Morse theory of harmonic forms

Morse Theory of Harmonic Forms

We consider the problem of whether it is possible to improve the Novikov inequalities for closed 1-forms, or any other inequalities of a similar nature , if we assume, additionally, that the given 1-form is harmonic with respect to some Riemannian metric. We show that, under suitable assumptions, it is impossible. We use a theorem of E.Calabi C], characterizing 1-forms which are harmonic with r...

Morse Theory

Development of Morse Theory

In this paper, we develop Morse theory, which allows us to determine topological information about manifolds using certain real-valued functions defined on the manifolds. We first prove the Morse lemma, which says that, near critical points, such functions can be written in a useful way that gives us topological information. We then show how the homotopy type of the manifold is related to the i...

Morse Theory on Graphs

Let Γ be a finite d-valent graph and G an n-dimensional torus. An " action " of G on Γ is defined by a map which assigns to each oriented edge, e, of Γ, a one-dimensional representation of G (or, alternatively, a weight, αe, in the weight lattice of G. For the assignment, e → αe, to be a schematic description of a " G-action " , these weights have to satisfy certain compatibility conditions: th...

Morse Theory on Meshes

In this report, we discuss two papers that deal with computing Morse function on triangulated manifolds. Axen [1] gives an algorithm for computing Morse function on a triangulated manifold of arbitrary dimension but it not practical because of its space requirement. Hence, he describes an algorithm for computing critical points and their Morse indices for a 2-manifold. Edelsbrunner et al. [2] d...