Designs in compact symmetric spaces and applications of great antipodal sets

TL;DR

This paper surveys the theory of designs in compact symmetric spaces, focusing on great antipodal sets, and explores their applications in topology and group theory.

Contribution

It provides a comprehensive overview of great antipodal sets in symmetric spaces and demonstrates their applications in topology and group theory.

Findings

Characterization of great antipodal sets in symmetric spaces

Connections between antipodal sets and 2-number in topology

Applications to group theory and topology

Abstract

The theory of designs is an important branch of combinatorial mathematics. It is well-known in the theory of designs that a finite subset of a sphere is a tight spherical 1-design if and only if it is a pair of antipodal points. On the other hand, antipodal sets and 2-number for a Riemannian manifold are introduced by B.-Y. Chen and T. Nagano in 1982. An antipodal set is called a great antipodal set if its cardinality is equal to the 2-number. The main purpose of this paper is to provide a survey on important results in compact symmetric spaces with great antipodal sets as the designs. In the last two sections of this paper, we present some important applications of 2-number and great antipodal sets to topology and group theory.