Related papers: Existence and deformability of topological Morse f…

To investigate the topological structure of Morse flows with a sink on the 2-sphere we use the planar tree as complete topological invariant of the flow. We give a list of all planar tree with at least 7 edges. We use a list of rooted…

Free actions of finite groups on spheres give rise to topological spherical space forms. The existence and classification problems for space forms have a long history in the geometry and topology of manifolds. In this article, we present a…

In this work, we design a nearly linear time discrete Morse theory based algorithm for computing homology groups of 2-manifolds, thereby establishing the fact that computing homology groups of 2-manifolds is remarkably easy. Unlike previous…

We study perfect discrete Morse functions on closed oriented n-dimensional manifolds. We show how to compose such functions on connected sums of closed oriented manifolds and how to decompose on connected sums of closed oriented surfaces.

This paper introduces a novel topology, referred to as the star topology, on finite graphs. By treating vertices and edges as points in a unified space, we explore continuous maps between Bare representations of a graph and their…

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…

It is well-known that a function on an open set in $\mathbb R^d$ is smooth if and only if it is arc-smooth, i.e., its composites with all smooth curves are smooth. In recent work, we extended this and related results (for instance, a real…

We present an algorithm for computing the main topological characteristics of three-dimensional bodies. The algorithm is based on a discretization of Morse theory and uses discrete analogs of smooth functions with only nondegenerate (Morse)…

A classical inequality, which is known for families of monotone functions, is generalized to a larger class of families of measurable functions. Moreover we characterize all the families of functions for which the equality holds. We apply…

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…

Poisson structures vanishing linearly on a set of smooth closed disjoint curves are generic in the set of all Poisson structures on a compact connected oriented surface. We construct a complete set of invariants classifying these structures…

The aim of this paper is to extend the Morse theory for geodesics to the conical manifolds. We define these manifolds as submanifolds of $\R^n$ with a finite number of conical singularities. To formulate a good Morse theory we must use an…

Examples of Morse functions with integrable gradient flows on some classical Riemannian manifolds are considered. In particular, we show that a generic height function on the symmetric embeddings of classical Lie groups and certain…

Fold maps are smooth maps at each singular point of which it is represented as the product map of a Morse function and the identity map. Round fold maps are, in short, such maps the sets of all singular points of which are embedded…

The paper contains a review on recent progress in the deformational properties of smooth maps from compact surfaces $M$ to a one-dimensional manifold $P$. It covers description of homotopy types of stabilizers and orbits of a large class of…

The (negative) gradient vector fields of Morse functions on a compact manifold provide an important example in dynamical system. In this note we prove two important properties of this kind of vector field: Connectedness of critical points…

Given a $C^{0}$-deformation of domains $D(t)$ on a manifold $M^{n}$, which allows the topological types of the domains $D(t)$ to change with $t$, in what cases are the entities in analysis continuous in $t$, so that analysis techniques…

This paper studies the homotopy-type of bi-filtrations of compact manifolds induced as the pre-image of filtrations of the plane for generic smooth functions f : M --> R^2. The primary goal of the paper is to allow for a simple description…

This re-certifying paper describes the details of the Morse homology of manifolds with boundary, introduced by the author before, in terms of handlebody decompositions.

Round fold maps are smooth maps on closed manifolds which are locally represented as the product maps of Morse functions and identity maps on open disks and whose singularity is realized as concentrically embedded spheres. The author…