The finite-holonomy normal model of a compact leaf

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The countable-choice principle used in the foliation pair). Let $F$ be a regular foliation, $L$ a compact leaf, $x\in L$, $T$ a local transversal at $x$ that is an embedded disk (Local transversals to a regular foliation), $H=\mathrm{Hol}(L,x)$ a finite holonomy group, $D\subseteq T$ an $H$-invariant open disk carrying the smooth finite action supplied by Finite holonomy acts on a small transverse disk, shrunk to a linearization disk by the construction below, and $p:\widehat{L}\to L$ the holonomy cover with $\mathrm{Deck}(p)\cong H$ acting by the covering-space action of The deck group of the holonomy cover is the holonomy group (The holonomy cover of a leaf).

In coordinates with $x=0$, write $f_h$ for these action maps and $A_h=D{f}_h(0)$. The chain rule gives $A_{gh}=A_g{A}_h$ (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$). Set $k(z)=\mathrm{\mid }H{\mathrm{\mid }}^{-1}{\sum }_{h\in H}{A}_h^{-1}{f}_h(z)\mathrm{.}$ Then $Dk(0)=I$, and reindexing the sum gives $k(f_g(z))=A_{g}k(z)$. The inverse function theorem gives a smooth inverse near zero (The Euclidean inverse function theorem). Intersect this inverse domain with its finitely many $H$-translates to keep it invariant and injective. Average the Euclidean inner product over the $A_h$; a sufficiently small ball for that inner product lies in the image of this domain and is $H$-invariant. Its inverse image under $k$ is the required smaller disk $D$. Thus the action on $D$ is conjugate to its linear derivative action. This is also the explicit construction in the transverse-disk lemma's Proof, step 3.1; its Statement alone asserts a smooth action, not a conjugacy. In transverse dimension zero $D=\{x\}$ and $H$ is trivial.

The finite-holonomy normal model of $(L,T,D,H)$ is the quotient $\mathcal{N}:=(\widehat{L}\times D)/H,h\cdot (\widehat{y},t):=(h\widehat{y},ht),$ with the diagonal $H$-action given by the deck action on $\widehat{L}$ and the holonomy action on $D$, together with the foliation ${\mathcal{F}}_{\mathcal{N}}$ obtained from the product foliation of $\widehat{L}\times D$ by the slices $\widehat{L}\times \{t\}$, which the diagonal action permutes. The quotient is a smooth foliated manifold: the diagonal action is free and a covering-space action and preserves the product foliation, so The quotient foliation under a free and properly discontinuous foliated action applies; the product carries its canonical product smooth structure (Products of smooth manifolds have a canonical product smooth structure), and the diagonal formula defines an action of the finite group $H$ (Left group actions, transitive actions, and faithful actions).

The central leaf of the model is the image of $\widehat{L}\times \{x\}$; it is canonically diffeomorphic to $L$, because $\widehat{L}\mathrm{/}H\cong L$ by the deck-group lemma and $x$ is fixed by the holonomy action. The leaves of ${\mathcal{F}}_{\mathcal{N}}$ are the images of the slices $\widehat{L}\times \{t\}$; a leaf represented by $t$ is $\widehat{L}\mathrm{/}{H}_t$, where $H_t\le H$ is the stabilizer of $t$. Its holonomy is the germ action of $H_t$: loops lift to paths in $\widehat{L}$ whose endpoints differ by elements of $H_t$, and every such element occurs by connectedness of $\widehat{L}$. The derivative action is faithful: if $A_h=I$, the conjugacy gives $h=\mathrm{i}\mathrm{d}$ as a germ. In the chosen linearized disk a nonidentity linear map cannot be the identity on an open neighborhood of $t$, so this germ action is faithful and the holonomy group is isomorphic to $H_t$. The slice $\widehat{L}\times \{t\}$ finitely covers its image because $H_t$ is finite. Every leaf of the model other than the central one is therefore finitely covered by the holonomy cover of $L$, and all leaves of the model are compact when $L$ is. The model realises $F$ near $L$ in the sense that the central leaf is $L$ and the local foliation near it is the one induced by the product foliation of $\widehat{L}\times D$; the descent to $F$ itself is proved in the normal-model map lemmas below.