C² plaque transport and finite transverse fences preserve C² regularity

By [F1] the given atlas is a $C^1$ foliation atlas. The chart-chain and homotopy argument of [F2] does not require transverse orientation: compose the transverse coordinate changes along a finite subdivision; a common refinement preserves the composite, and a finite rectangle subdivision of a leafwise homotopy changes paths only inside plaques, where transverse transport is unchanged. These statements use finite compact covers; they apply to germs of either orientation. Under the library concatenation convention, transport on the reversed loop gives the homomorphism. Thus clause (c) holds for the given atlas.

Let $\phi =(x,t):U\to {\mathbb{R}}^{n-1}\times \mathbb{R}$ be a chart of the given $C^2$ atlas, let $T,T^{\mathrm{\prime }}$ be $C^2$ local transversals through points $p,p^{\mathrm{\prime }}$ of one common plaque of $U$, and parametrize $T$ near $p$ and $T^{\mathrm{\prime }}$ near $p^{\mathrm{\prime }}$ by $C^2$ curves $\gamma :J\to U$ and ${\gamma }^{\mathrm{\prime }}:J^{\mathrm{\prime }}\to U$ with $\gamma (0)=p$, ${\gamma }^{\mathrm{\prime }}(0)=p^{\mathrm{\prime }}$, $(t\circ \gamma {)}^{\mathrm{\prime }}(0)\mathrm{\ne }0$ and $(t\circ {\gamma }^{\mathrm{\prime }}{)}^{\mathrm{\prime }}(0)\mathrm{\ne }0$. The plaques of $U$ are the level sets of $t$, so the plaque transport between $T$ and $T^{\mathrm{\prime }}$ matches points with equal $t$-coordinate.

Let $I_1,\dots ,I_m\subseteq \mathbb{R}$ be open intervals covering a compact parameter interval $[a,b]$, and let ${\gamma }_i:I_i\to M$ be $C^2$ traces that agree on $I_i\cap I_j$ for all $i,j$ (in particular on a collar neighbourhood of every seam), so that $\gamma (\theta ):={\gamma }_i(\theta )$ for $\theta \in I_i$ is a well-defined map on ${\bigcup }_i{I}_i\supseteq [a,b]$. At a parameter interior to some $I_i$ the glued map coincides on an open neighbourhood with the $C^2$ map ${\gamma }_i$, hence is $C^2$ there; at a parameter endpoint of $[a,b]$, restriction of any ${\gamma }_i$ whose interval contains that endpoint gives continuous one-sided derivatives of orders one and two, so $\gamma$ is $C^2$ on $[a,b]$ in the one-sided sense.

The function $s\mapsto t({\gamma }^{\mathrm{\prime }}(s))$ is $C^2$ with nonzero derivative at $0$, so by [F3] it has a $C^2$ local inverse $s=\sigma (z)$ near $z=t(p^{\mathrm{\prime }})$; likewise $\theta \mapsto t(\gamma (\theta ))$ has nonzero derivative at $0$ and is a $C^2$ local diffeomorphism. The single-chart transport written in the parameters of $T$ and $T^{\mathrm{\prime }}$ is therefore $\mathrm{\Theta }:=\sigma \circ t\circ \gamma$ near $\theta =0$, it is $C^2$ as a composite of $C^2$ maps, and ${\mathrm{\Theta }}^{\mathrm{\prime }}(0)=(t\circ \gamma {)}^{\mathrm{\prime }}(0)\mathrm{/}(t\circ {\gamma }^{\mathrm{\prime }}{)}^{\mathrm{\prime }}(0)\mathrm{\ne }0$. Hence the piece is a $C^2$ local diffeomorphism germ.

At every parameter the derivative of the glued trace equals the derivative of a piece defined on a neighbourhood of that parameter, and by hypothesis that derivative has nonzero transverse component in a foliation chart; consequently the glued trace is transverse to $F$ at every parameter, and at the endpoints its one-sided derivative equals the one-sided derivative of any piece containing that endpoint, so transversality persists there as well. This is clause (b).

Suppose a plaque transport meets the transversals $T_0,\dots ,T_m$ successively and the piece from $T_{i-1}$ to $T_i$ lies in the chart $U_i$. Inside $U_i$ step 2.1 exhibits that piece as a $C^2$ local diffeomorphism germ with nonzero derivative. If two consecutive pieces are computed in different charts, then on their common domain the transverse coordinates are related by the transition function $h$, which is a $C^2$ diffeomorphism by the atlas hypothesis, and composition with $h$ and with its inverse preserves both $C^2$ regularity and the nonvanishing of the derivative. A finite composition of $C^2$ local diffeomorphism germs with nonzero derivative is again such a germ, so every finite plaque transport between $C^2$ local transversals is a $C^2$ local diffeomorphism germ, which is clause (a).

Clause (a) is step 3.1, clause (b) is step 2.2, and clause (c) is step 1.1; the argument used finitely many charts, finitely many pieces and local $C^2$ inverses only, so no choice principle is invoked.