Related papers: Representations of Reeb spaces via simplified grap…

The Reeb graph is a construction which originated in Morse theory to study a real valued function defined on a topological space. More recently, it has been used in various applications to study noisy data which creates a desire to define a…

The cell complex structure is one of the most fundamental structures in topology and combinatorics, the Morse decomposition of a dynamical system analyzes the global gradient behavior, and the Reeb graph of a function is an elementary tool…

Previously, we have systematically constructed explicit real algebraic functions which are represented as the compositions of smooth real algebraic maps whose images are domains surrounded by hypersurfaces of degree 1 or 2 with canonical…

The Reeb space of a continuous function is the space of connected components of the level sets. In this paper we first prove that the Reeb space of a smooth function on a closed manifold with finitely many critical values has the structure…

We discuss smooth functions which are Morse on preimages of values not being local extrema. We call such a function internally Morse or I-Morse. The Reeb graph of a smooth function is the space of all connected components of preimages of…

Given a map $f:X \to M$ from a topological space $X$ to a metric space $M$, a decorated Reeb space consists of the Reeb space, together with an attribution function whose values recover geometric information lost during the construction of…

Fold maps are higher dimensional versions of Morse functions and fundamental and important tools in studying algebraic and differential topological properties of manifolds: as the theory established by Morse and the higher dimensional…

As a branch of algebraic and differential topology of manifolds, the theory of Morse functions and their higher dimensional versions or fold maps and its application to algebraic and differential topology of manifolds is fundamental,…

The theory of Morse functions and their higher dimensional versions or fold maps on manifolds and its application to geometric theory of manifolds is one of important branches of geometry and mathematics. Studies related to this was started…

For a connected locally path-connected topological space $X$ and a continuous function $f$ on it such that its Reeb graph $R_f$ is a finite topological graph, we show that the cycle rank of $R_f$, i.e., the first Betti number $b_1(R_f)$, in…

To investigate the topological structure of Morse functions on the projective plane we use the Reeb graphs. We describe it properties and prove that it is a complete topological invariant of simple Morse function on $\mathbb{R} P^2$. We…

In the singularity and differential topological theory of Morse functions and higher dimensional versions or fold maps and application to algebraic and differential topology of manifolds, constructing explicit fold maps and investigating…

Simple polyhedra are $2$-dimensional polyhedra and important objects in low-dimensional geometry and in the applications of {\it fold} maps, defined as smooth maps regarded as higher dimensional variants of Morse functions. For example,…

As a pioneering work we construct explicit real algebraic functions which may have both compact and non-compact preimages. The author has obtained explicit real algebraic functions with preimages satisfying some nice conditions. More…

This paper studies ways to represent an ordered topological vector space as a space of continuous functions, extending the classical representation theorems of Kadison and Schaefer. Particular emphasis is put on the class of semisimple…

Graph manifolds form important classes of $3$-dimensional closed and orientable manifolds. For example, {\it Seifert} manifolds are graph manifolds where hyperbolic manifolds are not. In applying singularity theory of differentiable maps to…

The Reeb graph $\mathcal{R}(f) $ is one of the fundamental invariants of a smooth function $f\colon M\to \mathbb{R} $ with isolated critical points. It is defined as the quotient space $M/_{\!\sim}$ of the closed manifold $M$ by a relation…

The present paper is on a new result on so-called realization problems of graphs as Reeb graphs of Morse functions with prescribed preimages. The Reeb graph of a smooth function is a graph obtained by identifying two points in the manifold…

In this paper, as a fundamental study on the theory of Morse functions and their higher dimensional versions or fold maps and applications to geometric theory of manifolds, which were started in 1950s by differential topologists such as…

Represented spaces form the general setting for the study of computability derived from Turing machines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented…