Pseudogroup

Definition

Given data

A topological space $M$.

Definition part

A pseudogroup $G$ over $M$ is a set of partial homeomorphisms between open sets in $G$ such that:

• When the composite of two elements of $G$ is well-defined, it is also an element of $G$
• The restriction of any $g\in G$ where $g$ to an open set contained inside its domain, is also in $G$. Thus, if $g$ maps $U$ homeomorphically to $V$, and $U_{\alpha }\subseteq U$, then the map $g_{\alpha }:U_{\alpha }\to g(U_{\alpha })$ obtained by restricting $g$ to $U_{\alpha }$, is also an element of $G$.
• If $U_{\alpha }$ form an open cover of $U$ and $g{|}_{U_{\alpha }}$ is in $G$ for all $\alpha$, then $g\in G$.

How they arise

Group acting on a manifold

One typical context in which a pseudogroup arises is relative to the action of a group on a manifold. Here, we define the elements of the pseudogroup as all possible restrictions of elements of the group, to open sets in the manifold.