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Germs of local diffeomorphisms at a point form a group

Statement

With the notation of Germs of local diffeomorphisms at a point: composition of representatives induces a well-defined binary operation on ${Diff}_{x} (M)$ ; this operation is associative, the germ of the identity is a two-sided identity, and every germ has a two-sided inverse. Hence ${Diff}_{x} (M)$ is a group.

Facts & Assumptions

Given: A smooth manifold $M$ and a point $x \in M$ , with ${Diff}_{x} (M)$ the set of germs at $x$ of local diffeomorphisms $(M, x) \to (M, x)$ .

[F1]

A germ of local diffeomorphisms at $x$ is an equivalence class of local diffeomorphisms $f : U \to V$ with $x \in U$ , $f (x) = x$ , two representatives being equivalent when they agree on a neighbourhood of $x$ ; the product of two germs is represented by the composite on the common domain, the identity germ is that of ${i d}_{M}$ , and the inverse germ is that of a local inverse (Germs of local diffeomorphisms at a point).

[F2]

A local diffeomorphism is a smooth map every point of which has an open neighbourhood mapped diffeomorphically onto an open set; in particular each local diffeomorphism has a smooth local inverse (Diffeomorphisms and local diffeomorphisms of manifolds).

[F3]

A group is a monoid in which every element is invertible, i.e. a set with an associative binary operation, a two-sided identity, and two-sided inverses (Group and abelian group, Subgroup).

Proof

technique · direct

1.1F1F2

Well-definedness. Let $f : U \to V$ and $f^{'} : U^{'} \to V^{'}$ be equivalent representatives of one germ at $x$ , and $g : W \to Z$ and $g^{'} : W^{'} \to Z^{'}$ equivalent representatives of another, all fixing $x$ . Choose an open neighbourhood $N \subseteq U \cap U^{'}$ of $x$ with $f ∣_{N} = f^{'} ∣_{N}$ . Then $g^{- 1} (N) \cap W$ and $g^{' - 1} (N) \cap W^{'}$ are open neighbourhoods of $x$ , because $g (x) = g^{'} (x) = x$ and both are smooth, hence continuous (Smooth maps are continuous), and on the intersection $P : = (g^{- 1} (N) \cap W) \cap (g^{' - 1} (N) \cap W^{'}) \cap Q$ , where $Q$ is an open neighborhood of $x$ on which $g = g^{'}$ , one has $f \circ g = f^{'} \circ g^{'}$ , since $g = g^{'}$ near $x$ and then $f = f^{'}$ on the common image. So the composite germ $[f] \cdot [g] : = [f \circ g]$ does not depend on the representatives.

1.2F1

Associativity and identity. Representatives of three germs all fix $x$ ; on a sufficiently small common neighbourhood the composites $(f \circ g) \circ h$ and $f \circ (g \circ h)$ agree, because composition of functions is associative. Likewise $f \circ i d = i d \circ f = f$ near $x$ , so the germ of ${i d}_{M}$ is a two-sided identity. Hence the operation is associative with a two-sided identity.

1.3F1F2construct

Inverses. Let $f : U \to V$ represent a germ in ${Diff}_{x} (M)$ , so $f (x) = x$ . By [F2] there is an open neighbourhood $A \subseteq U$ of $x$ such that $f ∣_{A}$ is a diffeomorphism onto the open set $f (A)$ ; the inverse $g : = (f ∣_{A})^{- 1} : f (A) \to A$ is a local diffeomorphism with $g (x) = x$ . Its germ $[g]$ satisfies $[f] \cdot [g] = [i d]$ and $[g] \cdot [f] = [i d]$ , because $f \circ g$ and $g \circ f$ are the identity on neighbourhoods of $x$ .

2.1F3step 1.1step 1.2step 1.3∎

Conclusion. The product is well defined (step 1.1), associative with two-sided identity (step 1.2), and every element is invertible (step 1.3). By [F3] the set ${Diff}_{x} (M)$ with this operation is a group.

Depends on

Used by

Cited to discharge well-definedness by Germs of local diffeomorphisms at a point.

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Leiden NCG seminar, Noncommutative Geometry of Foliations (2023 seminar notes) (standard reference, not scraped)
Eckhard Meinrenken, Lie Groupoids and Lie Algebroids, lecture notes (University of Toronto MAT1341, Fall 2017) (standard reference, not scraped)