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Rachel Roberts - One of the best experts on this subject based on the ideXlab platform.
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c 1 0 Foliation theory
Algebraic & Geometric Topology, 2019Co-Authors: William H. Kazez, Rachel RobertsAbstract:Transverse one dimensional Foliations play an important role in the study of codimension one Foliations. In \cite{KR2}, the authors introduced the notion of flow box decomposition of a 3-manifold $M$. This is a decomposition of $M$ that reflects both the structure of a given codimension one Foliation and that of a given transverse flow. In this paper, flow box decompositions are used to extend some classical Foliation results to Foliations that are not $C^2$. Enhancements of well-known results of Calegari on smoothing leaves, Dippolito on Denjoy blowup of leaves, and Tischler on approximations by fibrations are obtained. The methods developed are not intrinsically 3-dimensional techniques, and should generalize to prove corresponding results for codimension one Foliations in $n$-dimensional manifolds.
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$C^{1,0}$ Foliation theory
Algebraic & Geometric Topology, 2019Co-Authors: William H. Kazez, Rachel RobertsAbstract:Transverse one dimensional Foliations play an important role in the study of codimension one Foliations. In \cite{KR2}, the authors introduced the notion of flow box decomposition of a 3-manifold $M$. This is a decomposition of $M$ that reflects both the structure of a given codimension one Foliation and that of a given transverse flow. In this paper, flow box decompositions are used to extend some classical Foliation results to Foliations that are not $C^2$. Enhancements of well-known results of Calegari on smoothing leaves, Dippolito on Denjoy blowup of leaves, and Tischler on approximations by fibrations are obtained. The methods developed are not intrinsically 3-dimensional techniques, and should generalize to prove corresponding results for codimension one Foliations in $n$-dimensional manifolds.
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Approximating $C^{1,0}$-Foliations
arXiv: Geometric Topology, 2014Co-Authors: William H. Kazez, Rachel RobertsAbstract:We extend the Eliashberg-Thurston theorem on approximations of taut oriented $C^2$-Foliations of 3-manifolds by both positive and negative contact structures to a large class of taut oriented $C^{1,0}$-Foliations, where by $C^{1,0}$ Foliation, we mean a Foliation with continuous tangent plane field. These $C^{1,0}$-Foliations can therefore be approximated by weakly symplectically fillable, universally tight, contact structures. This allows applications of $C^2$-Foliation theory to contact topology and Floer theory to be generalized and extended to constructions of $C^{1,0}$-Foliations.
Stephane Druel - One of the best experts on this subject based on the ideXlab platform.
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on codimension 1 del pezzo Foliations on varieties with mild singularities
Mathematische Annalen, 2014Co-Authors: Carolina Araujo, Stephane DruelAbstract:In this paper we extend to the singular setting the theory of Fano Foliations developed in our previous paper (Araujo and Druel, Adv Math 238:70–118, 2013). A \(\mathbb {Q}\)-Fano Foliation on a complex projective variety \(X\) is a Foliation \(\fancyscript{F}\subsetneq T_X\) whose anti-canonical class \(-K_{\fancyscript{F}}\) is an ample \(\mathbb {Q}\)-Cartier divisor. In the spirit of Kobayashi–Ochiai Theorem, we prove that under some conditions the index \(i_{\fancyscript{F}}\) of a \(\mathbb {Q}\)-Fano Foliation is bounded by the rank \(r\) of \(\fancyscript{F}\), and classify the cases in which \(i_{\fancyscript{F}}=r\). Next we consider \(\mathbb {Q}\)-Fano Foliations \(\fancyscript{F}\) for which \(i_{\fancyscript{F}}=r-1\). These are called del Pezzo Foliations. We classify codimension 1 del Pezzo Foliations on mildly singular varieties.
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on codimension 1 del pezzo Foliations on varieties with mild singularities
arXiv: Algebraic Geometry, 2012Co-Authors: Carolina Araujo, Stephane DruelAbstract:In this paper we extend to the singular setting the theory of Fano Foliations developed in our previous paper. A Q-Fano Foliation on a complex projective variety X is a Foliation F whose anti-canonical class is an ample Q-Cartier divisor. In the spirit of Kobayashi-Ochiai Theorem, we prove that under some conditions the index i of a Q-Fano Foliation is bounded by the rank r of F, and classify the cases in which i=r. Next we consider Q-Fano Foliations F for which i=r-1. These are called del Pezzo Foliations. We classify codimension 1 del Pezzo Foliations on mildly singular varieties.
William H. Kazez - One of the best experts on this subject based on the ideXlab platform.
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c 1 0 Foliation theory
Algebraic & Geometric Topology, 2019Co-Authors: William H. Kazez, Rachel RobertsAbstract:Transverse one dimensional Foliations play an important role in the study of codimension one Foliations. In \cite{KR2}, the authors introduced the notion of flow box decomposition of a 3-manifold $M$. This is a decomposition of $M$ that reflects both the structure of a given codimension one Foliation and that of a given transverse flow. In this paper, flow box decompositions are used to extend some classical Foliation results to Foliations that are not $C^2$. Enhancements of well-known results of Calegari on smoothing leaves, Dippolito on Denjoy blowup of leaves, and Tischler on approximations by fibrations are obtained. The methods developed are not intrinsically 3-dimensional techniques, and should generalize to prove corresponding results for codimension one Foliations in $n$-dimensional manifolds.
