Related papers: Mapping Cone and Morse Theory

In this paper we study Morse homology and cohomology with local coefficients, i.e. "twisted" Morse homology and cohomology, on closed finite dimensional smooth manifolds. We prove a Morse theoretic version of Eilenberg's Theorem, and we…

The wedge by a smooth closed $\ell$-form induces the mapping cone de Rham cochain complex. This complex is quasi-isomorphic to the mapping cone Thom-Smale cochain complex. We give an instanton construction of the mapping cone Thom-Smale…

In this paper, we define a relative Morse complex for manifold with boundary using the handlebody decomposition of the manifold. We prove that the homology of the relative Morse complex is isomorphic to the relative singular homology.…

This paper brings together three distinct theories with the goal of quantifying shape textures with complex morphologies. Distance fields are central objects in shape representation, while topological data analysis uses algebraic topology…

Special generic maps are smooth maps at each singular point of which we can represent as $(x_1, \cdots, x_m) \mapsto (x_1,\cdots,x_{n-1},\sum_{k=n}^{m}{x_k}^2)$ for suitable coordinates. Morse functions with exactly two singular points on…

This article presents families of 7-dimensional closed and simply-connected manifolds and fold maps on them such that squares of 2nd integral cohomology classes may not be divisible by 2. Fold maps are higher dimensional versions of Morse…

Fold maps are fundamental tools in the theory of singularities of differentiable maps and its applications to geometry. They are higher dimensional variants of Morse functions. Classes of special generic maps and round fold maps are…

In this paper, we prove that discrete Morse functions on digraphs are flat Witten-Morse functions and Witten complexes of transitive digraphs approach to Morse complexes. We construct a chain complex consisting of the formal linear…

Given a compact Riemannian manifold $(M g)$ and Morse function $f:m\to \mathbb{R}$ whose gradient flow satisfies the Morse-Smale condition, (i.e. the stable and unstable manifolds of f intersect transversely) we construct a chain complex…

We discuss some applications of the Morse-Novikov theory to some problems in modern physics, where appears a non-exact closed 1-form $\omega$ (a multi-valued functional). We focus mainly our attention to the cohomology of the de Rham…

In 1960, Smale defined a filtration of a closed smooth manifold by the unstable manifolds of fixed points and closed orbits of a Morse-Smale vector field defined on it, and derived generalized Morse inequalities. This suggests that,…

We construct topological invariants, called abstract weak orbit spaces, of flows and homeomorphisms on topological spaces, to describe both gradient dynamics and recurrent dynamics. In particular, the abstract weak orbit spaces of flows on…

Computing homology and cohomology is at the heart of many recent works and a key issue for topological data analysis. Among homological objects, homology generators are useful to locate or understand holes (especially for geometric…

Stable fold maps are fundamental tools in a generalization of the theory of Morse functions on smooth manifolds and its application to studies of geometric properties of smooth manifolds. Round fold maps were introduced as stable fold maps…

Constructing Morse functions and their higher dimensional versions or fold maps is fundamental, important and challenging in investigating the topologies and the differentiable structures of differentiable manifolds via Morse functions,…

The Morse-Smale complex is an important tool for global topological analysis in various problems of computational geometry and topology. Algorithms for Morse-Smale complexes have been presented in case of piecewise linear manifolds.…

We exploit a key result from visual psychophysics---that individuals perceive shape qualitatively---to develop the use of a geometrical/topological "invariant'' (the Morse--Smale complex) relating image structure with surface structure.…

The purpose of this paper is to introduce a version of singular homology based on smooth mappings of manifolds with corners. Although variants of such a theory exists in the literature, we felt that certain points were not adequately…

Our objective is to develop a stratified Morse theory with tangential conditions. We define a continuous strata-wise smooth Morse function on an abstract stratified space by using control conditions and radiality assumptions on the gradient…

We consider the problem of efficiently computing a discrete Morse complex on simplicial complexes of arbitrary dimension and very large size. Based on a common graph-based formalism, we analyze existing data structures for simplicial…