The monodromy groupoid of a foliation

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 a smooth manifold $M$ with leaf-wise structure as in Leaves of a regular foliation, and let leafwise paths and leafwise homotopy relative to endpoints be as in Leafwise paths and leafwise homotopy relative to endpoints.

The monodromy groupoid $\mathrm{Mon}(F)$ is the groupoid with object set $M$ whose arrows from $x$ to $y$ are the leafwise homotopy classes relative to endpoints of leafwise paths from $x$ to $y$; there is no arrow from $x$ to $y$ when $x$ and $y$ lie in different leaves. Composition is induced by concatenation of leafwise paths: if $a$ is a leafwise path from $x$ to $y$ and $b$ a leafwise path from $y$ to $z$, the composite $[b]\circ [a]$ is the class of the concatenation of $b$ after $a$. The identity at $x$ is the class of the constant leafwise path at $x$, and the inverse of the class of $a$ is the class of the reversed path $a^{-1}$.

The groupoid laws have the following endpoint-fixed witnesses. If $A,B$ are homotopies of composable paths, their concatenation is $A(s,2t)$ for $t\le 1\mathrm{/}2$ and $B(s,2t-1)$ for $t\ge 1\mathrm{/}2$; the clauses agree at the common endpoint, so finite closed pasting makes this a leafwise homotopy. For any endpoint-fixing reparametrization $\varphi :I\to I$, $a((1-s)t+s\varphi (t))$ is an endpoint-fixed leafwise homotopy from $a$ to $a\circ \varphi$. This gives associativity and the two constant-path identities using the explicit reparametrizations in Loop classes form the group ${\pi }_1(X,x_0)$ under concatenation, proof steps 2.1–2.2; those formulas work also when the path endpoints differ. For $a:x\to y$, the path $a((1-s)\min (2t,2-2t))$ contracts $a\ast a^{-1}$ to the constant path at $x$; the same formula with $a^{-1}$ contracts $a^{-1}\ast a$ at $y$. All these maps stay in the single leaf of their paths, and the formulas and finite pasting establish continuity in $M$. Thus the displayed operations are well defined and satisfy all groupoid laws.

The groupoid is set-theoretic: no topology is imposed on the arrow set and no smooth structure on it is asserted here.