nLab global quotient orbifold

Idea

A global quotient orbifold is an orbifold that is represented by a G-space $X$: To the extent that orbifolds are identified with topological groupoids/Lie groupoids, a global quotient orbifold is represented by the action groupoid of $G$ acting on $X$. Understood as representing a topological stack/differentiable stack, this is the quotient stack of $X$ by $G$. Understood in (∞,1)-topos theory this is the homotopy quotient $X \sslash G$.

The orbifold cohomology of global quotient orbifolds is a version of equivariant cohomology of $X$.

The orbifolds that are Morita-equivalently global quotients of smooth manifolds by…

1. … an action of a Lie group are also called presentable orbifolds;

2. …the action of a discrete group are called good orbifolds,

3. … the action of a finite groups are called very good orbifolds .

Non-good orbifolds are also called bad orbifolds (Thurston 92, Ch. 13, Def. 13.2.3). It is conjectured (Adem, Leida & Ruan 2007, Conj. 1.55) that every orbifold is presentable.

Related concepts

• action groupoid, quotient stack, homotopy quotient

• orbifold K-theory, orbifold differential K-theory

• equivariant cohomology

References

Traditional orbifold cohomology

Traditionally, the cohomology of orbifolds has, by and large, been taken to be simply the ordinary cohomology of (the plain homotopy type of) the geometric realization of the topological/Lie groupoid corresponding to the orbifold.

For the global quotient orbifold of a G-space $X$, this is the ordinary cohomology of (the bare homotopy type of) the Borel construction $X \!\sslash\! G \;\simeq\; X \times_G E G$, hence is Borel cohomology (as opposed to finer versions of equivariant cohomology such as Bredon cohomology).

A dedicated account of this Borel cohomology of orbifolds, in the generality of twisted cohomology (i.e. with local coefficients) is in:

• Ieke Moerdijk, Dorette Pronk, Simplicial cohomology of orbifolds, Indagationes Mathematicae Volume 10, Issue 2, 1999, Pages 269-293 (doi:10.1016/S0019-3577(99)80021-4)

Moreover, since the orbifold’s isotropy groups $G_x$ are, by definition, finite groups, their classifying spaces $\ast \!\sslash\! G \simeq B G$ have purely torsion integral cohomology in positive degrees, and hence become indistinguishable from the point in rational cohomology (and more generally whenever their order is invertible in the coefficient ring).

Therefore, in the special case of rational/real/complex coefficients, the traditional orbifold Borel cohomology reduces further to an invariant of just (the homotopy type of) the naive quotient underlying an orbifold. For global quotient orbifolds this is the topological quotient space $X/G$.

In this form, as an invariant of just $X/G$, the real/complex/de Rham cohomology of orbifolds was originally introduced in

• Ichiro Satake, On a generalisation of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363 (doi:10.1073/pnas.42.6.359)

following analogous constructions in

• Walter Lewis Baily, On the quotient of an analytic manifold by a group of analytic homeomorphisms, PNAS 40 (9) 804-808 (1954) (doi:10.1073/pnas.40.9.804)

Since this traditional rational cohomology of orbifolds does, hence, not actually reflect the specific nature of orbifolds, a proposal for a finer notion of orbifold cohomology was famously introduced (motivated from orbifolds as target spaces in string theory, hence from orbifolding of 2d CFTs) in

• Weimin Chen, Yongbin Ruan, A New Cohomology Theory for Orbifold, Commun. Math. Phys. 248 (2004) 1-31 (arXiv:math/0004129)

However, Chen-Ruan cohomology of an orbifold $\mathcal{X}$ turns out to be just Borel cohomology with rational coefficients, hence is just Satake’s coarse cohomology – but applied to the inertia orbifold of $\mathcal{X}$. A review that makes this nicely explicit is (see p. 4 and 7):

• Emily Clader, Orbifolds and orbifold cohomology, 2014 (pdf)

Hence Chen-Ruan cohomology of a global quotient orbifold is equivalently the rational cohomology (real cohomology, complex cohomology) for the topological quotient space $AutMor(X\!\sslash\!G)/G$ of the space of automorphisms in the action groupoid by the $G$-conjugation action.

