Relative generic position for characteristic disk maps

Fix a unit tangent $\tau$ along $\mathrm{\partial }{D}^2$ and let $\beta =h^{\ast }\omega$; by [F1] the singularities of $\beta$ are exactly the critical points of the transverse functions, so it suffices to manipulate $\beta$. In a foliation chart with transverse coordinate $z$ one has $\omega =fdz$ with $f\mathrm{\ne }0$, and a modification of $\beta$ that does not change $\omega$ is the same as a modification of $d(z\circ h)$; the condition "$\beta \mathrm{\ne }0$" is chart-independent because $\beta$ is a globally defined $C^1$ $1$-form on $D^2$.

Leafwise boundary: a transverse field. Write $(\theta ,r)$ for a collar with inward coordinate $r\ge 0$. The compact image $h(\mathrm{\partial }{D}^2)$ has a neighborhood carrying a smooth field $V$ with $\omega (V)>0$: at each image point choose a constant field in a smooth ambient chart with positive evaluation; shrink its domain to retain positivity, take finitely many smaller compact cores covering the image, and patch those finitely many fields with nonnegative smooth bumps from [F3]. Their sum is smooth and has positive evaluation near the image. Its flow $\mathrm{\Phi }$ is jointly $C^2$ on a uniform short time interval there, by [F9] and compactness.

The transversal boundary case (a). If $h{\mathrm{\mid }}_{\mathrm{\partial }{D}^2}$ is a closed transversal, then by definition $\beta (\tau )\mathrm{\ne }0$ at every boundary point; since $\beta$ is continuous, [F2] gives ${\min }_{\mathrm{\partial }{D}^2}\mathrm{\mid }\beta (\tau )\mathrm{\mid }=m>0$, and by continuity of $\beta$ and compactness of $\mathrm{\partial }{D}^2$ there is a collar $C$ of $\mathrm{\partial }{D}^2$ with $\beta \mathrm{\ne }0$ on $C$, which is (a).

Leafwise boundary: normal derivative. Set $a(\theta )=\omega (dh({\mathrm{\partial }}_r))$ and $b(\theta )=\omega (V)>0$ at $h(\theta ,0)$. Choose one constant $c>0$ with $a+cb>0$ on the compact boundary. With a smooth cutoff $\chi (r)$ equal to one near zero and supported in $[0,\epsilon )$, define $h_1(\theta ,r)={\mathrm{\Phi }}_{cr\chi (r)}(h(\theta ,r))$, and keep $h$ outside this collar. This is $C^2$, fixed on the boundary, and there its characteristic covector has zero tangential component and normal component $a+cb>0$. Continuity and compactness therefore give a zero-free collar. Scaling the flow time gives a homotopy fixed on the boundary and supported in the chosen collar. Taking $\epsilon$ small makes the value displacement arbitrarily small, while the normal derivative change $cV$ need not be small. This proves (b) with $C^0$ closeness.

Preparation for (c). Enlarge the fixed collar $C_{+}$ slightly to an open collar $C\supset \overline{C_{+}}$ with compact closure on which $\beta \mathrm{\ne }0$, and let $D_1:=D^2\setminus C$, a compact disk contained in the interior of $D^2$; all singularities lie in $D_1$. Choose finitely many open disks $V_1,\dots ,V_N\subseteq \mathrm{int}{D}^2$, each with closure disjoint from $C_{+}$ and a compact core $K_i\subset V_i$ and with $h(\overline{V_i})$ contained in a single foliation chart $Q_i$ of $F$ with positive margin from its boundary, such that the interiors of the $K_i$ cover $D_1$; this is possible because $D_1$ is compact and $h$ is continuous. Since the conditions $h(\overline{V_i})\subseteq Q_i$ are open in the $C^2$ topology, there is a $C^2$ neighbourhood $\mathcal{V}\subseteq \mathcal{U}$ of $h$ such that every map in $\mathcal{V}$ still sends each $\overline{V_i}$ into $Q_i$ and is still regular on $\bar{C}$.

The local perturbation of (c). Fix $i$ and write the current map on $V_i$ as $x\mapsto {\chi }_i^{-1}(Y_i(x),u_i(x))$, where $u_i$ is the local transverse function. Choose a bump ${\rho }_i$ equal to $1$ near $K_i$ and supported in $V_i$ [F3], and for a parameter $a\in {\mathbb{R}}^2$ define the modified map on $V_i$ by replacing $u_i(x)$ with $u_i(x)+{\rho }_i(x)(a\cdot x)$, leaving the foliation coordinates $Y_i$ and the map outside $V_i$ unchanged; since ${\rho }_i$ is compactly supported in the interior of $V_i$, the result is a $C^2$ map on $D^2$ agreeing with the previous map near $\mathrm{\partial }{V}_i$ with all derivatives. On the open set where ${\rho }_i=1$ the new transverse function is $u_i+a\cdot x$, whose critical points are the solutions of $\mathrm{\nabla }{u}_i(x)=-a$; for small $a$ the map stays in $\mathcal{V}$.

