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We provide an analytic proof of Morse-type inequalities for vector fields determining a Morse decomposition with normally hyperbolic dynamics. In the demonstration we reduce the problem to the gradient case using a Morse–Bott Lyapunov function and then apply Schrödinger operator techniques. This yields an explicit construction of the cohomology complex of the manifold in terms of the invariant sets of the Morse decomposition associated with the vector field.

The goal of this work is to study the structure of the pure Morse complex of a graph, that is, the simplicial complex given by the set of all possible classes of discrete Morse functions (in Forman's sense) defined on it. First, we characterize the pure Morse complex of a tree and prove that it is collapsible. In order to study the general case, we consider all the spanning trees included in a given graph G and we express the pure Morse complex of G as the union of all pure Morse complexes corresponding to such trees.

Let G be a finite group and M be a compact G -manifold on which the G -action is semifree. For a specific local coefficient system we show that the Bredon cohomology of M can be expressed in terms of the classical equivariant cohomology of the fixed point sets. Using this we define Morse polynomial for an equivariant Morse function f defined on M . If M is a G -manifold such that codim M G ⩽2 and if f is an equivariant Morse function on M such that f | M G is also Morse then we show that the Morse polynomial of f completely reflects a G -CW structure of M .

This paper is focused on looking for links between the topology of a connected and non-compact surface with finitely many ends and any proper discrete Morse function which can be defined on it. More precisely, we study the non-compact surfaces which admit a proper discrete Morse function with a given number of critical elements. In particular, given any of these surfaces, we obtain an optimal discrete Morse function on it, that is, with the minimum possible number of critical elements.

Using the discrete Morse theory of R. Forman, we find a basis for the unique nonzero homology group of the complex of 2-connected graphs on n vertices. This answers a question of V. Vassiliev which arises in his study of knot invariants.

In this paper we study the existence of nontrivial 2π-periodic solutions of asymptotically linear Hamiltonian systems. We consider the case of resonance both at zero and at infinity, and we permit time-dependent asymptotic matrices. Our main tools are an infinite dimensional cohomology theory and a corresponding Morse theory recently constructed by W. Kryszewski and the first author. We develop a method to compute the new critical groups.

Milnor's treatment [6] of Morse's global theory of the calculus of variations for geodesics [7] is restated in the context of space-time geometry: it is seen as providing a link between the curvature and the causal structure of a stably causal globally hyperbolic Lorentzian manifold. An application is discussed.

Morse theory is a topological approach to the Calculus of Variations. It aims to relate the critical points of a functional to the topology of the function space on which the functional is defined. It is only directly applicable in special rather restrictive conditions, notably for problems involving one independent variable. However I will discuss a number of special examples, in some of which the Morse theory really works, and others in which it clearly fails but where nevertheless some aspects appear still to survive. These examples include those of physical interest and it would be interesting to investigate these further. One can make a number of speculations in this direction.

In this paper we use the equivariant Morse theory to give an estimate of the minimal number of central configurations in the N-body problem in ℝ3. In the case of equal masses we prove that the planar central configurations are saddle points for the potential energy. From this we deduce the presence of non-planar central configurations, for every N ≧ 4. The principal difficulty in applying Morse theory is that the potential function is defined on a manifold on which the group O(3) does not act freely. To avoid this problem the equivariant cohomology functor is applied in order to obtain the Morse inequalities.

It is a great pleasure to address this symposium in honor of my dear friend, teacher, and collaborator. I first met Chern in 1950, when he dropped in to visit Princeton for just one day and I sat near him at lunch. I don’t suppose that you remember this occasion, my dear friend, though I am sure I contrived to attract your attention by some impertinence or other. For I was immediately captivated by what you said and how you said it.