On the other hand, it was observed in (see p. 18)

• Ieke Moerdijk, Orbifolds as Groupoids: an Introduction, Alejandro Adem, Jack Morava, Yongbin Ruan (eds.) Orbifolds in Mathematics and Physics, Contemporary Math 310 , AMS (2002), 205–222 (arXiv:math.DG/0203100)

that for global quotient orbifolds Chen-Ruan cohomology indeed is equivalent to a $G$-equivariant Bredon cohomology of $X$ – for one specific choice of equivariant coefficient system (abelian sheaf on the orbit category of $G$), namely for $G/H \mapsto ClassFunctions(H)$.

Or rather, Moerdijk 02, p. 18 observes that the Chen-Ruan cohomology of a global quotient orbifold is equivalently the abelian sheaf cohomology of the naive quotient space $X/G$ with coefficients in the abelian sheaf whose stalk at $[x] \in X/G$ is the ring of class functions of the isotropy group at $x$; and then appeals to Theorem 5.5 in

• Hannu Honkasalo, Equivariant Alexander-Spanier cohomology for actions of compact Lie groups, Mathematica Scandinavica Vol. 67, No. 1 (1990), pp. 23-34 (jstor:24492569)

for the followup statement that the abelian sheaf cohomology of $X/G$ with coefficient sheaf $\underline{A}$ being “locally constant except for dependence on isotropy groups” is equivalently Bredon cohomology of $X$ with coefficients in $G/H \mapsto \underline{A}_x$ for $Isotr_x = H$. This identification of the coefficient systems is Prop. 6.5 b) in:

• Hannu Honkasalo, Equivariant Alexander-Spanier cohomology, Mathematica Scandinavia 63 (1988) 179-195 [doi:10.7146/math.scand.a-12232]

See also Section 4.3 of

• Richard Szabo, Alessandro Valentino, Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory, Commun. Math. Phys. 294 (2010) 647-702 [doi:10.1007/s00220-009-0975-1, arXiv:0710.2773]

In summary:

• Traditional orbifold cohomology theory is Borel cohomology of underlying Borel construction-spaces, and reduces rationally further to the rational cohomology of underlying naive quotient spaces.

• Chen-Ruan cohomology is just the latter rational cohomology but applied after passage to the inertia orbifold. This is equivalent to the Bredon cohomology of the original orbifold, for one specific equivariant coefficient-system.

This suggests, of course, that more of proper equivariant cohomology should be brought to bear on a theory of orbifold cohomology. A partial way to achieve this is to prove for a given equivariant cohomology-theory that it descends from an invariant of topological G-spaces to one of the associated global quotient orbifolds.

For topological equivariant K-theory this is the case, by

• Dorette Pronk, Laura Scull, Prop. 4.1 in: Translation Groupoids and Orbifold Bredon Cohomology, Canad. J. Math. 62(2010), 614-645 (arXiv:0705.3249, doi:10.4153/CJM-2010-024-1)

Therefore it makes sense to define orbifold K-theory for orbifolds $\mathcal{X}$ which are equivalent to a global quotient orbifold $\mathcal{X} \simeq \prec(X \!\sslash\! G)$ to be the $G$-equivariant K-theory of $X$: $K^\bullet(\mathcal{X}) \;\coloneqq\; K_G^\bullet(X) \,.$

This is the approach taken in

• Alejandro Adem, Yongbin Ruan, Section 3 of: Twisted Orbifold K-Theory, Commun. Math. Phys. 237 (2003) 533-556 (arXiv:math/0107168)

Exposition and review of traditional orbifold cohomology, with an emphasis on Chen-Ruan cohomology and orbifold K-theory, is in:

• Alejandro Adem, Johann Leida, Yongbin Ruan, Orbifolds and Stringy Topology, Cambridge Tracts in Mathematics 171 (2007) (doi:10.1017/CBO9780511543081, pdf)