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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

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 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.

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

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.

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.

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

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.

International audience; A section in the 2-jet space of Morse functions is not always homotopic to a holonomic section. We give a necessary condition for being the case and we discuss the sufficiency.

Let M be a smooth compact (orientable or not) surface with or without a boundary. Let \( \mathcal{D}_0 \) ⊂ Diff(M) be the group of diffeomorphisms homotopic to idM. Two smooth functions f, g: M → ℝ are called isotopic if f = h2 ℴ g ℴ h1 for some diffeomorphisms h1 ∈ \( \mathcal{D}_0 \) and h2 ∈ Diff+(ℝ). Let F be the space of Morse functions on M which are constant on each boundary component and have no critical points on the boundary. A criterion for two Morse functions from F to be isotopic is proved. For each Morse function f ∈ F, a collection of Morse local coordinates in disjoint circular neighborhoods of its critical points is constructed, which continuously and Diff(M)-equivariantly depends on f in C∞-topology on F (“uniform Morse lemma”). Applications of these results to the problem of describing the homotopy type of the space F are formulated.

Letf:M→S1be a Morse map of a closed manifoldMinto the circle, where a Morse map is a smooth map with only nondegenerate critical points. In this paper, we classify such maps up to fold cobordism. In the course of the classification, we get several fold cobordism invariants for such Morse maps. We also consider a slightly general situation where the source manifoldMhas boundary and the mapfrestricted to the boundary has no critical points. Letg: (Rm, 0) → (R2, 0),m≥ 2, be a generic smooth map germ, where the targetR2is oriented. Using the above-mentioned fold cobordism invariants, we show that the number of cusps with a prescribed index appearing in aC∞stable perturbation ofg, counted with signs, gives a topological invariant ofg.

We get a characterization theorem for equivalent discrete Morse functions defined on simplicial complexes in terms of their gradient vector field. As a consequence, we also characterize them in the 1-dimensional case by using critical elements.

In this paper, the concavity of closed geodesics proposed by M. Morse in 1930s is studied.

A relation for the algebraic number of Morse points for deformations of real hyperplane singularities is obtained.

We study codimension one smooth foliations with Morse type singularities on closed manifolds. We obtain a description of the manifold if there are more centers than saddles. This result relies on and extends previous results of Reeb for foliations having only centers, results of Wagneur for foliations with Morse singularities and results of Eells and Kuiper for manifolds admitting Morse functions with three singularities.

Morse decompositions provide inside information about the global asymptotic behavior of dynamical systems on compact metric spaces. Recently, the existence of Morse decompositions for nonautonomous dynamical systems was proved by restricting attention to the past or the future of the system, but in general, such a construction is not realizable for the entire time. In this article, it is shown that all-time Morse decompositions can be defined for linear systems on the projective space. Moreover, the dynamical properties are discussed and an analogue to the Theorem of Selgrade is proved.

The aim of this paper is to extend the Morse theory for geodesics to the conical manifolds. In a previous paper [15] we defined these manifolds as submanifolds of \({\mathbb{R}}^n\) with a finite number of conical singularities. To formulate a good Morse theory we use an appropriate definition of geodesic, introduced in the cited work. The main theorem of this paper (see Theorem 3.6, section 3) proofs that, although the energy is nonsmooth, we can find a continuous retraction of its sublevels in absence of critical points. So, we can give a good definition of index for isolated critical values and for isolated critical points. We prove that Morse relations hold and, at last, we give a definition of multiplicity of geodesics which is geometrical meaningful. In section 5 we compare our theory with the weak slope approach existing in literature. Some examples are also provided.

We introduce Morse homology for manifolds with boundary, and construct relative Morse homology sequences.

Using the notion of discrete Morse function introduced by R. Forman for finite cw-complexes, we generalize it to the infinite 2-dimensional case in order to get the corresponding version of the well-known discrete Morse inequalities on a non-compact triangulated 2-manifold without boundary and with finite homology. We also extend them for the more general case of a non-compact triangulated 2-pseudo-manifold with a finite number of critical simplices and finite homology.

The global asymptotic behavior of dynamical systems on compact metric spaces can be described via Morse decompositions. Their components, the so-called Morse sets, are obtained as intersections of attractors and repellers of the system. In this paper, new notions of attractor and repeller for nonautonomous dynamical systems are introduced which are designed to establish nonautonomous generalizations of the Morse decomposition. The dynamical properties of these decompositions are discussed, and nonautonomous Lyapunov functions which are constant on the Morse sets are constructed explicitly. Moreover, Morse decompositions of one-dimensional and linear systems are studied.

