# Equilateral $p$-gons in $\mathbb R^d$ and deformed spheres and mod $p$ Fadell-Husseini index

## TL;DR

This paper establishes the existence of certain equilateral polygons in Euclidean spaces and on spheres using topological methods, extending previous work with the Fadell-Husseini index.

## Contribution

It introduces the concept of r-equilateral m-gons and proves their existence in specific geometric and topological settings, advancing the application of topological tools in geometric problems.

## Key findings

- Existence of r-equilateral p-gons in re9d if r<d
- Existence of equilateral p-gons in images of injective maps from spheres
- Application of Fadell-Husseini index to geometric problems

## Abstract

We introduce the concept of $r$-equilateral $m$-gons. We prove the existence of $r$-equilateral $p$-gons in $\mathbb R^d$ if $r<d$ and the existence of equilateral $p$-gons in the image of continuous injective maps $f:S^d\to \mathbb R^{d+1}$. Our ideas are based mainly in the paper of Y. Soibelman \cite{soibelman}, in which the topological Borsuk number of $\mathbb{R}^2$ is calculated by means of topological methods and the paper of P. Blagojevi\'c and G. Ziegler \cite{blagojevictetrahedra} where Fadell-Husseini index is used for solving a problem related to the topological Borsuk problem for $\mathbb{R}^3$.