A vanishing cycle determines a nonzero limitwise-nullhomotopy class

Let $x={\sigma }_1(1)$; by [F1] the trace map is jointly $C^2$, its point tracks are transverse, and $[{\sigma }_1]\mathrm{\ne }1$ in ${\pi }_1(L_1)$, while near $t=1$ the trace annulus is a one-sided transverse fence for ${\sigma }_1$ because $S^1\times [0,1]$ is compact and the tracks are transverse.

Cover the compact annulus by finitely many foliation charts and subdivide it into rectangles contained in single charts; in each rectangle plaque coordinates identify the upper loop with the normal displacement of the lower loop up to a path inside a plaque (Flat charts for a distribution, Plaques of a flat chart), the identifications agree on shared edges, and the C² plaque transport of specified charts preserves the regularity (lem-c2-plaque-transport-and-transverse-fences-preserve-c2-regularity), so for every sufficiently small positive parameter the trace loop ${\sigma }_t$ is leafwise homotopic to the corresponding normal displacement of ${\sigma }_1$.

For $t<1$ the loop ${\sigma }_t$ is closed and null-homotopic on its leaf by [F1], so the leafwise homotopic displaced loop of ${\sigma }_1$ is closed and null-homotopic as well; closedness of all sufficiently small positive displacements is exactly triviality of the one-sided holonomy germ of $[{\sigma }_1]$, and null-homotopy of those displacements is the predicate $Q_j$, so $[{\sigma }_1]$ is a nonzero element of ${\mathrm{\Pi }}_1^j(L_1,x)$ by [F1] and the class-level definition of the limitwise-nullhomotopy subgroup, with only the standing countable choice used.