Publisher Summary This chapter describes the degenerate duality, catastrophes, and saddle functionals. The statement of a single stationary principle may appear in different but equivalent forms. It is a preliminary study of the transformations that may be needed to pass between equivalent variational principles. The ideas are fixed by concentrating on the finite-dimensional case so that one can make a precise connection with the critical point theory of Morse functions, and this approach also allows relating objective directly with the classification of degenerate stationary points, which is a concern of elementary catastrophe theory. Saddle functions appear as one of the forms from which stationary principles may be generated. A diffeomorphism can be found to express any Morse function of two variables, quadratic or not, exactly as one of the three standard forms near a stationary point. There is no smooth coordinate transformation that can convert one of the three standard Morse functions into either of the other two.

It is shown that the Morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics Hamiltonian. Some of the implications of modern ideas in mathematics for supersymmetric theories are discussed.

The reader who is interested only in Morse theory for singular spaces or for nonproper Morse function may skip this chapter. We will consider the Morse theory of a composition $$ X\xrightarrow{R}Z\xrightarrow{f}\mathbb{R} $$ which will eventually be used (in Part II, Sects. 5.1 and 5.1*, with Z = ℂℙ n ) to prove a conjecture of Deligne [Dl] concerning Lefschetz hyperplane theorems for a variety X and an algebraic map π: X → ℂℙ n . We will approximate the function f by a Morse function, although the composition fπ: X → ℝ is not Morse (or even Morse-Bott) in any reasonable sense. All attempts to prove Deligne’s conjecture by approximating fπ by a Morse function seem to end in failure because one loses curvature estimates on the Morse index of fπ. Instead, we are forced to “relativize” the Morse theory of f. Our main result is stated in Sect. 9.8.

We define a relative Morse index of nondegenerate critical points of the symplectic action functional, in the case of paths connecting two Lagrangian submanifolds.

On stable properties of the solution set of an ordinary differential equation Elementary properties of flows The Morse index Continuation Bibliography.

This chapter contains the estimates on the connectivity of the Morse data which are necessary for the proofs of the main applications. Since the Morse data is the product of the tangential Morse data with the normal Morse data, this requires estimates on both. The estimates on tangential Morse data are made in terms of the remarkable properties of the Levi form of the Morse function (see the appendix, Sect. 4.A). The normal Morse data is analyzed (inductively) by applying the entire apparatus of Morse theory to the complex link, which is a complex analytic space of smaller dimension.

Summary The entire body may be approximated with a closed surface having the homotopy type of a sphere. The Morse inequalities then yield certain relationships between the local potential extrema and the saddle points appearing on body surface potential maps (BSPMs). In particular, a list of possible extremal configurations is presented.

Throughout this chapter we will fix an arrangement A = {A1A2,…,Am} of affine subspaces of ℝ n , let P denote the corresponding partially ordered set of flats which are the intersections of the affine spaces, let T denote the unique maximal element of P corresponding to ℝ n , and let K(P)denote the order complex of P. We denote by M = ℝ n − ∪ A the space of interest, i.e., the complement of the affine subspaces in A.

In this chapter we analyze the normal Morse data at a critical point p∈Z of a function f1: Z → ℝ under the assumption that there exists a second function f2: Z → ℝ such that the map (f1,f2): Z → ℝ2 has a nondegenerate critical point at p (see below).

Publisher Summary This chapter discusses the Morse theory for Hamiltonian systems. The chapter presents the differential equation. The chapter presents infinite dimensional case Bott's Morse theory for the functions with nondegenerate critical manifolds. A critical circle is nondegenerate if the utility of all points is one.

