Vanishing cycles of a codimension-one foliation

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $F$ be a $C^2$ transversely oriented codimension-one regular foliation. A vanishing cycle supported on a leaf $L_1$ is a jointly $C^2$ family of loops ${\sigma }_t:S^1\to M$, $t\in [0,1]$, such that: (i) each ${\sigma }_t$ lies in one leaf $L_t$ of $F$; (ii) $[{\sigma }_1]$ is nonzero in ${\pi }_1(L_1)$; (iii) ${\sigma }_t$ is null-homotopic in $L_t$ for every $t<1$; and (iv) for each $\theta \in S^1$, $t\mapsto {\sigma }_t(\theta )$ is transverse to $F$. The nearby loops in (iii) have trivial holonomy because they are null-homotopic. Closure of this transverse family makes the supporting loop's holonomy the identity on the side approached by the family: its return map fixes every sufficiently close parameter on that side. Thus the supported loop is a nonlimit cycle on the approached side; its opposite-side holonomy may be nonidentity. The supported class instead determines a nonzero element of the distinct subgroup ${\mathrm{\Pi }}_1^j(L_1)$ on the side approached by the family, as proved in A vanishing cycle determines a nonzero limitwise-nullhomotopy class. Here jointly $C^2$ means the trace map $S^1\times [0,1]\to M$ is $C^2$, with one-sided derivatives at the parameter endpoints; transversality requires its parameter derivative to have nonzero normal component. Smooth foliation data with a smooth trace are included as a special case.