C. Camacho, B. Scardua
We study codimension one smooth foliations with Morse type singularities on closed ma-nifolds. We obtain a description of the manifold in case the number of centers in greater then the number of saddles. This result relies on and extends previous results of Reeb (for foliations having only centers) and Ells-Kuiper (for manifolds admitting Morse functions with three singularities).
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Morse foliations of codimension one on the sphere S^3 are studied and the existence of special components for these foliations is derived. As a corollary the instability of Morse foliations can be proven in almost all cases.
On a homology of foliations defined by non-singular Morse-Smale flows
本文提出由非奇异 Morse-Smale 流定义的一维叶状结构的同调定义，并展示了计算与 Seifert 纤维化相关的此类叶状结构同调的方法。
Let $M$ be a smooth manifold and let $\F$ be a codimension one, $C^\infty$ foliation on $M$, with isolated singularities of Morse type. The study and classification of pairs $(M,\F)$ is a challenging (and difficult) problem. In this setting, a classical result due to Reeb \cite{Reeb} states that a manifold admitting a foliation with exactly two center-type singularities is a sphere. In particular this is true if the foliation is given by a function. Along these lines a result due to Eells and Kuiper \cite{Ku-Ee} classify manifolds having a real-valued function admitting exactly three non-degen…
We show that a Bott-Morse foliation in dimension 3 admits a linear, singular, Poisson structure of rank 2 with Bott-Morse singularities. We provide the Poisson bivectors for each type of singular component, and compute the symplectic forms of the characteristic distribution.
The idea of Lichnerowicz or Morse-Novikov cohomology groups of a manifold has been utilized by many researchers to study important properties and invariants of a manifold. Morse-Novikov cohomology is defined using the differential $d_ω=d+ω\wedge$, where $ω$ is a closed $1$-form. We study Morse-Novikov cohomology relative to a foliation on a manifold and its homotopy invariance and then extend it to more general type of forms on a Riemannian foliation. We study the Laplacian and Hodge decompositions for the corresponding differential operators on reduced leafwise Morse-Novikov complexes. In the…
We give an upper bound for the Reidemeister-Singer distance between two Heegaard splittings in terms of the genera and the number of cusp points of the product map of Morse functions for the splittings. It suggests that a certain development in singularity theory may lead to the best possible bound for the Reidemeister-Singer distance.
We study codimension one foliations with singularities defined locally by Bott-Morse functions on closed oriented manifolds. We carry to this setting the classical concepts of holonomy of invariant sets and stability, and prove a stability theorem in the spirit of the local stability theorem of Reeb. This yields, among other things, a good topological understanding of the leaves one may have around a center-type component of the singular set, and also of the topology of its basin. The stability theorem further allows the description of the topology of the boundary of the basin and how the topo…
Vector bundle automorphisms preserving Morse-Bott foliations
本文研究了保持Morse-Bott叶状结构不变的微分同胚群，证明了其同伦类型的计算可转化为三个独立的子群，并给出了相关计算实例。
Topology of singular foliations of closed 1-forms on orbifolds
研究闭合1形式诱导的奇异叶状结构的拓扑性质，确定叶的紧致性条件及共存机制，扩展Calabi在orbifold上的经典结果。
Homotopy types of diffeomorphisms groups of simplest Morse-Bott foliations on lens spaces, 2
本文证明透镜空间上最简单 Morse-Bott 叶状结构的保叶微分同胚群到不交换奇异圆的叶状微分同胚群的包含映射是同伦等价。
Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary
本文研究实心 Klein 瓶及其扭曲丛上的 Morse–Bott Klein 瓶叶状结构，并确定相关叶状保持与逐叶保持微分同胚群的同伦型。
Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces, 1
本文研究实心环体及粘合所得透镜空间上的极 Morse-Bott 叶状结构，证明边界固定且保叶微分同胚群可缩，并计算透镜空间上相应保叶微分同胚群的同伦型。
Conic singularities and immediate transversality
本文证明了即时横截性对所有具有 $C^1$ 锥奇点的紧致子流形均成立，将该性质从 Morse 理论推广至一般情况，并借此探讨了锥奇点问题。
We define the Milnor number -- as the intersection number of two holomorphic sections -- of a one-dimensional holomorphic foliation $\mathscr{F}$ with respect to a compact connected component $C$ of its singular set. Under certain conditions, we prove that the Milnor number of $\mathscr{F}$ on a three-dimensional manifold with respect to $C$ is invariant by $C^1$ topological equivalences.
We study holomorphic foliations with an affine homogeneous transverse structure. We give a friendly characterization of the case of transversely affine foliations in terms of matrix valued pairs of differential forms. This leads naturally to the study of the case of foliations with singularities. A first extension theorem is then proved in the generic singularities framework.
Stratified Morse critical points and Brasselet number on non-degenerate locally tame singularities
本文证明在非退化局部驯服完备交奇点族上，Brasselet数之差与Milnor纤维正则部分上Morse临界点数目相关，连接了拓扑与几何性质，并导出基于Newton多面体的公式。
This paper presents two existence h-principles, the first for conformal symplectic structures on closed manifolds, and the second for leafwise conformal symplectic structures on foliated manifolds with non empty boundary. The latter h-principle allows to linearly deform certain codimension-$1$ foliations to contact structures. These results are essentially applications of the Borman-Eliashberg-Murphy h-principle for overtwisted contact structures and of the Eliashberg-Murphy symplectization of cobordisms, together with tools pertaining to foliated Morse theory, which are elaborated here.
The object of this survey is to give an overview on the topology of singularities of holomorphic foliation germs on $(\mathbb C^2,0)$.
By a gradient-like flow on a closed orientable surface $M$, we mean a closed 1-form $β$ defined on $M$ punctured at a finite set of points (sources and sinks of $β$) such that there exists a Morse function $f$ on $M$, called an energy function of $β$, whose critical points coincide with equilibria of $β$, and the pair $(f,β)$ has a canonical form near each critical point of $f$. Let $\mathcal{B}=\mathcal{B}(β_0)$ be the space of all gradient-like flows on $M$ having the same types of local singularities as a flow $β_0$, and $\mathcal{F}=\mathcal{F}(f_0)$ the space of all Morse functions on $M$…
We prove that for any convex globally hyperbolic maximal (GHM) anti-de Sitter (AdS) 3-dimensional space-time $N$ with particles (cone singularities of angles less than $π$ along time-like curves), the complement of the convex core in $N$ admits a unique foliation by constant Gauss curvature surfaces. This extends, and provides a new proof of, a result of \cite{BBZ2}. We also describe a parametrization of the space of convex GHM AdS metrics on a given manifold, with particles of given angles, by the product of two copies of the Teichmüller space of hyperbolic metrics with cone singularities of …
Lefschetz fibrations on the Milnor fibers of cusp and simple elliptic singularities
研究显示，通过构造S^1参数的genus-一Lefschetz纤维结构，揭示了尖点和简单椭圆奇点的Milnor纤维与Lawson叶状结构的关系，并证明其可粘合成K3表面。
We present existence results for certain singular 2-dimensional foliations on 4-manifolds. The singularities can be chosen to be simple, e.g. the same as those that appear in Lefschetz pencils. There seems to be a wealth of such creatures on most 4-manifolds. In certain cases, one can prescribe surfaces to be transverse or be leaves of these foliations. The purpose of this paper is to offer objects, hoping for a future theory to be developed on them. For example, foliations that are taut might offer genus bounds for embedded surfaces (Kronheimer's conjecture).
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