nLab
pseudogroup

Contents

Contents

Idea

The definition of a smooth manifold is, as usual, given by

A smooth nn-manifold is a topological space MM equipped with an open cover U αU_\alpha with homeomorphisms ϕ α:U α n\phi_\alpha: U_\alpha \to \mathbb{R}^n such that the transition maps ϕ βϕ α 1:ϕ α(U αU β)ϕ β(U αU β)\phi_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U_\beta) \to \phi_\beta(U_\alpha \cap U_\beta) are smooth.

If we replace n\mathbb{R}^n with some other topological space XX (eg. n\mathbb{C}^n), and “smooth” with some other suitable restriction (eg. analytic), then we get a different kind of manifold. A pseudogroup is a structure we put on XX to specify what transition maps are allowed. The usage of pseudogroups to define the corresponding notion of manifolds is described at manifolds.

Definition

Definition

Let XX be a topological space (or locale) and let HH be the groupoid whose objects are the open subsets of XX and morphisms are the homeomorphisms between them. A pseudogroup (also called transformation pseudogroup) on XX is a wide sub-groupoid GG of HH (ie. GG contains all open subsets of XX) satisfying the sheaf property:

  • If g:UVg: U \to V is a homeomorphism and U αU_\alpha is a cover of UU, then gg is in GG if and only if each restriction g| U α:U αg(U α)g|_{U_\alpha}: U_\alpha \to g(U_\alpha) is in GG.

Note that in particular the restriction of a map in GG remains in GG.

Example

We have the following correspondence between common pseudogroups and kinds of manifolds (the precise correspondence is defined in manifold):

Topological spaceMorphismsManifold
real n-dimensional space n\mathbb{R}^nsmooth maps(n-dimensional) smooth manifolds
complex n-dimensional space n\mathbb{C}^nanalytic functions(n-dimensional) complex manifolds
n-dimensional upper half-space H n={(x 1,,x n) n:x 10}H^n = \{(x_1, \ldots, x_n) \in \mathbb{R}^n: x_1 \geq 0\}smooth mapsmanifolds with boundary
the nn-cube I n=[0,1] nI^n = [0, 1]^nsmooth mapsmanifold with corners
Definition

More generally, an abstract pseudogroup is a sub-inverse semigroup of the inverse semigroup of partial bijections of a set (or, even more generally, local automorphisms of an object in more general context).

Properties

Relation to effective étale stacks

In the language of topological groupoids/Lie groupoids and their associated geometric stacks, pseudogroups correspond to effective étale stacks. (Lawson-Lenz 11, Carchedi 12, section 2).

References

The concept was introduced in

  • Élie Cartan, Sur la structure des groupes infinis de transformation (suite) Ann. Sci. École Norm. Sup. (3), 22:219–308, 1905 (Numdam)

  • Élie Cartan, Les groupes de transformations continus, infinis, simples. Annales Scientifiques de l’École Normale Supérieure 26: 93–161, 1909 (Numdam)

  • St. Golab, Über den Begriff der “Pseudogruppe von Transformationen” Mathematische Annalen 116: 768–780 (1939)

The history of the concept is reviewed in

  • B. L. Reinhart

who attributes the term “pseudogroup” to

  • Veblen, Whitehead 1932

The relation of pseudogroups to Lie groupoids originates with Haefliger, see at Haefliger groupoid.

Further discussion of the relation of psuedogroups to étale groupoids/étale stacks includes

A textbook account is in

  • Alan Paterson, Groupoids, Inverse Semigroups, and their Operator Algebras

See also

Abstract pseudogroups are studied in the inverse semigroup literature, e.g.

  • Mark V. Lawson, Inverse semigroups: the theory of partial symmetries, World Scientific, 1998.

Last revised on August 7, 2016 at 10:31:10. See the history of this page for a list of all contributions to it.