Sard makes the core nondegenerate. The gradient map $\mathrm{\nabla }{u}_i:V_i\to {\mathbb{R}}^2$ is a $C^1$ map, so by [F4] its set of critical values is null in ${\mathbb{R}}^2$. A null set has empty interior: if a null set contained a closed square $Q$ of side $L\le 1$, nullity would give a sequence of closed cubes covering $Q$ with total area at most $L^2\mathrm{/}4$; thickening the $n$-th cube by ${\delta }_n=\min (1,L^2\mathrm{/}(1024\cdot 2^n({\mathrm{\ell }}_n+1)))$ on each side makes a cover by open cubes whose total area exceeds $L^2\mathrm{/}4$ by at most ${\sum }_n(4{\mathrm{\ell }}_n{\delta }_n+4{\delta }_n^2)<L^2\mathrm{/}64$, hence has total area below $17L^2\mathrm{/}64$; by compactness of $Q$ [F5] finitely many of them cover $Q$ with total area $V_f<L^2\mathrm{/}2$, and [F5] turns this finite cover into a grid of $Q$ whose cells meeting $Q$ (that is, all cells) have total area below $V_f+L^2\mathrm{/}2<L^2$, contradicting that the cells of a grid of $Q$ have total area $L^2$. Hence the critical values of $\mathrm{\nabla }{u}_i$ have empty interior and arbitrarily small vectors $-a$ are regular values, so all solutions of $\mathrm{\nabla }{u}_i(x)=-a$ in $V_i$ have invertible Hessian $D^2{u}_i(x)$.

Preservation of the earlier cores and the fixed charts. At the moment core $i$ has been treated, its singularities are the finite set $\mathrm{\nabla }{u}_i^{-1}(-a)\cap K_i$ (finite because $D^2{u}_i$ is invertible at each solution, so the solutions are isolated, and $K_i$ is compact): they are nondegenerate, and on the compact complement of small isolating disks the gradient of the new transverse function is bounded away from zero. This property, "all singularities in $K_i$ are nondegenerate and isolated", is open in the $C^2$ topology: near each singularity the Hessian determinant stays nonzero, and on the compact remainder the gradient norm stays positive. Since there are only finitely many earlier cores and finitely many chart conditions, the regular value $-a$ in step 5.1 may be chosen arbitrarily small, and the perturbation in chart $i$ then preserves every earlier core and every fixed chart inclusion; moreover each earlier core property in turn is preserved by all later perturbations for the same reason.

Finiteness, interiority and the homotopy. After the finitely many steps, every point of $D_1$ lies in the interior of some core $K_i$; at the end all singularities in each $K_i$ remain nondegenerate (their positions may move), because all later perturbations preserve this property, and there are none in the collar $C$. Hence $\mathrm{Sing}(g^{\ast }\omega )$ is a finite set of interior nondegenerate points. Scaling the finitely many parameters $a_i$ linearly from $0$ to their chosen values and concatenating the resulting homotopies gives a homotopy from $h$ to $g$ that fixes an open neighbourhood of the original collar $C_{+}$ and keeps every intermediate map $C^2$.

A nondegenerate singularity is a center or a saddle. Let $p$ be a nondegenerate singularity with local transverse function $u$ and Hessian $H=D^{2}u(p)$, and translate so that $p=0$ and $u(0)=0$. If $H$ is definite, then by [F7] each ray $t\mapsto u(t\xi )$ is strictly monotone in $t$ near $0$; for a small positive level $c$ (or negative, according to the sign of $H$) every ray meets $\{u=c\}$ in exactly one point near $0$, by the intermediate value theorem, and the resulting radius is continuous in $\xi$; the levels are therefore small closed curves around $0$, so the singularity is a center. If $H$ is indefinite, diagonalize $H$ linearly to assume $u_{xx}(0)>0>u_{yy}(0)$; the map $(x,y)\mapsto u_x(x,y)$ has invertible $x$-derivative $u_{xx}(0)$ at the origin, so [F6] solves $u_x=0$ locally as a $C^1$ curve $x=\eta (y)$ with $\eta (0)={\eta }^{\mathrm{\prime }}(0)=0$. Put $b(y):=u(\eta (y),y)$; then $b$ is $C^2$ with $b^{\mathrm{\prime }}(y)=u_y(\eta (y),y)$ and $b^{\mathrm{\prime }\mathrm{\prime }}(0)=u_{yy}(0)<0$, and Taylor's theorem with [F8] applied twice in the $x$ variable gives $u(x,y)-b(y)=(x-\eta (y){)}^{2}A(x,y)$ and $b(0)-b(y)=y^{2}B(y)$ with $A,B$ continuous and $A(0,0)>0$, $B(0,0)>0$. The changes $X=\mathrm{sign}(x-\eta (y))\sqrt{u(x,y)-b(y)}$ and $Y=\mathrm{sign}(y)\sqrt{b(0)-b(y)}$ are continuous and strictly monotone in $x$ and $y$ respectively near the origin, hence define local coordinates there, and in them $u-u(0)=X^2-Y^2$; the level sets of $u$ therefore have the four-sector saddle picture.

By steps 2.1, 2.2, 7.1 and 8.1 assertions (a), (b) and (c) hold. The construction selects only finitely many objects at each stage (finitely many charts, finitely many bumps, finitely many arbitrarily small regular values), so the proof's own choices are finite and need no choice principle; the stated hypothesis ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited from the cooriented smooth-distribution interface used to speak of the foliation, its defining form and its flat charts, exactly as recorded in Transversely oriented codimension-one foliations. Nothing here separates distinct singularities into distinct leaves, since in a nonproper foliation two different transverse coordinates may lie in the same leaf; this is why no such separation is asserted.