The multiple points of maps from sphere to Euclidean space

TL;DR

This paper establishes conditions for the existence of multiple points in maps from spheres to Euclidean spaces, using algebraic topology tools, and extends classical theorems to new settings involving symmetry and orthogonality.

Contribution

It introduces new sufficient conditions for multiple points in sphere-to-Euclidean maps and generalizes Hopf's theorem to include orthogonal points with the same image.

Findings

Existence of multiple points under certain symmetry conditions.

Generalization of Hopf's theorem to orthogonal points.

Conditions when multiple points are linearly dependent or independent.

Abstract

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 more detailed relative positional relationship of multiple points. It is proved that for a continuous real value function $f: S^m\rightarrow \mathbb{R}$ such that $f(-p)=-f(p)$, if $m+1$ is a power of $2$, then there are $m+1$ points $p_1, \ldots, p_{m+1}$ in $S^m$ such that $f(p_1)=\cdots=f(p_{m+1})$, where $p_1, \ldots, p_{m+1}$ are linearly dependent and any $m$ points of $p_1, \ldots, p_{m+1}$ are linearly independent. As a generalization of Hopf's theorem, we also prove that for any continuous map $f: S^m\rightarrow \mathbb{R}^d$, if $m> d$, then there exists a pair of mutually orthogonal points having the same image in addition to the antipodal points.