We present a new approach to Morse and Novikov theories, based on the deRham Federer theory of currents, using the finite volume flow technique of Harvey and Lawson [HL]. In the Morse case, we construct a noncompact analogue of the Morse complex, relating a Morse function to the cohomology with compact forward supports of the manifold. This complex is then used in Novikov theory, to obtain a geometric realization of the Novikov Complex as a complex of currents and a new characterization of Novikov Homology as cohomology with compact forward supports. Two natural ``backward-forward'' dualities are also established: a Lambda duality over the Novikov Ring and a Topological Vector Space duality over the reals.

LetM2 be a closed orientable surface with curvatureK and γ ⊂ M a closed geodesic. The Morse index of γ is the index of the critical point γ for the length functional on the space of closed curves, i.e., the number of negative eigenvalues (counted with multiplicity) of the second derivative of length. Since the second derivative of length at γ in the direction of a normal variation un is − ∫ γ uLγ u where Lγ u = u′′ +Ku, the Morse index is the number of negative eigenvalues of Lγ . (By convention, an eigenfunction φ with eigenvalue λ of Lγ is a solution of Lγ φ + λφ = 0.) Note that if λ = 0, then φ (or φn) is a (normal) Jacobi field. γ is stable if the index is zero. The index of a noncompact geodesic is the dimension of a maximal vector space of compactly supported variations for which the second derivative of length is negative definite. We also say that such a geodesic is stable if the index is 0. We give in this paper bounds for the Morse indices of a large class of simple geodesics on a surface with a generic metric. To our knowledge these bounds are the first that use only the generic hypothesis on the metric.

In 1974, Mark Goresky and Robert MacPherson began their development of intersection homology theory (see [24] in these volumes), and their first paper on this topic appeared in 1980; see [12]. At that time, they were missing a fundamental tool which was available for the study of smooth manifolds; they had no Morse Theory for stratified spaces. Goresky and MacPherson wished to have a Stratified Morse Theory to allow them to prove a Lefschetz hyperplane theorem for the intersection homology of complex singular spaces, just as ordinary Morse Theory yields the Lefschetz Hyperplane Theorem for ordinary homology of complex manifolds ([34], §7). The time was ripe for a stratified version of Morse Theory. In 1970, Mather had given a rigorous proof of Thom’s first isotopy lemma [33]; this result says that proper, stratified, submersions are locally-trivial fibrations. In 1973, Morse functions on singular spaces had been defined by Lazzeri in [25], and the density and stability of Morse functions under perturbations had been proved in [37]. We shall recall these definitions and results in Section 2. What was missing was the analog of the fundamental result of Morse Theory, a theorem describing how the topology of a space is related to the critical points of a proper Morse function. In [16], Goresky and MacPherson proved such a theorem for stratified spaces. Suppose that M is a smooth manifold, that X is a Whitney stratified subset ofM , and that f : X → R is a proper function which is the restriction of a smooth function on M . For all v ∈ R, let X≤v := f−1((−∞, v]). Suppose that a, b ∈ R, a < b, and f−1([a, b]) contains a single (stratified) critical point, p, which is non-degenerate (see Definition 2.3) and contained in the open set f−1((a, b)). Let S be the stratum containing p. Then, the Main Theorem of Stratified Morse Theory (see Theorem 2.16) says that the topological space X≤b is obtained from the space X≤a by attaching a space A to X≤a along a subspace B ⊆ A, where the pair (A,B), the Morse data, is the product of the tangential Morse data of f at p and the normal Morse data of f at p. This result is especially powerful in the complex analytic case, where the normal Morse data depends on the stratum S, but not on the point p or on the particular Morse function f . Detailed proofs of these results appeared in the 1988 book Stratified Morse Theory [16]; we present a summary of a number of these results in Section 2. Even before the appearance of [16], Goresky and MacPherson published two papers, Stratified Morse Theory [15] and Morse Theory and Intersection Homology Theory [14], which contained announcements of many of the fundamental definitions and results of Stratified Morse Theory. In addition, these two papers showed that Stratified Morse Theory has a number of important applications to complex analytic spaces, including homotopy results, the desired Lefschetz Hyperplane Theorem for intersection homology, and the first proof that the (shifted) nearby cycles of a perverse sheaf are again perverse. We shall discuss these results and others in Section 3.

By using Morse theory, we study the existence and multiplicity of nontrivial solutions for a class of p -sublinear p -Laplacian elliptic equations.

