∞-Lie theory

# Contents

## Idea

Where an ordinary manifold is a space locally modeled on Cartesian spaces, an orbifold is, more generally, a space that is locally modeled on smooth action groupoids (=”stack quotients”) of a finite group acting on a Cartesian space.

This turns out to be equivalent (Moerdijk, Mordijk-Pronk) to saying that an orbifold is a proper étale Lie groupoid. (Morita equivalent Lie groupoids correspond to the same orbifolds.)

The word orbifold was invented in (Thurston 1992), while the original name was $V$-manifold (Satake), and was taken in a more restrictive sense, assuming that the actions of finite groups on the charts are always effective. Nowdays we call such orbifolds effective and those which are global quotients by a finite group global quotient orbifolds.

There is also a notion of finite stabilizers in algebraic geometry. A singular variety is called an (algebraic) orbifold if it has only so-called orbifold singularities.

## Definition

An orbifold is a stack presented by an orbifold groupoid.

## Properties

### General

• One can consider a bicategory of proper étale Lie groupoids and the orbifolds will be the objects of certain bicategorical localization of this bicategory (a result of Moerdijk and Pronk).

• Equivalently, every orbifold is globally a quotient of a smooth manifold by an action of finite-dimensional Lie group with finite stabilizers in each point.

### (Co)homology

It has been noticed that the topological invariants of the underlying topological space of an orbifold as a topological space with an orbifold structure are not appropriate, but have to be corrected leading to orbifold Euler characteristics, orbifold cohomology etc. One of the constructions which is useful in this respect is the inertia orbifold (the inertia stack of the original orbifold) which gives rise to “twisted sectors” in Hilbert space of a quantum field theory on the orbifold, and also to twisted sectors in the appropriate cohomology spaces. A further generalization gives multitwisted sectors.

## Examples

Orbifolds are in differential geometry what Deligne-Mumford stacks are in algebraic geometry. In topology one also speaks of orbispaces.

Orbifolds may be regarded as a kind of stratified spaces.

## References

Original sources on orbifolds include

• I. Satake, On a generalisation of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363.
• I. Satake, The Gauss–Bonnet theorem for $V$-manifolds, J. Math. Soc. Japan 9 (1957), 464–492.

• William Thurston, Three-dimensional geometry and topology, preliminary draft, University of Minnesota, Minnesota, (1992) which in completed and revised form is available as his book: The Geometry and Topology of Three-Manifolds; in particular the orbifold discussion is in chapter 13.

Discussion of orbifold as Lie groupoids/differentiable stacks is in

Orbifolds often appear as moduli spaces in differential geometric setting:

The generalization of orbifolds to weighted branched manifolds is discussed in

• Dusa McDuff, Groupoids, branched manifolds and multisections, J. Symplectic Geom. Volume 4, Number 3 (2006), 259-315 (project euclid)