Related papers: Existence of 2-parameter crossings, with applicati…

We classify the path-components of the space of circle-valued Morse functions on compact surfaces: two Morse functions $f, g: M\to S^1$ belong to same path-component of this space if and only if they are homotopic and have equal numbers of…

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…

We investigate the problem of the realization of a given graph as the Reeb graph $\mathcal{R}(f)$ of a smooth function $f\colon M\rightarrow \mathbb{R}$ with finitely many critical points, where $M$ is a closed manifold. We show that for…

The ambient framed bordism class of the connecting manifold of two consecutive critical points of a Morse-Smale function is estimated by means of a certain Hopf invariant. Applications include new examples of non-smoothable Poincare duality…

We offer a new proof that two closed oriented 4-manifolds are cobordant if their signatures agree, in the spirit of Lickorish's proof that all closed oriented 3-manifolds bound 4-manifolds. Where Lickorish uses Heegaard splittings we use…

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…

We define a notion of Morse function and establish Morse theory-like theorems over offsets of any compact set in a Euclidean space at regular values of their distance function. Using non-smooth analysis and tools from geometric measure…

Given a Morse function f on a closed manifold M with distinct critical values, and given a field F, there is a canonical complex, called the Morse-Barannikov complex, which is equivalent to any Morse complex associated with f and whose form…

2-stratifolds are a generalization of 2-manifolds in that there are disjoint simple closed curves where several sheets meet. They arise in the study of categorical invariants of 3-manifolds and may have applications to topological data…

We construct random Morse functions on surfaces by random walk and compute related distributions. We study the space of Morse functions through these random variables. We consider subspaces characterized by the surfaces with boundary…

We present new explicit decompositions of manifolds via so-called fold maps into lower dimensional spaces. Fold maps form a nice class of so-called generic maps, generalizing Morse functions naturally. To understand the topologies and the…

Let $f$ be a real- or circle-valued Morse function on a compact surface M having exactly $n>0$ critical points. Denote by $O$ the orbit of $f$ with respect to the right action of the group of diffeomorphisms of $M$. We show that the…

This paper was motivated by work of Arnold where he explains how to count "snakes", i.e. Morse functions on the real axis with prescribed behavior at infinity. This leads immediately to a count of excellent Morse functions on the circle,…

Fold maps are higher dimensional versions of Morse functions, which play important roles in the studies of smooth manifolds, and such general maps also have been fundamental tools in the studies of smooth manifolds by using generic maps. In…

We address the homotopy theory of 2-crossed modules of commutative algebras, which are equivalent to simplicial commutative algebras with Moore complex of length two. In particular, we construct for maps of 2-crossed modules a homotopy…

We study how to construct explicit deformations of generic smooth maps from closed $n$--dimensional manifolds $M$ with $n \geq 2$ to the $2$--sphere $S^2$ and show that every smooth map $M \to S^2$ is homotopic to a $C^\infty$ stable map…

To investigate the topological structure of Morse flows on the 2-disk we use the planar graphs as destinguished graph of the flow. We assume, that the flow is transversal to the boundary of the 2-disk. We give a list of all planar graph…

A fold map is a smooth map at each singular point of which it is represented as the product map of a Morse function and the identity map on an open ball. A special generic map is a fold map such that the Morse function can be taken as a…

A knotted surface in the 4-sphere may be described by means of a hyperbolic diagram that captures the 0-section of a special Morse function, called a hyperbolic decomposition. We show that every hyperbolic decomposition of a knotted surface…

Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then…