Pseudogroup

In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating however from the geometric approach of Sophus Lie to investigate symmetries of differential equations, rather than out of abstract algebra (such as quasigroup, for example). The modern theory of pseudogroups was developed by Élie Cartan in the early 1900s. A pseudogroup imposes several conditions on a sets of homeomorphisms (respectively, diffeomorphisms) defined on open sets U of a given Euclidean space or more generally of a fixed topological space (respectively, smooth manifold). Since two homeomorphisms h : U → V and g : V → W compose to a homeomorphism from U to W, one asks that the pseudogroup is closed under composition and inversion. However, unlike those for a group, the axioms defining a pseudogroup are not purely algebraic; the further requirements are related to the possibility of restricting and of patching homeomorphisms (similar to the gluing axiom for sections of a sheaf). More precisely, a pseudogroup on a topological space S is a collection Γ of homeomorphisms between open subsets of S satisfying the following properties: The domains of the elements g in Γ cover S ("cover"). The restriction of an element g in Γ to any open set contained in its domain is also in Γ ("restriction"). The composition g ○ h of two elements of Γ, when defined, is in Γ ("composition"). The inverse of an element of g is in Γ ("inverse"). The property of lying in Γ is local, i.e. if g : U → V is a homeomorphism between open sets of S and U is covered by open sets Ui with g restricted to Ui lying in Γ for each i, then g also lies in Γ ("local"). As a consequence the identity homeomorphism of any open subset of S lies in Γ. Similarly, a pseudogroup on a smooth manifold X is defined as a collection Γ of diffeomorphisms between open subsets of X satisfying analogous properties (where we replace homeomorphisms with diffeomorphisms).