The holonomy cover 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 with base point $x$, let $T$ be a local transversal at $x$, let ${\rho }_x:{\pi }_1(L,x)\to {\mathrm{Diff}}_x(T)$ be the holonomy representation and let $K=\ker {\rho }_x$ (The holonomy representation and the holonomy group of a leaf). The holonomy cover of $L$ relative to $T$ is the connected covering $p:\widehat{L}\to L$ with $p_{\ast }{\pi }_1(\widehat{L},\widehat{x})=K$ supplied by The covering of a leaf associated with the holonomy kernel exists, equipped with a base point $\widehat{x}$ over $x$. It exists and, by that lemma, is unique up to an isomorphism over $L$: the construction takes the universal cover $\tilde{L}\to L$, identifies ${\pi }_1(L,x)$ with its deck group and puts $\widehat{L}=\tilde{L}\mathrm{/}K$, so the fibre of $p$ over $z\in L$ is the set of $K$-orbits in the universal-cover fibre over $z$.

By construction ${\pi }_1(\widehat{L},\widehat{x})\cong K$ and the covering $p:\widehat{L}\to L$ is the covering associated with the kernel of the holonomy representation; the covering class of $p$ is the leaf-level input for the finite-holonomy normal model of the Reeb stability pair, which consumes the holonomy cover rather than the universal cover. The kernel $K$ is independent of the choice of the local transversal $T$, because replacing $T$ conjugates ${\rho }_x$ and conjugation preserves kernels, so the holonomy cover of $L$ does not depend on $T$ up to isomorphism over $L$.