Holonomy of a C¹ foliation is a representation into C¹ transverse germs

(Transport along a chart chain.) Let $a:I\to L$ be a leafwise loop at $x$. Cover the compact image $a(I)$ by finitely many foliation charts and subdivide $I$ so that each subinterval is mapped by $a$ into a single chart of the cover. Shrink the transversal $T$ so that all the finitely many transitions between consecutive charts are defined on the successive images of $T$; each crossing transports $T$ along the transverse coordinate change $h_{\beta \alpha }$, a one-dimensional $C^1$ local diffeomorphism [F1]. Composing the finitely many resulting germs at $x$ gives an element ${\mathrm{\Phi }}_a\in {\mathrm{Diff}}_x^1(T)$ [F2].

(Independence of the chain.) Two chains of charts for the same loop admit a common refinement by foliation charts. Inserting an intermediate chart replaces one transition germ $h$ by a composite $h=h_2\circ h_1$ of the two induced transverse transitions, and composition in ${\mathrm{Diff}}_x^1(T)$ is associative [F2], so the composite germ does not change. Hence ${\mathrm{\Phi }}_a$ is well defined, independently of the chosen cover, subdivision and chart chain.

(Invariance under leafwise homotopy.) Let $a_s$, $s\in [0,1]$, be a homotopy of leafwise loops at $x$ relative to the endpoints. The parameter square is compact, so it is subdivided into finitely many small rectangles each of which is carried by the homotopy into a single foliation chart [F1]. Within a chart the transverse coordinate is constant along plaques, so moving the path across a rectangle does not change the transverse transport germ; hence the transports along the two boundary paths of each rectangle agree, and gluing the rectangles along their edges shows that the transport along $a_0$ equals that along $a_1$. Therefore ${\mathrm{\Phi }}_a$ depends only on the class $[a]\in {\pi }_1(L,x)$ [F3].

(Homomorphism and orientation.) For composable loops $\alpha ,\beta$ the concatenation $\alpha \ast \beta$ travels along $\alpha$ first and then along $\beta$, so the transport satisfies ${\mathrm{\Phi }}_{\alpha \ast \beta }={\mathrm{\Phi }}_{\beta }\circ {\mathrm{\Phi }}_{\alpha }$; defining ${\rho }_x([\alpha ]):={\mathrm{\Phi }}_{{\alpha }^{-1}}$ on the reversed loop therefore gives ${\rho }_x([\alpha ][\beta ])={\rho }_x([\alpha ])\circ {\rho }_x([\beta ])$, so ${\rho }_x$ is a homomorphism [F2, F3, step 2.1]. Transverse orientability makes every transverse transition increasing, so every transport germ has positive derivative; the same holds for the reversed loop, whence the image lies in ${\mathrm{Diff}}_x^{1,+}(T)$ [F1, F2].

Plaque transport along leafwise loops therefore defines a well-defined homomorphism ${\rho }_x:{\pi }_1(L,x)\to {\mathrm{Diff}}_x^{1,+}(T)$ independent of chart chains and invariant under leafwise homotopies relative to endpoints, as claimed.