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We construct a discrete Morse function which induces both a specified gradient vector field and homological sequence on a given tree. After reviewing the basics of discrete Morse theory, we provide an algorithm to construct a discrete Morse function on a tree inducing a desired gradient vector field and homological sequence. We prove that our algorithm is correct, and conclude with an example to illustrate its use.

Let N be a closed oriented k-dimensional submanifold of the (k+2)-dimensional sphere; denote its complement by C(N). Denote by x the 1-dimensional cohomology class in C(N), dual to N. The Morse-Novikov number of C(N) is by definition the minimal possible number of critical points of a regular Morse map f from C(N) to a circle, such that f belongs to x. In the first part of this paper we study the case when N is the twist frame spun knot associated to an m-knot K. We obtain a formula which relates the Morse-Novikov numbers of N and K and generalizes the classical results of D. Roseman and E.C. Zeeman about fibrations of spun knots. In the second part we apply the obtained results to the computation of Morse-Novikov numbers of surface-links in 4-sphere., 13 pages

Let $M$ be a complex manifold of dimension $n$ with smooth connected boundary $X$. Assume that $\overline M$ admits a holomorphic $S^1$-action preserving the boundary $X$ and the $S^1$-action is transversal on $X$. We show that the $\overline\partial$-Neumann Laplacian on $M$ is transversally elliptic and as a consequence, the $m$-th Fourier component of the $q$-th Dolbeault cohomology group $H^q_m(\overline M)$ is finite dimensional, for every $m\in\mathbb Z$ and every $q=0,1,\ldots,n$. This enables us to define $\sum^{n}_{j=0}(-1)^j{\rm dim\,}H^q_m(\overline M)$ the $m$-th Fourier component of the Euler characteristic on $M$ and to study large $m$-behavior of $H^q_m(\overline M)$. In this paper, we establish an index formula for $\sum^{n}_{j=0}(-1)^j{\rm dim\,}H^q_m(\overline M)$ and Morse inequalities for $H^q_m(\overline M)$., Comment: 39 pages

Topological methods are important tools for data analysis, and recently receiving more and more attention in vector field visualization. In this paper, we give an introductory description to some important topological methods in vector field visualization. Besides traditional methods of vector field topology, space-time method and finite-time Lyapunov exponent, we also include in this survey Hodge decomposition, combinatorial vector field topology, Morse decomposition, and robustness, etc. In addition to familiar numerical techniques, more and more combinatorial tools emerge in vector field visualization. The numerical methods often rely on error-prone interpolations and interpolations, while combinatorial techniques produce robust but coarse features. In this survey, we clarify the relevant concepts and hope to guide future topological research in vector field visualization.

In this work, we give new existence and multiplicity results for the solutions of the prescription problem for the Webster scalar curvature on a 3-dimensional Pseudo Hermitian Manifold. The critical points of prescribed functions verify mixed conditions. We establish some Morse Inequalities at Infinity and a Poincare–Hopf type formula to give a lower bound on the number of solutions as well as an upper bound for the Morse index of such solutions.

A Morse–Smale function on a compact Riemannian manifold can be used to define an associated Morse complex. Its homology is isomorphic to the singular homology of the manifold, which coincides with the singular homology of the Conley index of the manifold. In this paper, we consider an analogous Morse complex for isolated invariant sets of certain semilinear parabolic equations. It is shown that its homology is isomorphic to the singular homology of the Conley index of the isolated invariant set.

In this paper, we first develop the concept of Lyapunov graph to weighted Lyapunov graph (abbreviated as WLG) for nonsingular Morse-Smale flows (abbreviated as NMS flows) on $S^3$. WLG is quite sensitive to NMS flows on $S^3$. For instance, WLG detect the indexed links of NMS flows. Then we use WLG and some other tools to describe nonsingular Morse-Smale flows without heteroclinic trajectories connecting saddle orbits (abbreviated as behavior $0$ NMS flows). It mainly contains the following several directions: &nbsp 1. we use WLG to list behavior $0$ NMS flows on $S^3$; &nbsp 2. with the help of WLG, comparing with Wada's algorithm, we provide a direct description about the (indexed) link of behavior $0$ NMS flows; &nbsp 3. to overcome the weakness that WLG can't decide topologically equivalent class, we give a simplified Umanskii Theorem to decide when two behavior $0$ NMS flows on $S^3$ are topological equivalence; &nbsp 4. under these theories, we classify (up to topological equivalence) all behavior 0 NMS flows on $S^3$ with periodic orbits number no more than 4.