The new 3 3 - and 4 4 -manifold invariants recently constructed by Ozsváth and Szabó are based on a Floer theory associated with Heegaard diagrams. The following notes try to give an accessible introduction to their work. In the first part we begin by outlining traditional Morse theory, using the Heegaard diagram of a 3 3 -manifold as an example. We then describe Witten’s approach to Morse theory and how this led to Floer theory. Finally, we discuss Lagrangian Floer homology. In the second part, we define the Heegaard Floer complexes, explaining how they arise as a special case of Lagrangian Floer theory. We then briefly describe some applications, in particular the new 4 4 -manifold invariant, which is conjecturally just the Seiberg–Witten invariant.

In this work we prove some multiplicity results for solutions of a system of elliptic quasilinear equations, involving the p-Laplace operator (p > 2). The proof are based on variational and topological arguments and makes use of new perturbation results in Morse theory for the Banach space [Formula: see text].

In this paper we study the Morse functions on Grassmann manifolds. It is shown that many well-known calibrations can be used to construct degenerate Morse functions on Grassmann manifolds. In the study we use Clifford algebras and Lie groups.

If K is a finite simplicial complex and h is an injective map from the vertices of K to ℝ, we show how to extend h to a discrete Morse function in the sense of Forman [Forman 02] in a reasonably efficient manner so that the resulting discrete Morse function mirrors the large-scale behavior of h. A concrete algorithm is given for the case where K is a subcomplex of ℝ3.

We deal with compact K��hler manifolds $M$ acted on by a compact Lie group $K$ of isometries, whose complexification $K^\C$ has exactly one open and one closed orbit in $M$. If the $K$-action is Hamiltonian, we obtain results on the cohomology and the $K$-equivariant cohomology of $M$., 9 pages

It is often said that the Morse-Bott Lemma can be viewed as a “parameterized” Morse Lemma, and its proof should follow from the differentiability of the methods used to prove the Morse Lemma. The goal of this expository paper is to fill in the details. We present Palais' proof of the Morse Lemma using Moser's path method, which yields the necessary differentiability.

We obtain conditions under which the Morse functions defined on surfaces are smooth equivalent and functions with isolated critical (singular) points are topologically equivalent.

Let M be a closed orientable surface and let ϕ be a C1-flow on M with set of singularities compact countable. In this paper, we prove the Morse conjecture for ϕ: if ϕ is topologically transitive then it is metrically transitive.

In Morse Theory for Cell Complexes, we presented a discrete Morse theory that can be applied to general cell complexes. In particular, we defined the notion of a discrete Morse function, along with its associated set of critical cells. We also constructed a discrete Morse cocomplex, built from the critical cells and the gradient paths between them, which has the same cohomology as the underlying cell complex. In this paper we show how various cohomological operations are induced by maps between Morse cocomplexes. For example, given three discrete Morse functions, we construct a map from the tensor product of the first two Morse cocomplexes to the third Morse cocomplex which induces the cup product on cohomology. All maps are constructed by counting certain configurations of gradient paths. This work is closely related to the corresponding formulas in the smooth category as presented by Betz and Cohen and by Fukaya.

In [7, 8, 9], we developed a combinatorial Morse theory which can be applied to any CW complex. In [25, 26] Novikov presented a generalization of classical Morse theory in which the Morse function is replaced by a closed 1-forms. In this paper we extend our combinatorial Morse theory to include a combinatorial analog of Novikov's theory. Along the way we introduce the notion of a combinatorial differential form which is well-suited to our work, and which may have other applications.

From the work of C. Conley [Isolated invariant sets and the Morse index. in: Conference Board Math. Sci., No. 38. Providence: AMS; 1978], it is known that a compact metric space with a fixed flow can be decomposed into a finite collection of mutually disjoint compact invariant subsets (an attractor–repeller pair, or more generally, a Morse decomposition is such a decomposition). Here we shall display the intrinsic relation between the existence of these decompositions and that of Liapounov functions.

This paper is a glossary of notions and methods related to the topological theory of affine plane arrangements, including braid groups, configuration spaces, order complexes, stratified Morse theory, simplicial resolutions, complexes of graphs, Orlik-Solomon rings, Salvetti complexes, matroids, Spanier-Whitehead duality, twisted homology groups, monodromy theory, and multidimensional hypergeometric functions. The emphasis is upon making the presentation as geometric as possible. Applications and analogies in differential topology are outlined, and some recent results of the theory are presented.

