Homotopy commutativity in symmetric spaces

TL;DR

This paper investigates the homotopy commutativity of loop spaces in symmetric spaces, extending previous results and identifying which spaces exhibit this property, with a focus on Hermitian symmetric spaces.

Contribution

The authors extend prior work on homotopy commutativity of loop spaces, showing that most irreducible symmetric spaces, except  3P^3, are not homotopy commutative.

Findings

Loop spaces of all irreducible symmetric spaces except  3P^3 are not homotopy commutative.

Extended previous results of Ganea and collaborators on Hermitian symmetric spaces.

Identified specific exceptions in the class of symmetric spaces regarding homotopy commutativity.

Abstract

We extend the former results of Ganea and the two of the authors with Takeda on the homotopy commutativity of the loop spaces of Hermitian symmetric spaces such that the loop spaces of all irreducible symmetric spaces but $\mathbb{C}P^3$ are not homotopy commutative.