Germs of local diffeomorphisms at a point

Let $M$ and $N$ be smooth manifolds (Smooth manifolds and their smooth charts) and let $x\in M$ and $y\in N$. A germ of local diffeomorphisms from $(M,x)$ to $(N,y)$ is an equivalence class of local diffeomorphisms $f:U\to V$ (Diffeomorphisms and local diffeomorphisms of manifolds) with $U$ open in $M$, $x\in U$, $V$ open in $N$, $y\in V$ and $f(x)=y$, two such maps $f:U\to V$ and $f^{\mathrm{\prime }}:U^{\mathrm{\prime }}\to V^{\mathrm{\prime }}$ being equivalent when they agree on some open neighbourhood $W\subseteq U\cap U^{\mathrm{\prime }}$ of $x$. This is the germ-of-maps relation of The germ of a smooth function at a point, read for local diffeomorphisms instead of functions; the class of $f$ is written ${\mathrm{germ}}_x(f)$, or simply $\mathrm{germ}(f)$ when $x$ is understood.

When $M=N$ and $x=y$, write ${\mathrm{Diff}}_x(M)$ for the set of germs of local diffeomorphisms $(M,x)\to (M,x)$. For two germs $[f],[g]\in {\mathrm{Diff}}_x(M)$ represented by local diffeomorphisms $f:U\to V$ and $g:U^{\mathrm{\prime }}\to V^{\mathrm{\prime }}$ with $f(x)=g(x)=x$, the composite $f\circ g$ is defined on $g^{-1}(U)\cap U^{\mathrm{\prime }}$, an open neighbourhood of $x$, and is again a local diffeomorphism fixing $x$; the germ of this composite is declared to be the product $[f]\cdot [g]$. The germ of ${\mathrm{i}\mathrm{d}}_M$ is declared to be the identity, and the germ of a local inverse of a representative is declared to be its inverse. That these declarations are well defined and satisfy the group axioms is the content of Germs of local diffeomorphisms at a point form a group ↗; in particular ${\mathrm{Diff}}_x(M)$ is a group under this operation, and it is the group of germs used for transverse diffeomorphisms on this page.