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$C^{1,0}$ Foliation theory
Algebraic & Geometric Topology, 2019Co-Authors: William H. Kazez, Rachel RobertsAbstract:Transverse one dimensional Foliations play an important role in the study of codimension one Foliations. In \cite{KR2}, the authors introduced the notion of flow box decomposition of a 3-manifold $M$. This is a decomposition of $M$ that reflects both the structure of a given codimension one Foliation and that of a given transverse flow. In this paper, flow box decompositions are used to extend some classical Foliation results to Foliations that are not $C^2$. Enhancements of well-known results of Calegari on smoothing leaves, Dippolito on Denjoy blowup of leaves, and Tischler on approximations by fibrations are obtained. The methods developed are not intrinsically 3-dimensional techniques, and should generalize to prove corresponding results for codimension one Foliations in $n$-dimensional manifolds.
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Approximating $C^{1,0}$-Foliations
arXiv: Geometric Topology, 2014Co-Authors: William H. Kazez, Rachel RobertsAbstract:We extend the Eliashberg-Thurston theorem on approximations of taut oriented $C^2$-Foliations of 3-manifolds by both positive and negative contact structures to a large class of taut oriented $C^{1,0}$-Foliations, where by $C^{1,0}$ Foliation, we mean a Foliation with continuous tangent plane field. These $C^{1,0}$-Foliations can therefore be approximated by weakly symplectically fillable, universally tight, contact structures. This allows applications of $C^2$-Foliation theory to contact topology and Floer theory to be generalized and extended to constructions of $C^{1,0}$-Foliations.
Danny Calegari - One of the best experts on this subject based on the ideXlab platform.
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The Gromov norm and Foliations
arXiv: Geometric Topology, 2000Co-Authors: Danny CalegariAbstract:We define a norm on the homology of a foliated manifold, which refines and majorizes the usual Gromov norm on homology. This norm depends in an upper semi-continuous way on the underlying Foliation, in the geometric topology, and can therefore be used to study the question of which Foliations arise as geometric limits of isotopy classes of other Foliations. We show that this norm is non-trivial -- i.e. it distinguishes certain taut Foliations of a given hyperbolic 3-manifold. In fact, modulo some standard conjectures, this norm should precisely detect amongst taut Foliations of closed atoroidal 3-manifolds, those whose universal covers branch in both directions. A homotopy-theoretic refinement gives even more information in certain cases; in particular, we show that a taut Foliation whose leaf space in the universal cover branches in at most one direction cannot be the geometric limit of a sequence of isotopies of a fixed taut Foliation whose leaf space in the universal cover branches in both directions. Our technology also lets us produce examples of taut Foliations which cannot be made transverse to certain geodesic triangulations of hyperbolic 3-manifolds, even after passing to a finite cover. Finally, our norm can be extended to actions of fundamental groups of manifolds on order trees, where it has similar upper semi-continuity properties.
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The Gromov norm and Foliations
Geometric and Functional Analysis, 2000Co-Authors: Danny CalegariAbstract:We define a norm on the homology of a foliated manifold, which refines and majorizes the usual Gromov norm on homology. This norm depends in an upper semi-continuous way on the underlying Foliation, in the geometric topology. We show that this norm is non-trivial — i.e. it distinguishes certain taut Foliations of a given hyperbolic 3-manifold.¶Using a homotopy-theoretic refinement, we show that a taut Foliation whose leaf space branches in at most one direction cannot be the geometric limit of a sequence of isotopies of a fixed taut Foliation whose leaf space branches in both directions. Our technology also lets us produce examples of taut Foliations which cannot be made transverse to certain geodesic triangulations of hyperbolic 3-manifolds, even after passing to a finite cover.¶Finally, our norm can be extended to actions of fundamental groups of manifolds on order trees, where it has similar upper semi-continuity properties.
Marco Zambon - One of the best experts on this subject based on the ideXlab platform.
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hausdorff morita equivalence of singular Foliations
Annals of Global Analysis and Geometry, 2019Co-Authors: Alfonso Garmendia, Marco ZambonAbstract:We introduce a notion of equivalence for singular Foliations—understood as suitable families of vector fields—that preserves their transverse geometry. Associated with every singular Foliation, there is a holonomy groupoid, by the work of Androulidakis–Skandalis. We show that our notion of equivalence is compatible with this assignment, and as a consequence, we obtain several invariants. Further, we show that it unifies some of the notions of transverse equivalence for regular Foliations that appeared in the 1980s.
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Holonomy transformations for singular Foliations
Advances in Mathematics, 2014Co-Authors: Iakovos Androulidakis, Marco ZambonAbstract:Abstract In order to understand the linearization problem around a leaf of a singular Foliation, we extend the familiar holonomy map from the case of regular Foliations to the case of singular Foliations. To this aim we introduce the notion of holonomy transformation. Unlike the regular case, holonomy transformations cannot be attached to classes of paths in the Foliation, but rather to elements of the holonomy groupoid of the singular Foliation.
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Holonomy transformations for singular Foliations
arXiv: Differential Geometry, 2012Co-Authors: Iakovos Androulidakis, Marco ZambonAbstract:In order to understand the linearization problem around a leaf of a singular Foliation, we extend the familiar holonomy map from the case of regular Foliations to the case of singular Foliations. To this aim we introduce the notion of holonomy transformation. Unlike the regular case, holonomy transformations can not be attached to classes of paths in the Foliation, but rather to elements of the holonomy groupoid of the singular Foliation. This assignment is injective. Holonomy transformations allow us to link the linearization problem with the compactness of the isotropy group of the holonomy groupoid, as well as with the linearization problem for proper Lie groupoids.
