The multiple points of maps from sphere to Euclidean space

2021/09/23 by Jun Wang, Xuezhi Zhao, Wang, Jun +1
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.2109.11575

Abstract

In this paper, we obtain some sufficient conditions to guarantee the existence of multiple points of maps from S^m to 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→ ℝ such that f(-p)=-f(p), if m+1 is a power of 2, then there are m+1 points p₁, …, p_m+1 in S^m such that f(p₁)=⋯=f(p_m+1), where p₁, …, p_m+1 are linearly dependent and any m points of p₁, …, p_m+1 are linearly independent. As a generalization of Hopf's theorem, we also prove that for any continuous map f: S^m→ ℝ^d, if m> d, then there exists a pair of mutually orthogonal points having the same image in addition to the antipodal points.

Related

• Multiple point spaces of finite holomorphic maps
• Functions on the sphere with critical points in pairs and orthogonal geodesic chords
• Hopf-type theorems for convex surfaces
• Rational maps from Euclidean Configuration Spaces to Spheres
• Mobius transformations, the Caratheodory metric, and the objects of complex analysis and potential theory in multiply connected domains