THE K-RING OF E_6/Spin(10)

TL;DR

This paper computes the K-theory ring of the coset space E_6/Spin(10), identifies its tangent bundle class, and demonstrates its immersion in 53-dimensional Euclidean space.

Contribution

It determines the K-ring of E_6/Spin(10), a significant new calculation in the topology of exceptional Lie groups and their homogeneous spaces.

Findings

K-ring of E_6/Spin(10) explicitly computed

Tangent bundle class identified in KO-theory

Space can be immersed in R^{53}

Abstract

Let $\mathrm E_6$ denote the simply-connected compact exceptional Lie group of rank 6. The Lie group $\mathrm Spin(10)$ naturally embeds in $\mathrm E_6$, corresponding to the inclusion of the Dynkin diagrams. We determine the K-ring of the coset space ${\mathrm E}_6/\mathrm Spin(10)$. We identify the class of the tangent bundle of ${\mathrm E}_6/\mathrm Spin(10)$ in $KO({\mathrm E}_6/\mathrm Spin(10))$. As an application we show that ${\mathrm E}_6/\mathrm Spin(10)$ can be immersed in the Euclidean space $\mathbb R^{53}$.