Gap of codimension one foliations (Q2869282)

Let \(\mathcal F\) be a transversely oriented foliation on a closed manifold \(M\). Suppose that \(\mathcal F\) is of class \(C^{r,0_+}\) in the following sense: there is a foliated atlas so that the changes of coordinates are of class \(C^r\) in the direction of the leaves, their leafwise partial derivatives of order \(\leq r\) are continuous on each coordinated open set, and \(M\) has a differentiable structure so that \(T\mathcal F\) becomes a \(C^0\) vector subbundle of \(TM\). It is said that \(\mathcal F\) is taut if every leaf meets a closed transversal. The depth of the leaves is inductively defined as follows: the depth \(0\) leaves are the compact ones, and, for some integer \(k\geq1\), a leaf \(L\) is at depth \(k\) if \(\overline{L}\setminus L\) consists of leaves at depth \(<k\). Leaves that don't satisfy this condition are said to be at infinite depth. The depth of \(\mathcal F\), denoted by \(\text{depth}(\mathcal{F})\), is the top depth of its leaves.NEWLINENEWLINEFix a one-dimensional foliation \(\mathcal{F}^\perp\) transverse to \(\mathcal F\). For all leaf \(L\) of \(\mathcal F\), each component of its inverse image to the unit tangent bundle of \(\mathcal{F}^\perp\) is called a side of \(L\). Two leaves \(L_1\) and \(L_2\) of \(\mathcal F\) are called equivalent if \(L_1=L_2\), or there exists an embedding \(\phi:L_1\times[0,1]\to M\) such that \(\phi(L_1\times\{i\})=L_i\) (\(i=1,2\)) and \(\phi(\{x\}\times[0,1])\) (\(x\in L_1\)) is contained in a leaf of \(\mathcal{F}^\perp\). If moreover \(\phi_*(\partial/\partial t)|_{t=0}\) defines a side \(\widetilde{L}_1\) of \(L_1\), then it is said that \(L_1\) is equivalent to \(L_2\) through \(\widetilde{L}_1\).NEWLINENEWLINESuppose that \(\mathcal F\) is of finite depth. Then the author constructs a modification \(\widetilde{\mathcal F}\) of \(\mathcal F\) on \(M\), satisfying the same conditions as \(\mathcal F\), and moreover so that: (1) the number of equivalence classes of leaves of \(\widetilde{\mathcal F}\) is finite; (2) if a leaf \(L_1\) of \(\widetilde{\mathcal F}\) is equivalent to another leaf \(L_2\) via an embedding \(\phi:L_1\times[0,1]\to M\), then \(\widetilde{\mathcal F}|_{\phi(L_1\times[0,1])}\) is a product foliation; and (3) \(\text{depth}(\widetilde{\mathcal F})\leq\text{depth}(\mathcal{F})\).NEWLINENEWLINELet \(L_i^0\) be representatives of the equivalence classes of depth \(0\) leaves of \(\widetilde{\mathcal F}\). Let \(\widehat M\) be the union of the path-metric completions of the components of \(M\setminus\bigcup_iL^0_i\), and let \(\widehat{\mathcal F}\) be the foliation on \(\widehat M\) induced by \(\mathcal F\). \(\widehat{\mathcal F}\) has the same depth as \(\mathcal F\), and each \(L^0_i\) is covered by two leaves of \(\widehat{\mathcal F}\), denoted by \(L_i^{0\pm}\). Let \(\widehat{G}(\widehat{\mathcal F})\) be the directed graph whose vertices are the classes of leaves of \(\widehat{\mathcal F}\), and so that there is an edge from a vertex \(v_1\) to another vertex \(v_2\) if there are representatives \(L_1\) and \(L_2\) of \(v_1\) and \(v_2\) such that \(L_2\subset\overline{L_1}\setminus L_1\), and there does not exist any leaf \(L\) with \(L\subset\overline{L_1}\setminus L_1\) and \(L_2\subset\overline{L}\setminus L\). Now let \(G(\widetilde{F})\) be the graph obtained from \(\widehat{G}(\widehat{\mathcal F})\) by identifying the vertices \([L_i^{0+}]\) and \([L_i^{0-}]\) for each \(i\). The length of an edge of \(G(\widetilde{F})\) from a vertex \(v\) to a vertex \(v'\) is defined as the depth of the representatives of \(v'\) minus the depth of the representatives of \(v\). Then the gap of \(\widetilde{\mathcal F}\), denoted by \(\text{gap}(\widetilde{\mathcal F})\), is the maximum length of the edges of \(G(\widetilde{F})\), being \(0\) if there are no edges.NEWLINENEWLINEOn the other hand, the author introduces the concept of a \(0\)-twisted double \(K\) of a non-cable knot in \(S^3\). Given such a \(K\), let \(S^3(K,0)\) be the manifold obtained from \(S^3\) by performing \(0\)-surgery along \(K\), let \(\Sigma^{(n)}(K,0)\) be the \(n\)-fold cyclic covering space of \(S^3(K,0)\), and let \(\mathcal F\) be a foliation as above on \(\Sigma^{(n)}(K,0)\). The main theorem of the paper states the following. If \(\mathcal F\) has exactly one depth \(0\) leaf corresponding to a generator of \(H_2(S^3(K,0))\cong\mathbb{Z}\), and \(G(\widetilde{F})\) is a tree, then \(\text{depth}(\mathcal{F})\geq\frac{1}{2}(n+\text{gap}(\widetilde{\mathcal F}))\).