If a complex analytic function, $f$, has a stratified isolated critical point, then it is known that the cohomology of the Milnor fibre of $f$ has a direct sum decomposition in terms of the normal Morse data to the strata. We use microlocal Morse theory to obtain the same result under the weakened hypothesis that the vanishing cycles along $f$ have isolated support. We also investigate an index-theoretic proof of this fact., Comment: 18 pages

We construct explicit Morse functions on Grassmannian manifolds, and use them to find explicit taut embeddings of the quadratic hypersurfaces in complex projective spaces into Euclidean spaces.

We study the existence of L2 holomorphic sections of invariant line bundles over Galois coverings. We show that the von Neumann dimension of the space of L2 holomorphic sections is bounded below under weak curvature conditions. We also give criteria for a compact complex space with isolated singularities and some related strongly pseudoconcave manifolds to be Moishezon. As applications we prove the stability of the previous Moishezon pseudoconcave manifolds under perturbation of complex structures as well as weak Lefschetz theorems.

In this paper we show the existence of non-minimal critical points of the Yang-Mills functional over a certain family of 4-manifolds { M 2 g : g = 0 , 1 , 2 , … } \{ M_{2g} : g=0,1,2, \ldots \} with generic S U ( 2 ) SU(2) -invariant metrics using Morse and homotopy theoretic methods. These manifolds are acted on fixed point freely by the Lie group S U ( 2 ) SU(2) with quotient a compact Riemann surface of even genus. We use a version of invariant Morse theory for the Yang-Mills functional used by Parker in A Morse theory for equivariant Yang-Mills, Duke Math. J. 66-2 (1992), 337–356 and Råde in Compactness theorems for invariant connections, submitted for publication.

We study the shape (in the sense of Borsuk) of attractors of continuous semi-dynamical systems on general metric spaces. We show, in particular, that the natural inclusion of the global attractor into the state space is a shape equivalence. This and other results of the paper are used to develop an elementary Morse theory of an attractor. © 2000 John Wiley & Sons, Inc.

We investigate the conjugacy of Morse functions on closed surfaces. By using cellular decompositions of surfaces, we formulate a criterion for the conjugacy of Morse functions. We establish a criterion for the conjugacy of mappings into a circle with nondegenerate critical points.

In this paper we construct a Universal chain complex, counting zeros of closed 1-forms on a manifold. The Universal complex is a refinement of the well known Novikov complex; it relates the homotopy type of the manifold, after a suitable noncommutative localization, with the numbers of zeros of different indices which may have closed 1-forms within a given cohomology class. The main theorem of the paper generalizes the result of a joint paper with A. Ranicki, which treats the special case of closed 1-forms having integral cohomology classes. The present paper also describes a number of new inequalities, giving topological lower bounds on the minimum number of zeros of closed 1-forms. In particular, such estimates are provided by the homology of flat line bundles with monodromy described by complex numbers, which are not Dirichlet units.

We describe the Morse groups of the nearby cycles sheaves on the nilcones in three classical symmetric spaces., Comment: 16 pages, AMSLaTeX

Given an S 1 S^1 -invariant Morse function f f and an S 1 S^1 -invariant Riemannian metric g g , a family of finite dimensional subcomplexes ( Ω ~ i n v , s m ∗ ( M , t ) , D ( t ) ) (\widetilde \Omega ^*_{inv,sm}(M,t), D(t)) , t ∈ [ 0 , ∞ ) t\in [0,\infty ) , of the Witten deformation of the S 1 S^1 -equivariant de Rham complex is constructed, by studying the asymptotic behavior of the spectrum of the corresponding Laplacian Δ ~ k ( t ) = D k ∗ ( t ) D k ( t ) + D k − 1 ( t ) D k − 1 ∗ ( t ) \widetilde \Delta ^k(t)=D^*_k(t)D_k(t)+D_{k-1}(t)D^*_{k-1}(t) as t → ∞ t\to \infty . In fact the spectrum of Δ ~ k ( t ) \widetilde \Delta ^k(t) can be separated into the small eigenvalues, finite eigenvalues and the large eigenvalues. Then one obtains ( Ω ~ i n v , s m ∗ ( M , t ) , D ( t ) ) ( \widetilde \Omega ^*_{inv,sm}(M,t),D(t)) as the complex of eigenforms corresponding to the small eigenvalues of Δ ~ ( t ) \widetilde \Delta (t) . This permits us to verify the S 1 S^1 -equivariant Morse inequalities. Moreover suppose f f is self-indexing and ( f , g ) (f,g) satisfies the Morse-Smale condition, then it is shown that this family of subcomplexes converges as t → ∞ t\to \infty to a geometric complex which is induced by ( f , g ) (f,g) and calculates the S 1 S^1 -equivariant cohomology of M M .