Related papers: The Borsuk-Ulam theorem for n-valued maps between …

Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the…

In this paper, we present a new qualitative extension of the Hopf theorem (and a generalization of Borsuk-Ulam theorem), concerning continuous maps $f$ from a compact Riemannian manifold $M$ of dimension $n$ to $\mathbb{R}^n$. We remove the…

Taking an elementary and straightforward approach, we develop the concept of a regular value for a smooth map f: O -> P between smooth orbifolds O and P. We show that Sard's theorem holds and that the inverse image of a regular value is a…

In combinatorial problems it is sometimes possible to define a $G$-equivariant mapping from a space $X$ of configurations of a system to a Euclidean space $\mathbb{R}^m$ for which a coincidence of the image of this mapping with an…

The moduli space of planar polygons with generic side lengths is a closed, smooth manifold. Mapping a polygon to its reflected image across the $X$-axis defines a fixed-point-free involution on these moduli spaces, making them into free…

Let M be a Seifert manifold which belongs to the geometry Flat. In this work we determine all the free involutions {\tau} on M, and the Borsuk-Ulam indice of (M,{\tau}).

A well known consequence of the Borsuk-Ulam theorem is that if the $d$-dimensional sphere $S^d$ is covered with less than $d+2$ open sets, then there is a set containing a pair of antipodal points. In this paper we provide lower and upper…

For each sapphire Sol $3$-manifold, we classify the free involutions. For each triple $(M, \tau; R^n)$ where $M$ is a sapphire Sol $3$-manifold and $\tau$ is a free involution, we show if $(M, \tau; R^n)$ has the Borsuk-Ulam property or…

Using equivariant obstruction theory we construct equivariant maps from certain classifying spaces to representation spheres for cyclic groups, product of elementary Abelian groups and dihedral groups. Restricting them to finite skeleta…

We give several construction methods and use them to produce many examples of proper biharmonic maps including biharmonic tori of any dimension in Euclidean spheres (Theorem 2.2, Corollaries 2.3, 2.4, and 2.6), biharmonic maps between…

Let $M$ be a topological space that admits a free involution $\tau$, and let $N$ be a topological space. A homotopy class $\beta \in [ M,N ]$ is said to have the Borsuk-Ulam property with respect to $\tau$ if for every representative map…

For each integer q>0 there is a cohomology theory such that the zero cohomology group of a manifold N of dimension n is a certain group of cobordism classes of proper fold maps of manifolds of dimension n+q into N. We prove a splitting…

We consider spaces with free involutions that satisfy the Borsuk - Ulam theorems (BUT-spaces). There are several equivalent definitions for BUT-spaces that can be considered as their properties. Our main technical tool is Yang's…

In this paper, we apply techniques from equivariant geometry to prove that a generalized Bour's theorem holds for surfaces that are invariant under the action of a one-parameter group of isometries of a three-dimensional Riemannian…

A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is a topological disk. In this paper we present a bijective link between unicellular maps on a non-orientable surface and…

Let $M$ be a topological space that admits a free involution $\tau$, and let $N$ be a topological space. A homotopy class $\beta \in [ M,N ]$ is said to have {\it the Borsuk-Ulam property with respect to $\tau$} if for every representative…

We show that the size of codes in projective space controls structural results for zeros of odd maps from spheres to Euclidean space. In fact, this relation is given through the topology of the space of probability measures on the sphere…

We introduce and study a new family of extensions for the Borsuk-Ulam and topological Radon type theorems. The defining idea for this new family is to replace requirements of the form `a subset that is large in some sense goes to a…

We consider spaces of plane curves in the setting of algebraic geometry and of singularity theory. On one hand there are the complete linear systems, on the other we consider unfolding spaces of bivariate polynomials of Brieskorn-Pham type.…

In this paper we consider several generalizations of the Borsuk-Ulam theorem for G-spaces and apply these results to Tucker type lemmas for G-simplicial complexes and PL-manifolds.