Morse Theory without Non-Degeneracy

Morse Theory for Riemannian Geodesics without Nondegeneracy Assumptions

Let f ∈ C(M,R) be a functional defined on a Hilbert manifold M. It is well known that if f is a Morse functional (i.e. every critical point of f is nondegenerate) and f satisfies the so called Palais–Smale condition, the Morse relations hold. More precisely, let x ∈ M be a critical point of f , and m(x, f) denote the Morse index at x (i.e. the maximal dimension of the subspaces of TxM where the...

Morse Theory

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...

Linking and Morse Theory

A. In this paper we use Morse theory and the gradient flow of a Morse-Smale function to compute the linking number of a two-component link L in S 3 , by counting the signed number of gradient flow lines passing through each component of L. We will also use three Morse-Smale functions and their gradient flows, to compute Milnor's triple linking number of three-component links by counting ...