Morse theory offers a natural and mathematically-sound tool for shape analysis and understanding. It allows studying the behavior of a scalar function defined on a manifold. Starting from a Morse function, we can decompose the domain of the function into meaningful regions associated with the critical points of the function. Such decompositions, called Morse complexes, provide a segmentation of a shape and are extensively used in terrain modeling and in scientific visualization. Discrete Morse theory, a combinatorial counterpart of smooth Morse theory defined over cell complexes, provides an excellent basis for computing Morse complexes in a robust and efficient way. Moreover, since a discrete Morse complex computed over a given complex has the same homology as the original one, but fewer cells, discrete Morse theory is a fundamental tool for efficiently detecting holes in shapes through homology and persistent homology. In this survey, we review, classify and analyze algorithms for computing and simplifying Morse complexes in the context of such applications with an emphasis on discrete Morse theory and on algorithms based on it.

This paper presents a fundamental lemma concerning the index of a zero of a vector field. Based on this lemma one obtains a procedure of constructing a higher dimensional smooth sphere map from a lower dimensional one and gives an explicit formula for smooth sphere map with a given topological degree. Also based on this lemma, a new proof of the generalized Morse index formula is presented for smooth vector fields on a manifold with boundary.

Orientadora: Profa. Dra. Mariana Rodrigues da Silveira Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , 2018. A Teoria de Morse é baseada na obtenção de informações topológicas de uma variedade M diferenciável por meio de uma função real f : M !R com apenas pontos críticos não degenerados. Nesta dissertação estudamos uma adaptação desta teoria para funções com imagem no círculo S1. Este estudo é realizado considerando o recobrimento cíclico infinito de M induzido pelo recobrimento universal R sobre S1. Mostramos que, assim como no caso Morse, informações topológicas de M podem ser recuperadas através de um complexo de cadeias construído a partir dos pontos críticos de f . Morse theory is based on recovering topological information about a smooth manifold M using a real valued function f : M ! R with a finite number of nondegenarate critical points. In this work we study an adaptation of this theory for circle valued maps. This study is done considering the infinite cyclic covering of M induced by the universal covering R of S1. We prove that, as in the Morse case, topological information of M can be recovered using a chain complex generated by the critical points of f .

The first author in recent work with D. Gay developed the notion of a Morse structure on an open book as a tool for studying closed contact 3-manifolds. We extend the notion of Morse structure to extendable partial open books in order to study contact 3-manifolds with convex boundary.

Homotopy connectedness theorems for complex submanifolds of homogeneous spaces (sometimes referred to as theorems of Barth-Lefshetz type) have been established by a number of authors. Morse Theory on the space of paths lead to an elegant proof of homotopy connectedness theorems for complex submanifolds of Hermitian symmetric spaces. In this work we extend this proof to a larger class of compact complex homogeneous spaces.

We lay the foundations of a Morse homology on the space of connections on a principal G-bundle over a compact manifold Y, based on a newly defined gauge-invariant functional on . While the critical points of correspond to Yang–Mills connections on P, its L2-gradient gives rise to a novel system of elliptic equations. This contrasts previous approaches to a study of the Yang–Mills functional via a parabolic gradient flow. We carry out the analytical details of our programme in the case of a compact two-dimensional base manifold Y. We furthermore discuss its relation to the well-developed parabolic Morse homology over closed surfaces. Finally, an application of our elliptic theory is given to three-dimensional product manifolds .

In [J. Topol. Anal. 6 (2014), 305–338], we have developed a homology theory (Morse–Conley–Floer homology) for isolated invariant sets of arbitrary flows on finite-dimensional manifolds. In this paper, we investigate functoriality and duality of this homology theory. As a preliminary, we investigate functoriality in Morse homology. Functoriality for Morse homology of closed manifolds is known, but the proofs use isomorphisms to other homology theories. We give direct proofs by analyzing appropriate moduli spaces. The notions of isolated map and flow map allow the results to generalize to local Morse homology and Morse–Conley–Floer homology. We prove Poincaré-type duality statements for local Morse homology and Morse–Conley–Floer homology.

