Homotopy commutativity in Hermitian symmetric spaces

TL;DR

This paper investigates the homotopy commutativity of loop spaces in Hermitian symmetric spaces, extending Ganea's result from complex projective spaces to a broader class of spaces, and also explores homotopy nilpotency of flag manifolds.

Contribution

It generalizes Ganea's theorem to all irreducible Hermitian symmetric spaces except C3, providing new insights into their loop space properties and homotopy nilpotency.

Findings

Loop spaces of all irreducible Hermitian symmetric spaces except C3 are not homotopy commutative.

The results determine the homotopy nilpotency of loop spaces of flag manifolds.

Ganea's criterion is extended to a wider class of symmetric spaces.

Abstract

Ganea proved that the loop space of $\mathbb{C}P^n$ is homotopy commutative if and only if $n=3$. We generalize this result to that the loop spaces of all irreducible Hermitian symmetric spaces but $\mathbb{C}P^3$ are not homotopy commutative. The computation also applies to determining the homotopy nilpotency of the loop spaces of flag manifolds.