Related papers: Borsuk-Ulam Type Theorems and Mountain Climbing Pr…

We work within the framework of a program aimed at exploring various extended versions for theorems from a class containing Borsuk-Ulam type theorems, some fixed point theorems, the KKM lemma, Radon, Tverberg, and Helly theorems. In this…

In this paper we study the problems of the following kind: For a pair of topological spaces $X$ and $Y$ find sufficient conditions that under every continuous map $f : X\to Y$ a pair of sufficiently distant points is mapped to a single…

In this paper we study variations of the Hopf theorem concerning continuous maps $f$ of a compact Riemannian manifold $M$ of dimension $n$ to $\mathbb{R}^n$. We investigate the case when $M$ is a closed convex $n$-dimensional surface and…

A generalization of the Borsuk-Ulam theorem to Stiefel manifolds is considered. This theorem is applied to derive bounds on $d$ that guarantee-for a given set of $m$ measures in $\mathbb{R}^d$-the existence of $k$ mutually orthogonal…

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…

Borsuk-Ulam's theorem is a useful tool of algebraic topology. It states that for any continuous mapping $f$ from the $n$-sphere to the $n$-dimensional Euclidean space, there exists a pair of antipodal points such that $f(x)=f(-x)$. As for…

B\'ar\'any's "topological Tverberg conjecture" from 1976 states that any continuous map of an $N$-simplex $\Delta_N$ to $\mathbb{R}^d$, for $N\ge(d+1)(r-1)$, maps points from $r$ disjoint faces in $\Delta_N$ to the same point in…

We observe that the classical Borsuk-Ulam theorem has an easy generalization to maps from an n-manifold M^n to R^n. We point out a geometric corollary.

In this paper, we obtain some sufficient conditions to guarantee the existence of multiple points of maps from $S^m$ to $\mathbb{R}^d$. Our main tool is the ideal-valued index of $G$-space defined by E. Fadell and S. Husseini. We obtain…

In this work we analysed the validity of a type of Borsuk-Ulam theorem for multimaps between surfaces. We developed an algebraic technique involving braid groups to study this problem for $n$-valued maps. As a first application we described…

We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two metric spaces $X$ and~$Y$ can be lower bounded by the…

Let (X, t, S) be a triple, where S is a compact, connected surface without boundary, and t is a free cellular involution on a CW-complex X. The triple (X, t, S) is said to satisfy the Borsuk-Ulam property if for every continuous map…

These informal notes, not intended for publication, provide an approach to the Borsuk--Ulam theorem via Stokes' theorem, in a similar spirit to Lima's proof of the Brouwer fixed point theorem. They are intended to be accessible to anyone…

In this paper, we address the longstanding question of whether expansive homeomorphisms can exist within convex bodies in Euclidean spaces. Utilizing fundamental tools from topology, including the Borsuk-Ulam theorem and Brouwer's…

In this paper some results on the topology of the space of $k$-flats in $\mathbb R^n$ are proved, similar to the Borsuk-Ulam theorem on coverings of sphere. Some corollaries on common transversals for families of compact sets in $\mathbb…

Suppose that $f_1,\ldots ,f_m : S(V)\to R$ are $m$ ($\geq 1$) continuous functions defined on the unit sphere in a Euclidean vector space $V$ of dimension $m+1$ satisfying $f_i(-v)=-f_i(v)$ for all $v\in S(V)$. The classical Borsuk-Ulam…

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…

We introduce the {\em $\mu$-topological stability}. This is a type of stability depending on the measure $\mu$ different from the set-valued approach \cite{lm}. We prove that the map $f$ is $m_p$-topologically stable if and only if $p$ 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…

We develop a lifting theory for the exponential map of semi-Riemannian manifolds that overcomes the classical obstruction caused by its singularities. We show that every smooth path in the manifold admits, up to a nondecreasing…