We generalize Cohen & Jones & Segal's flow category whose objects are the critical points of a Morse function and whose morphisms are the Morse moduli spaces between the critical points to an n-category. The n-category construction involves repeatedly doing Morse theory on Morse moduli spaces for which we have to construct a class of suitable Morse functions. It turns out to be an `almost strict' n-category, i.e. it is a strict n-category `up to canonical isomorphisms'., 31 pages, 3 figures, 14 diagrams; In accordance with the journal's copyright, I am making the preprint version of the published paper available on the arXiv

In this paper, we give the explicit bounds for the data of objects involved in some basic theorems of singularity theory: the inverse, implicit and rank theorems for Lipschitz mappings, the splitting lemma and the Morse lemma, the density and openness of Morse functions. We expect that the results will make singularities more applicable and will be useful for numerical analysis and some fields of computing.

The main aim of this paper is to show some specific connections between linear dynamic and graphs. Precisely, the Morse decomposition of a linear flow on the Grassmannians induces a directed graph. We apply the results appearing in Ayala et al. (2006, 2005) [2] , [3] and Colonius et al. (2002) [4] and compute the associated graphs for linear flows in dimensions two and three.

In this paper, the existence of nontrivial solutions is obtained for a class of Kirchhoff type problems with Dirichlet boundary conditions by computing the critical groups and Morse theory.

Morse theory inspired several robust and well-grounded tools in discrete function analysis, geometric modeling and visualization. Such techniques need to adapt the original differential concepts of Morse theory in a discrete setting, generally using either piecewise-linear (PL) approximations or Formanʼs combinatorial formulation. The former carries the intuition behind Morse critical sets, while the latter avoids numerical integrations. Formanʼs gradients can be constructed from a scalar function using greedy strategies, although the relation with its PL gradient is not straightforward. This work relates the critical sets of both approaches. It proves that the greedy construction on two-dimensional meshes actually builds an adjacent critical cell for each PL critical vertex. Moreover, the constructed gradient is globally aligned with the PL gradient. Those results allow adapting the many works in PL Morse theory for triangulated surfaces to Formanʼs combinatorial setting with low algorithmic complexity.

The main theorem of discrete Morse theory states that a finite, regular CW complex X equipped with a discrete Morse function is homotopy equivalent to a CW complex that has one d -cell for each critical cell in X of index d . We prove, using the terminology of discrete Morse matchings, a version of this theorem that works for infinite complexes, provided the Morse matching induces finitely many equivalence classes of rays in the Hasse diagram. We work in the class of h-regular posets, introduced by Minian, which is strictly larger than the class of face posets of regular CW complexes. A homological version of the theorem for cellular posets is also given.

It is proved that every discrete Morse function in the sense of Forman on a finite regular CW complex can be represented by a polyhedral Morse function in the sense of Banchoff on an appropriate embedding in Euclidean space of the barycentric subdivision of the CW complex; such a representation preserves critical points. The proof is stated in terms of discrete Morse functions on posets.

We introduce an algorithm for construction of the Morse hierarchy, i.e., a hierarchy of Morse decompositions of a piecewise constant vector field on a surface driven by stability of the Morse sets with respect to perturbation of the vector field. Our approach builds upon earlier work on stable Morse decompositions, which can be used to obtain Morse sets of user-prescribed stability. More stable Morse decompositions are coarser, i.e., they consist of larger Morse sets. In this work, we develop an algorithm for tracking the growth of Morse sets and topological events (mergers) that they undergo as their stability is gradually increased. The resulting Morse hierarchy can be explored interactively. We provide examples demonstrating that it can provide a useful coarse overview of the vector field topology.

Using variational methods and Morse theory, we obtain some existence results of multiple solutions for certain semilinear problems associated with general Dirichlet forms.

Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse functions on $M$ having fixed number of critical points of each index, moreover at least $\chi(M)+1$ critical points are labeled by different labels (enumerated). A notion of a skew cylindric-polyhedral complex, which generalizes the notion of a polyhedral complex, is introduced. The skew cylindric-polyhedral complex $\mathbb{\widetilde K}$ (the "complex of framed Morse functions"), associated with the space $F$, is defined. In the case when $M=S^2$, the polyhedron $\mathbb{\widetilde K}$ is finite; its Euler characteristic is evaluated and the Morse inequalities for its Betti numbers are obtained. A relation between the homotopy types of the polyhedron $\mathbb{\widetilde K}$ and the space $F$ of Morse functions, endowed with the $C^\infty$-topology, is indicated., Comment: 32 pages, in Russian

We develop a relative variant of the Morse theory for Lipschitz functionals defined on closed subsets of a Banach manifold. We prove the invariance of topological characteristics of functionals under uniform deformations.

In this work we use Morse theory to study the generating frames of transnormal embeddings of S m in R n . In general we start with M as a smooth compact connected m-manifold without boundary and f : M −→ R as a function whose critical points are non-degenerate. Using Morse Inequalities, we show that the generating frame of a transnormal embedding of S 1 splits into two transitive sets. Also we give a detailed proof for Robertson’s conclusion [13] regarding r-transnormal embeddings of S m , m> 1.

The paper is concerned with the Morse equation for flows in a representation of a compact Lie group. As a consequence of this equation we give a relationship between the equivariant Conley index of an isolated invariant set of the flow given by .x = −∇f(x) and the gradient equivariant degree of ∇f. Some multiplicity results are also presented.

This paper is concerned with a Morse theory of attractors for finite-dimensional nonsmooth dynamical systems described by differential inclusions with upper semi-continuous righthand sides. We first show that all open attractor neighborhoods of an attractor share the same homotopy type. Then based on this basic fact we introduce the concept of homology index for Morse sets and establish Morse inequalities and Morse equation by using smooth Morse–Lyapunov functions.

Conditional extremal curves in a complete Riemannian manifold M are defined as the critical points of the squared L 2 distance between the tangent vector field of a curve and a so-called prior vector field. We prove that this L 2 distance satisfies the Palais–Smale condition on the space of absolutely continuous curves joining two submanifolds of M , and thus establish the existence of critical points. We also prove a Morse index theorem in the case where the two submanifolds are single points, and use the Morse inequalities to place lower bounds on the number of critical points of each index.

In this mostly expository paper we describe applications of Morse theory to moduli spaces of Higgs bundles. The moduli spaces are finite-dimensional analytic varieties but they arise as quotients of infinite-dimensional spaces. There are natural functions for Morse theory on both the infinite-dimensional spaces and the finite-dimensional quotients. The first comes from the Yang–Mills–Higgs energy, while the second is provided by the Hitchin function. After describing what Higgs bundles are, we explore these functions and how they may be used to extract topological information about the moduli spaces.

In this paper, we introduce the concept of dynamic Morse decomposition for an action of a semigroup of homeomorphisms. Conley has shown in [5, Sec. 7] that the concepts of Morse decomposition and dynamic Morse decompositions are equivalent for flows in metric spaces. Here, we show that a Morse decomposition for an action of a semigroup of homeomorphisms of a compact topological space is a dynamic Morse decomposition. We also define Morse decompositions and dynamic Morse decompositions for control systems on manifolds. Under certain condition, we show that the concept of dynamic Morse decomposition for control system is equivalent to the concept of Morse decomposition.

Using the three critical points theorem, Clark's theorem and the Morse theory, multiple periodic solutions for non-linear difference systems involving the p-Laplacian are obtained by variational methods.

This work is focused on the links between Formanʼs discrete Morse theory and graph theory. More precisely, we are interested on putting the optimization of a discrete Morse function in terms of matching theory. It can be done by describing the process of cancellation of pairs of critical simplices by means of obtaining Morse matchings on the corresponding Hasse diagram with a greater number of edges using the combinatorial notion of transference.

