In [16, 17], Pajitnov considers the closed orbit structure of generic gradient flows of circle-valued Morse functions. It turns out that the torsion of a chain homotopy equivalence between the Novikov complex and the completed simplicial chain complex of the universal cover detects the eta function of the flow. This eta function counts the closed orbits and reduces to the logarithm of the zeta ...

Let M be a compact oriented simply-connected manifold of dimension at least 8. Assume M is equipped with a torsion-free semi-free circle action with isolated fixed points. We prove M has a perfect invariant Morse-Smale function. The major ingredient in the proof is a new cancellation theorem for the invariant Morse theory.

In classical Morse theory the number and type (index) of critical points of a smooth function on a manifold are related to topological invariants of that manifold through the Morse inequalities. There the index of a critical point is the number of negative eigenvalues that the Hessian matrix has on that tangent plane. Here deenitions of \critical point" and \index" are given that are suitable f...

In [18, 20], Pajitnov considers the closed orbit structure of generic gradient flows of circle-valued Morse functions. It turns out that the torsion of a chain homotopy equivalence between the Novikov complex and the completed simplicial chain complex of the universal cover detects the eta function of the flow. This eta function counts the closed orbits and reduces to the logarithm of the zeta ...

Let M be a closed, oriented, n-dimensional manifold. In this paper we give a Morse theoretic description of the string topology operations introduced by Chas and Sullivan, and extended by the first author, Jones, Godin, and others. We do this by studying maps from surfaces with cylindrical ends to M , such that on the cylinders, they satisfy the gradient flow equation of a Morse function on the...

The notion of Reeb graph is part of the Morse theory which provides an analysis of the relationship between the topology and the geometrical information of a space given by a suitable function, a Morse function. Until recently, the main drawback for using Reeb graph analysis was the incapacity of constructing adequate Morse functions. Before the work of Ni et al. [2004], the used Morse function...

Morse inequalities for diffeomorphisms of a compact manifold were first proved by Smale [21] under the assumption that the nonwandering set is finite. We call these the integral Morse inequalities. They were generalized by Zeeman in an unpublished work cited in [25] to diffeomorphisms with a hyperbolic chain recurrent set that is axiom-A-diffeomorphisms which satisfy the no cycle condition. For...

Let f be a smooth Morse function on an infinite dimensional separable Hilbert manifold, all of whose critical points have infinite Morse index and co-index. For any critical point x choose an integer a(x) arbitrarily. Then there exists a Riemannian structure on M such that the corresponding gradient flow of f has the following property: for any pair of critical points x, y, the unstable manifol...

Classical Morse Theory [8] considers the topological changes of the level sets Mh = {x ∈ M | f(x) = h } of a smooth function f defined on a manifold M as the height h varies. At critical points, where the gradient of f vanishes, the topology changes. These changes can be classified locally, and they can be related to global topological properties of M . Between critical values, the level sets v...

– The ambient framed bordism class of the connecting manifold of two consecutive critical points of a Morse–Smale function is estimated by means of a certain Hopf invariant. Applications include new examples of non-smoothable Poincaré duality spaces as well as an extension of the Morse complex.  2002 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ. – La classe ambiante de bordisme stab...