Gap of codimension one foliations
type: Item
instance of: scholarly article
zbMATH Open document ID: 1280.57023
author: Hiroko Murai
publication date: 2014-01-03T00:00:00Z
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 6242566
zbMATH Keywords: codimension one foliation
zbMATH Keywords: depth of foliations
zbMATH Keywords: gap of foliations
zbMATH Keywords: \(0\)-twisted cover
zbMATH Keywords: cable knot
MaRDI profile type: Publication
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title: Gap of codimension one foliations
published in: Kobe Journal of Mathematics
review text: 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{\m…
reviewed by: Jesús A. Álvarez López
description: scientific article; zbMATH DE number 6242566