cohomology of a weighted Grassmann orbifold has no $p$-torsion. We compute the equivariant $K$-theory ring of weighted Grassmann orbifolds with rational coefficients. We introduce divisive weighted Grassmann orbifolds and show that they have invariant cell structures. We calculate the equivariant cohomology ring, equivariant $K$-theory ring and equivariant cobordism ring of a divisive weighted Grassma…
We introduce equivariant factorization homology, extending the axiomatic framework of Ayala-Francis to encompass multiplicative invariants of manifolds equipped with finite group actions. Examples of such equivariant factorization homology theories include Bredon equivariant homology and (twisted versions of) Hochschild homology. Our main result is that equivariant factorization homology satisfies an equivariant version of tensor excision, and is uniquely characterised by this property. We also discuss applications to representation theory, such as constructions of categorical braid group acti…
Equivariant cohomology of slice groupoids
本文通过构造Weil或Cartan模型之间的自然链映射，证明了切片邻域的等变上同调与切片上的等变上同调同构，并引入切片群胚概念，推广等变理论至此类群胚，证明等变上同调在Morita等价下不变。
Starting with a $\mathbb{C}^*$-valued cocycle on the global quotient orbifold $X // G$, we apply transgression techniques for 2-gerbes, as developed by Lupercio and Uribe, to construct a gerbe on the orbifold loop space $\mathcal{L}(X//G)$. This gives a loop based definition of twisted equivariant Tate K-theory for finite groups that conforms to the definition that Luecke provides for compact, connected Lie groups. We relate our construction to Ganter's work on Moonshine and Huan's quasi-elliptic cohomology.
Twisted factorial Grothendieck polynomials and equivariant $K$-theory of weighted Grassmann orbifolds
本文研究加权格拉斯曼 orbifold 的等效K-理论中的Schubert类，引入扭曲阶乘Grothendieck多项式并证明其代表这些类，同时给出结构常数和限制公式。
Toric orbifolds are a topological generalization of projective toric varieties associated to simplicial fans. We introduce some sufficient conditions on the combinatorial data associated to a toric orbifold to ensure the existence of an invariant cell structure on it and call such a toric orbifold retractable. In this paper, our main goal is to study equivariant cohomology theories of retractable toric orbifolds. Our results extend the corresponding results on divisive weighted projective spaces.
We introduce an equivariant Pontrjagin-Thom construction which identifies equivariant cohomotopy classes with certain fixed point bordism classes. This provides a concrete geometric model for equivariant cohomotopy which works for any compact Lie group G. In the special case when G is finite or a torus, we show that our construction recovers the construction of Wasserman, providing a new perspective on equivariant bordism. We connect the results with bordisms of global quotient orbifolds, utilizing the machinery of Gepner-Henriques to describe bordisms of framed orbifolds in terms of equivaria…
In this paper we introduce the concept of Deligne cohomology of an orbifold. We prove that the third Deligne cohomology group of a smooth étale groupoid classify gerbes with connection over the groupoid. We argue that the $B$-field and the discrete torsion in type II superstring theories are special kinds of gerbes with connection, and finally, for each one of them, using Deligne cohomology we construct a flat line bundle over the inertia groupoid, namely a Ruan inner local in the case of an orbifold.
In this paper, we study Hamiltonian R-actions on symplectic orbifolds [M/S], where R and S are tori. We prove an injectivity theorem and generalize Tolman-Weitsman's proof of the GKM theorem in this setting. The main example is the symplectic reduction X//S of a Hamiltonian T-manifold X by a subtorus S of T. This includes the class of symplectic toric orbifolds. We define the equivariant Chen-Ruan cohomology ring and use the above results to establish a combinatorial method of computing this equivariant Chen-Ruan cohomology in terms of orbifold fixed point data.
We introduce the notion of generalized orbifold Euler characteristic associated to an arbitrary group, and study its properties. We then calculate generating functions of higher order (p-primary) orbifold Euler characteristic of symmetric products of a G-manifold M. As a corollary, we obtain a formula for the number of conjugacy classes of d-tuples of mutually commuting elements (of order powers of p) in the wreath product G wreath S_n in terms of corresponding numbers of G. As a topological application, we present generating functions of Euler characteristic of equivariant Morava K-theories o…
We describe the torus-equivariant cohomology of weighted partial flag orbifolds ${\mathrm{w}}Σ$ of type $A$. We establish counterparts of several results known for the partial flag variety that collectively constitute what we refer to as ``Schubert Calculus on ${\mathrm{w}}Σ$''. For the weighed Schubert classes in ${\mathrm{w}}Σ$, we give the Chevalley's formula. In addition, we define the weighted analogue of double Schubert polynomials and give the corresponding Chevalley--Monk's formula.
