December 1, 2001
``An orbifold is a space which is locally modeled on the quotient of a vector space by a finite group.'' This sentence is so easily said or written that more than one person has missed some of the subtleties hidden by orbifolds. Orbifolds were first introduced by Satake under the name ``V-manifold'' and rediscovered by Thurston who called them ``orbifolds''. Both of them used only faithful actions to define their orbifolds (these are the so called reduced orbifolds). This intuitive restriction is very unnatural mathematically if we want for example to study suborbifolds. Orbispaces are to topological spaces what orbifolds are to manifolds. They have been defined by Haefliger via topological groupoids. He defined the homotopy and (co)homology groups of an orbispace to be those of the classifying space of the topological groupoid. However, his definition of morphisms is complicated and technical. Ruan and Chen have tried to reformulate the definitions, but they don't seem treat the non-reduced case in a satisfactory way. Our idea is to define orbispaces and orbifolds directly via their classifying spaces. This approach allows us to easily define morphisms of orbispaces and orbifolds with all the desired good properties.
Similar papers 1
March 11, 2002
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.
January 20, 2014
In this paper, we give an accessible introduction to the theory of orbispaces via groupoids. We define a certain class of topological groupoids, which we call orbigroupoids. Each orbigroupoid represents an orbispace, but just as with orbifolds and Lie groupoids, this representation is not unique: orbispaces are Morita equivalence classes of orbigroupoids. We show how to formalize this equivalence by defining the category of orbispaces as a bicatecory of fractions from the cat...
September 25, 2013
The purpose of this thesis is to use the language of orbifold groupoids to describe the geometry and topology of orbifolds, highlighting advantages and disadvantages of this language as they arise.
April 9, 2015
We describe various equivalent ways of associating to an orbifold, or more generally a higher \'etale 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. I...
September 3, 2007
Orbifold groupoids have been recently widely used to represent both effective and ineffective orbifolds. We show that every orbifold groupoid can be faithfully represented on a continuous family of finite dimensional Hilbert spaces. As a consequence we obtain the result that every orbifold groupoid is Morita equivalent to the translation groupoid of an action of a bundle of compact topological groups.
July 18, 2013
We classify orthogonal actions of finite groups on Euclidean vector spaces for which the corresponding quotient space is a topological, homological or Lipschitz manifold, possibly with boundary. In particular, our results answer the question of when the underlying space of an orbifold is a manifold.
October 2, 2006
This is the second of a series of papers which are devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the construction of a set of algebraic invariants -- the homotopy groups, and (2) an analog of CW-complex theory. As a corollary of this machinery, the classical Whitehead theorem which asserts that a weak homotopy equivalence is a homotopy equivalence ...
January 31, 2007
Given a topological group G, its orbit category Orb_G has the transitive G-spaces G/H as objects and the G-equivariant maps between them as morphisms. A well known theorem of Elmendorf then states that the category of G-spaces and the category of contravariant functors Func(Orb_G,Spaces) have equivalent homotopy theories. We extend this result to the context of orbispaces, with the role of Orb_G now played by a category whose objects are topological groups and whose morphisms...
November 20, 2000
This is a survey article on the recent development of "stringy geometry and topology of orbifolds", a new subject of mathematics motivated by orbifold string theory.
June 25, 2008
The first goal of this survey paper is to argue that if orbifolds are groupoids, then the collection of orbifolds and their maps has to be thought of as a 2-category. Compare this with the classical definition of Satake and Thurston of orbifolds as a 1-category of sets with extra structure and/or with the "modern" definition of orbifolds as proper etale Lie groupoids up to Morita equivalence. The second goal is to describe two complementary ways of thinking of orbifolds as ...