In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions: Δ2u = K(x)up, u > 0 in Ω, Δu = u = 0 on ∂Ω, where Ω is a smooth domain in ℝn,n≥5, and p+1=2nn−4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers lower bounds for the number of solutions. As further applications of this Morse theoretical approach, we prove more existence results.

Вивчаються гомотопiчнi iнварiанти схрещених i гiльбертових комплексiв. Цi iнварiанти використовуютьcя для пiдрахунку точних значень чисел Морса гладких кобордизмiв. We study the homotopy invariants of crossed and Hilbert complexes. These invariants are applied to the calculation of exact values of Morse numbers of smooth cobordisms.

The Witten deformation is a method proposed by Witten which, given a function on a smooth compact manifold M, allows to prove the Morse inequalities. Witten’s proof of the Morse inequalities is analytical and can thus be applied to situations where the Thom-Smale method is not accessible. In these notes we generalise the Witten deformation to certain singular Riemannian manifolds X which are metric models for singular algebraic curves, and functions on X which we call admissible Morse functions. They are particular examples of stratified Morse functions in the sense of the theory developed by Goresky and MacPherson. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

On a smooth closed n -manifold, we consider Morse forms with wedge-product zero; we call such forms collinear. This is an equivalence relation. Collinearity classes are classified by the underlying foliation; so, in other words, we study the set of Morse forms that define the same foliation. We describe the set of the ranks of such forms and show how it is related to the structure of the foliation and the manifold.

In this Note we generalise the Witten deformation to even dimensional Riemannian manifolds with cone-like singularities X and certain functions f , which we call admissible Morse functions. As a corollary we get Morse inequalities for the L 2 -Betti numbers of X . The contribution of a singular point p of X to the Morse inequalities can be expressed in terms of the intersection cohomology of the local Morse datum of f at p . The definition of the class of functions which we study here is inspired by stratified Morse theory as developed by Goresky and MacPherson. However the setting here is different since the spaces considered here are manifolds with cone-like singularities instead of Whitney stratified spaces.

In 1963–64, Palais and Smale have introduced a compactness condition, namely condition (C), on real functions of class C 1 defined on a Riemannian manifold modeled upon a Hilbert space, in order to extend Morse theory to this frame and study nonlinear partial differential equations. This condition and some of its variants have been essential in the development of critical point theory on Banach spaces or Banach manifolds, and are referred as Palais–Smale-type conditions. The paper describes their evolution.

Let be a smooth, compact, not necessarily orientable surface with (maybe empty) boundary, and let be the space of Morse functions on that are constant on each component of the boundary and have no critical points at the boundary. The notion of framing is defined for a Morse function . In the case of an orientable surface this is a closed 1-form on with punctures at the critical points of local minimum and maximum of such that in a neighbourhood of each critical point the pair has a canonical form in a suitable local coordinate chart and the 2-form does not vanish on punctured at the critical points and defines there a positive orientation. Each Morse function on is shown to have a framing, and the space endowed with the -topology is homotopy equivalent to the space of framed Morse functions. The results obtained make it possible to reduce the problem of describing the homotopy type of to the simpler problem of finding the homotopy type of . As a solution of the latter, an analogue of the parametric -principle is stated for the space . Bibliography: 41 titles.

We consider nonlinear elliptic Neumann problems driven by the p-Laplacian. Using variational techniques together with Morse theory (in particular, critical groups and the Poincaré-Hopf formula), we prove some multiplicity results: either three or four distinct nontrivial solutions are guaranteed, depending on the geometry and smoothness of the nonlinear term.

In a previous Note the author gave a generalisation of Witten's proof of the Morse inequalities to the model of a singular complex algebraic curve X and a stratified Morse function f . In this Note a geometric interpretation of the complex of eigenforms of the Witten Laplacian corresponding to small eigenvalues is provided in terms of an appropriate subcomplex of the complex of unstable cells of critical points of f . To cite this article: U. Ludwig, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Given a symmetric tensor on a real vector bundle of dimension two, we construct a space where this tensor corresponds to a scalar function. We prove that under certain regularity conditions such a space and the corresponding scalar function are smooth. We study the topology of this space for the case of surfaces and produce a version of Morse inequalities for symmetric tensors. We apply our results to the geometry of surfaces.