In this note, we study equivariant versions of Stolz' $R$-groups, the positive scalar curvature structure groups $R^{\rm spin}_n(X)^G$, for proper actions of discrete groups $G$. We define the concept of a fundamental groupoid functor for a $G$-space, encapsulating all the fundamental group information of all the fixed point sets and their relations. We construct classifying spaces for fundamental groupoid functors. As a geometric result, we show that Stolz' equivariant $R$-group $R^{\rm spin}_n(X)^G$ depends only on the fundamental groupoid functor of the reference space $X$. The proof covers…
The main objects of this paper are torus orbifolds that have exactly two fixed points. We study the equivariant topological type of these orbifolds and consider when we can use the results of the paper [DKS] (arXiv:1809.03678) to compute its integral equivariant cohomology, in terms of generators and relations, coming from the corresponding orbifold torus graph.
Let G be a compact Lie group acting on a smooth manifold M. In this paper, we consider Meinrenken's G-equivariant bundle gerbe connections on M as objects in a 2-groupoid. We prove this 2-category is equivalent to the 2-groupoid of gerbe connections on the differential quotient stack associated to M, and isomorphism classes of G-equivariant gerbe connections are classified by degree three differential equivariant cohomology. Finally, we consider the existence and uniqueness of conjugation-equivariant gerbe connections on compact semisimple Lie groups.
We study the GKM theory for a equivariant stratified space having orbifold structures in tis successive quotients. Then, we introduce the notion of an \emph{almost simple polytope}, as well as a \emph{divisive toric variety} generalizing the concept of a divisive weighted projective space. We employ the GKM theory to compute the generalized equivariant cohomology theories of toric varieties associated to almost simple polytopes and divisive toric varieties.
This very speculative sketch suggests that a theory of fundamental groupoids for tensor triangulated categories could be used to describe the ring of integers as the singular fiber in a family of ring-spectra parametrized by a structure space for the stable homotopy category, and that Bousfield localization might be part of a theory of `nearby' cycles for stacks or orbifolds.
We introduce a new construction, the isotropy groupoid, to organize the orbit data for split $Γ$-spaces. We show that equivariant principal $G$-bundles over split $Γ$-CW complexes $X$ can be effectively classified by means of representations of their isotropy groupoids. For instance, if the quotient complex $A=Γ\backslash X$ is a graph, with all edge stabilizers toral subgroups of $Γ$, we obtain a purely combinatorial classification of bundles with structural group $G$ a compact connected Lie group. If $G$ is abelian, our approach gives combinatorial and geometric descriptions of some results …
The concept of orbifolds should unify differential geometry with equivariant homotopy theory, so that orbifold cohomology should unify differential cohomology with proper equivariant cohomology theory. Despite the prominent role that orbifolds have come to play in mathematics and mathematical physics, especially in string theory, the formulation of a general theory of orbifolds reflecting this unification has remained an open problem. Here we present a natural theory argued to achieve this. We give both a general abstract axiomatization in higher topos theory, as well as concrete models for or…
We use equivariant methods to establish basic properties of orbifold K-theory. We introduce the notion of twisted orbifold K-theory in the presence of discrete torsion, and show how it can be explicitly computed for global quotients.
We describe various equivalent ways of associating to an orbifold, or more generally a higher étale differentiable stack, a weak homotopy type. Some of these ways extend to arbitrary higher stacks on the site of smooth manifolds, and we show that for a differentiable stack X arising from a Lie groupoid G, the weak homotopy type of X agrees with that of BG. Using this machinery, we are able to find new presentations for the weak homotopy type of certain classifying spaces. In particular, we give a new presentation for the Borel construction of an almost free action of a Lie group G on a smooth …
Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact étale Lie groupoid $M$, we show that the corresponding category of gerbe modules, if it is non-trivial, is equivalent to the category of finitely generated projective modules over an Azumaya algebra on $M$. This result can be seen as an equivariant Serre-Swan theorem for twisted vector bundles.
We present a comparative study of certain invariants defined for group actions and their analogues defined for orbifolds. In particular, we prove that Fadell's equivariant category for $G$-spaces coincides with the Lusternik-Schnirelmann category for orbifolds when the group is finite.
Cyclification of Orbifolds
本文在凝聚高阶拓扑斯理论中将轨形环化严格构造为模叠上的基变换，证明Ganter/Huan扩展惯性群胚是其模型，并应用于ADE-轨形等变Tate-椭圆上同调的扭曲及其与椭圆M5-膜亏格的联系。
We describe a bicategory $(\mathcal{R}ed\,\mathcal{O}rb)$ of reduced orbifolds in the framework of classical differential geometry (i.e. without any explicit reference to notions of Lie groupoids or differentiable stacks, but only using orbifold atlases, local lifts and changes of charts). In order to construct such a bicategory, we first define a $2$-category $(\mathcal{R}ed\,\mathcal{A}tl)$ whose objects are reduced orbifold atlases (on any paracompact, second countable, Hausdorff topological space). The definition of morphisms is obtained as a slight modification of a definition by A. Pohl,…
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