Related papers: Fiber connected, indefinite Morse 2-functions on c…

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…

For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse…

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…

Let M be a smooth connected compact surface, P be either the real line R^1 or the circle S^1. For a subset X of M denote by D(M,X) the group of diffeomorphisms of M fixed on X. In this note we consider a special class F of smooth maps…

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…

{\it Fold maps} are fundamental tools in generalizing the theory of Morse functions and its application to studies of geometric properties of manifolds. One of the fundamental and important problems in the theory of fold maps is to…

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…

The lifting problem for continuous bi-equivariant maps and bi-equivariant covering homotopies is considered, which leads to the notion of a bi-equivariant fibration. An intrinsic characteristic of a bi-equivariant Hurewicz fibration is…

We study properties of the continuation map for the Morse fundamental group $\pi_1^\text{Morse}(f,\ast)$ associated to a Morse-Smale pair $(f,g)$ on a manifold $M$. We get a morphism between $\pi_1^\text{Morse}(f_1,\ast_1)$ and…

The Reeb space of a smooth function is a topological and combinatoric object and fundamental and important in understanding topological and geometric properties of the manifold of the domain. It is the graph and a topological space endowed…

This paper is concerned with developing a 2-dimensional analogue of the notion of an ordinary discrete fibration. A definition is proposed, and it is shown that such discrete 2-fibrations correspond via a 2-equivalence to certain…

We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning…

We study a secondary invariant, called the Meyer function, on the fundamental group of the complement of the dual variety of a smooth projective variety. This invariant have played an important role when studying the local signatures of…

We show that the category of graphs has the structure of a 2-category with homotopy as the 2-cells. We then develop an explicit description of homotopies for finite graphs, in terms of what we call `spider moves'. We then create a category…

We show that for every complete Riemannian surface $M$ diffeomorphic to a sphere with $k \geq 0$ holes there exists a Morse function $f:M \rightarrow \mathbb{R}$, which is constant on each connected component of the boundary of $M$ and has…

Work of numerous authors has shown that any smooth, orientable, closed 4-manifold may be described as a loop of Morse functions on a surface, a loop in the cut complex, a loop in the pants complex, or as a multisection. In this paper, we…

We consider the maximal number of arbitrary points in a special fibre that can be simultaneously approached by points in one sequence of general fibres. Several results about this topological invariant and their applications describe the…

Considering non-constant holomorphic maps $\beta_{i}:S_{i}\to S_{0}$, $i\in\{1,2\}$, between non-compact Riemann surfaces for which it is associated its fiber product $S_{1}\times_{(\beta_{1},\beta_{2})}S_{2}$. With this setting, in this…

We define a class of multiparameter persistence modules that arise from a one-parameter family of functions on a topological space and prove that these persistence modules are stable. We show that this construction can produce…

Closed (and simply-connected) manifolds whose dimensions are larger than 4 are central geometric objects in classical algebraic topology and differential topology. They have been classified via algebraic and abstract objects. On the other…