Global analysis, analysis on manifolds Definition

Pseudogroup

Builds on Topological space Homeomorphism

Definition

Let $X$ be a topological space. A pseudogroup on $X$ is a set $\Gamma$ of homeomorphisms $f \colon U \to V$ between open subsets $U, V$ of $X$ with these properties.

The identity map of $X$ is in $\Gamma$ .
If $f \colon U \to V$ is in $\Gamma$ , then $f^{-1} \colon V \to U$ is in $\Gamma$ .
If $f \colon U \to V$ and $h \colon U' \to V'$ are in $\Gamma$ , then the composite $h \circ f$ , defined on the open set $f^{-1}(V \cap U')$ , is in $\Gamma$ .
If $f \colon U \to V$ is in $\Gamma$ and $W \subseteq U$ is open, then the restriction $f|_W \colon W \to f(W)$ is in $\Gamma$ .
Let $f \colon U \to V$ be a homeomorphism between open subsets of $X$ , and let $U = \bigcup_i U_i$ be a cover by open sets. If each restriction $f|_{U_i}$ is in $\Gamma$ , then $f$ is in $\Gamma$ .

Remarks

Example: the set of all $C^\infty$ diffeomorphisms between open subsets of $\R^n$ is a pseudogroup on $\R^n$ . Each clause holds because being a diffeomorphism is preserved by inversion, composition and restriction, and it is a local property, which gives clause 5. The same holds for real-analytic diffeomorphisms, for biholomorphic maps between open subsets of $\C^n$ and for symplectomorphisms between open subsets of $\R^{2n}$ .

Non-example: on $\R$ , let $\Gamma$ be the set of restrictions to open sets of the bijective affine maps $x \mapsto ax + b$ , $a \ne 0$ . It satisfies clauses 1 to 4 but fails clause 5. The map on $(0, 1) \cup (2, 3)$ that is the identity on $(0, 1)$ and $x \mapsto x + 10$ on $(2, 3)$ is affine on each piece but is not the restriction of a single affine map.

A pseudogroup generated by a set of local homeomorphisms is the smallest pseudogroup containing them. A $\Gamma$ -structure on a space, such as a smooth structure, is an atlas whose transition maps lie in $\Gamma$ .

Reading path

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Homeomorphism concept
Topological space concept
Pseudogroup definition

Tag	`K8YRDXE4WC`
URI	`m2://K8YRDXE4WC`
Kind	definition
Writing	W0 stub
MSC	58H05, 57R30
Defines	Pseudogroup
Uses	Topological space, Homeomorphism

Pseudogroup

Definition

Remarks

Reading path

Further reading

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