We are now in a position to identify the homotopy Morse data (Part I, Sect. 3.3) for Morse functions on complex analytic varieties. Recall that homotopy Morse data is a pair (A, B) which is homotopy equivalent to some choice of Morse data. The importance of homotopy Morse data is the following: Suppose the pair (A, B) is homotopy Morse data for the Morse function f: X → ℝ at the critical point p, with critical value v= f(p). Suppose v ∈(a, b) and that the closed interval [a, b] contains no other critical values of f. Then there exists a continuous map h: B → X≤a such that X≤b is homotopy equivalent to the adjunction space X≤a⋃ B A (see Part I, Sect. 3.3). The identification of homotopy Morse data uses the deepest results of Part I and of Part II, Sect. 2: Theorem 3.5.4 of Part I says that local Morse data is Morse data and Theorem 3.7 of Part I says that local Morse data is the product of tangential Morse data with normal Morse data. So, homotopy Morse data is the product of the homotopy type of the tangential Morse data with the homotopy type of the normal Morse data. The homotopy type of tangential Morse data was identified classically by Morse, Thom, and Bott. It is the pair (D λ ,∂D λ ), where λ denotes the Morse index of the restriction of f to the stratum which contains the critical point p. So, the problem of identifying the homotopy Morse data is reduced to the problem of identifying the homotopy type of the normal Morse data. This was carried out in Sect. 2. In this short chapter we summarize those results.

Theorem 1 fills a gap, pointed out by" Fischer in [2], in the proof of a theorem of Andreotti and Grauert in [1]. Let X be a reduced, Hausdorff, q-convex complex space, with countable basis. g(X) denotes the set of real valued, infinitely differentiable functions onX. Let K be the compact set of X and q~ the continuous function on X, strongly q-convex on X K , respectively, as stated in the definition of [1] and [2]; we can suppose that q~ 8(X). We shall prove the following

We investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space. The results are applied to an infinite dimensional problem involving the symplectίc action function.

An infinite dimensional Morse lemma is proved using the deformation lemma from singularity theory. It is shown that the versions of the Morse lemmas due to Palais and Tromba are special cases. An infinite dimensional splitting lemma is proved. The relationship of the work here to other approaches in the literature in discussed.

0. Introduction. Let M be a differentiable manifold of class C°°. By a Morse function / on M, we mean a differentiable function / on M having only non-degenerate critical points. A well-known topological result of Reeb states that if M* is compact and there is a Morse function / on M having exactly 2 critical points, then M is homeomorphic to an ?^-sphere, S (see, for example, [3], p. 25). In a recent paper, [4], Nomizu and Rodriguez found a geometric characterization of a Euclidean ^-sphere SaR in terms of the critical point behavior of a certain class of functions Lp, p e R , on M. In that case, if p e R, x e M, then Lp(x) = (d(x, p))\ where d is the Euclidean distance function. Nomizu and Rodriguez proved that if M {n ^ 2) is a connected, complete Riemannian manifold isometrically immersed in R such that every Morse function of the form Lp, p e R , has index 0 or n at any of its critical points, then M is embedded as a Euclidean subspace, R, or a Euclidean ^-sphere, S. This result includes the following: if M is compact such that every Morse function of the form Lp has exactly 2 critical points, then M = S. In this paper, we prove results analogous to those of Nomizu and Rodriguez for a submanifold M of hyperbolic space, H, the spaceform of constant sectional curvature —1. For p € H, x e M, we define the function Lp(x) to be the distance in H from p to x. We then define the concept of a focal point of (M, x) and prove an Index Theorem for Lp which states that the index of Lp at a non-degenerate critical point x is equal to the number of focal points of {M, x) on the geodesic in H from x to p. In section 2, we prove that a metric sphere S c H can be characterized by the condition that every Morse function of the form Lp, p e H, has exactly 2 critical points. In section 3, we give an example which shows that a result analo

In this paper we deal with the existence and multiplicity of nontrivial solutions to a class of operator equation. By using infinite dimensional Morse theory, we establish some conditions which guarantee that the equation has many nontrivial solutions.

We introduce combinatorial multivector fields, associate with them multivalued dynamics and study their topological features. Our combinatorial multivector fields generalize combinatorial vector fields of Forman. We define isolated invariant sets, Conley index, attractors, repellers and Morse decompositions. We provide a topological characterization of attractors and repellers and prove Morse inequalities. The generalization aims at algorithmic analysis of dynamical systems through combinatorialization of flows given by differential equations and through sampling dynamics in physical and numerical experiments. We provide a prototype algorithm for such applications.