The holonomy representation and the holonomy group of a leaf

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $F$ be a regular foliation of $M$, let $L$ be a leaf (Leaves of a regular foliation), let $x\in L$, and let $T$ be a local transversal to $F$ at $x$ (Local transversals to a regular foliation). Equip $L$ with the unique intrinsic smooth manifold structure of its maximal connected integral manifold of $TF$ (Regular foliations and integrable distributions correspond, Existence and uniqueness of maximal connected integral manifolds). This topology, rather than the ambient subspace topology, defines ${\pi }_1(L,x)$. The leaf inclusion is smooth and continuous, so intrinsic leaf loops and endpoint-fixed homotopies are leafwise paths and homotopies in $M$.

Since $T$ is a smooth manifold, the germs at $x$ of local diffeomorphisms $T\to T$ form the group ${\mathrm{Diff}}_x(T)$ (Germs of local diffeomorphisms at a point, Germs of local diffeomorphisms at a point form a group).

The holonomy representation of $L$ at $x$ relative to $T$ is ${\rho }_x:{\pi }_1(L,x)\to {\mathrm{Diff}}_x(T),{\rho }_x([a]):=h_{a^{-1}}(T,T),$ where $a$ is a based loop in the intrinsic leaf topology (Based loops and the fundamental group, Leafwise paths and leafwise homotopy relative to endpoints). The inverse is essential: the library product $[a][b]=[a\ast b]$ traverses $a$ first, whereas ordinary composition of germs applies the rightmost map first. Homotopy invariance and reversal therefore give ${\rho }_x([a][b])=(h_b\circ h_a{)}^{-1}=h_a^{-1}\circ h_b^{-1}={\rho }_x([a])\circ {\rho }_x([b])$ (Holonomy depends only on leafwise homotopy relative to endpoints, Holonomy respects path concatenation and reversal). Thus ${\rho }_x$ is a homomorphism. The unreversed map $[a]\mapsto h_a(T,T)$ is an antihomomorphism with the same image and kernel. The holonomy group of $L$ at $x$ is the image $\mathrm{Hol}(L,x):={\rho }_x({\pi }_1(L,x))\le {\mathrm{Diff}}_x(T),$ a subgroup of the group of germs in the sense of Subgroup and Group and abelian group.

Replacing $T$ by another local transversal $T_1$ at $x$ replaces ${\rho }_x$ by a conjugate homomorphism: with $\alpha$ the germ of the transport across the plaque at $x$ from $T_1$ to $T$, one has ${\rho }_x^{T_1}([a])={\alpha }^{-1}\circ {\rho }_x^T([a])\circ \alpha$ for every leaf loop $a$. Indeed, viewing the loop $a$ as the concatenation of the constant path at $x$, then $a$, then the constant path at $x$, the concatenation law gives exactly this formula, with the constant-path germs supplying $\alpha$ and ${\alpha }^{-1}$ (Holonomy respects path concatenation and reversal). It follows that the kernel of ${\rho }_x$ and the conjugacy class of the holonomy group are intrinsic to the leaf and do not depend on the choice of the local transversal $T$. The representative ${\rho }_x$ itself does depend on $T$.