C¹ foliation charts preserve plaque equivalence and transverse orientation

(Plaques and intrinsic topology.) Around a point of an overlap, restrict to product boxes in both charts. A transverse transition is a local diffeomorphism, so a connected sufficiently small piece of one slice lies in exactly one slice of the other chart. The resulting plaque-coordinate transitions are $C^1$ with $C^1$ inverse. Thus relatively open plaque pieces give compatible local charts for the intrinsic leaf topology. The inclusion into $M$ is continuous; disjoint ambient neighborhoods separate distinct leaf points, so this topology is Hausdorff. On a compact leaf finitely many plaque charts cover it; the union of their countable Euclidean bases is a countable base. A path in any plaque joining two points is covered by finitely many smaller plaque charts, so refinement preserves its plaque-chain class. Compatible atlas changes have a common local product refinement and preserve the same classes. Reflexivity, reversal and concatenation of finite chains give the equivalence-relation axioms. Plaques themselves depend on the chart domains.

(Transition signs are locally constant.) Work on a connected transverse interval in an overlap of charts $\alpha ,\beta$. $h_{\beta \alpha }$ is a $C^1$ local diffeomorphism of intervals, so its derivative is continuous and nowhere zero [F2]; a continuous nowhere-zero function on an interval has constant sign, since otherwise the intermediate value theorem would give a zero [F3]. Consequently ${\sigma }_{\beta \alpha }(p):=\mathrm{sign}{h}_{\beta \alpha }^{\mathrm{\prime }}(t_{\alpha }(p))\in \{1,-1\}$ is locally constant on the overlap, though its values on different components may differ.

(Cocycle and invariance of co-orientation.) Pointwise on a triple overlap the transitions compose, and The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$ gives ${\sigma }_{\gamma \alpha }={\sigma }_{\gamma \beta }{\sigma }_{\beta \alpha }$, so $\sigma$ is a $\{1,-1\}$-valued cocycle; changing the signed transverse coordinate of chart $\alpha$ means replacing $t_{\alpha }$ by ${\epsilon }_{\alpha }{t}_{\alpha }$ with a locally constant function ${\epsilon }_{\alpha }:U_{\alpha }\to \{1,-1\}$, working on componentwise product refinements so that the signed coordinates remain charts, and the new transition signs are ${\epsilon }_{\beta }{\sigma }_{\beta \alpha }{\epsilon }_{\alpha }$, that is, $\sigma$ changes by the coboundary determined by $\epsilon$. Therefore the existence of signs $\epsilon$ with ${\epsilon }_{\beta }{\sigma }_{\beta \alpha }{\epsilon }_{\alpha }=1$ pointwise on all overlaps — the condition that the atlas can be signed so that all transitions are increasing, which is exactly transverse orientability — is invariant under the choice of signed transverse coordinates, and the leaf decomposition is invariant by step 1.1. For a compatible new atlas, transfer the transverse orientation on every small old/new product overlap: the cross-transition derivative has constant sign locally, and the cocycle identity makes these signs agree where overlaps meet. They orient the transverse coordinate in every refined new chart. Conversely transfer an orientation back to the old atlas. Hence existence of coorientation is independent of the compatible atlas.

The intrinsic leaf charts and Hausdorff topology are supplied by step 1.1; compact leaves are second countable. The plaque-chain classes are independent of refinement and compatible atlas changes, and coorientation is independent of those atlas changes and of signed-coordinate choices.