C¹ germs of local diffeomorphisms at a point

Let $T$ be a one-dimensional manifold equipped with a $C^1$ atlas: its charts are homeomorphisms onto open subsets of $\mathbb{R}$ whose transition maps are $C^1$ with $C^1$ inverses, and "of class $C^1$" for maps of $T$ means of class $C^1$ in these charts, which is exactly the Euclidean notion of Continuously differentiable maps, local inverses, and local diffeomorphisms. Fix $x\in T$.

A $C^1$ local diffeomorphism of $T$ at $x$ fixing $x$ is a map $f:U\to T$ defined on an open neighbourhood $U\subseteq T$ of $x$ such that $f(U)$ is open, $f:U\to f(U)$ is a bijection, $f(x)=x$, and both $f$ and $f^{-1}:f(U)\to U$ are of class $C^1$. Two such local diffeomorphisms $f:U\to T$ and $g:V\to T$ define the same $C^1$ germ at $x$ when they agree on some neighbourhood of $x$ contained in $U\cap V$; write $f{\sim }_{x}g$ for this relation.

The equivalence classes of ${\sim }_x$ are the $C^1$ germs of local diffeomorphisms of $T$ at $x$, and their set is denoted ${\mathrm{Diff}}_x^1(T)$. Composition of representatives induces a binary operation ${\mathrm{Diff}}_x^1(T)\times {\mathrm{Diff}}_x^1(T)\to {\mathrm{Diff}}_x^1(T),$ the class of $f\circ g$ being independent of the chosen representatives, and with this operation ${\mathrm{Diff}}_x^1(T)$ is a group (Group and abelian group) whose identity is the germ of ${\mathrm{i}\mathrm{d}}_T$. Well-definedness of the operation, associativity, the two-sided identity and two-sided inverses are verified in C¹ germs of local diffeomorphisms form a group ↗.

Finally suppose an orientation of a neighbourhood of $x$ is fixed, represented by a chart $t$ at $x$ with $t(x)=0$; write $\tilde{f}$ for the coordinate expression of a representative $f$. The sign of the derivative ${\tilde{f}}^{\mathrm{\prime }}(0)$ is independent of the positively oriented chart and of the representative of the germ, so the germs whose representatives have ${\tilde{f}}^{\mathrm{\prime }}(0)>0$ in such a chart are well defined. They form a subgroup ${\mathrm{Diff}}_x^{1,+}(T)\le {\mathrm{Diff}}_x^1(T)$ (Subgroup), the orientation-preserving $C^1$ germs at $x$; both the invariance of the sign and the subgroup property are proved in C¹ germs of local diffeomorphisms form a group ↗.

Only the case $\dim T=1$ is used in this pair, and there only for the transverse coordinate of a codimension-one foliation, where the sign of the derivative is the transverse orientation datum.