What question did this study set out to answer?

This research aims to explore the classical topological invariants of homogeneous spaces using K-theory for simply connected compact simple Lie groups.

July 10, 2026Open Access

The classical topological invariants of homogeneous spaces

JJJ. H. JonesExactech (United States)DRDmitriy RumyninUniversity of Warwick ATAdam R. ThomasUniversity of Warwick

Key Points

This research aims to explore the classical topological invariants of homogeneous spaces using K-theory for simply connected compact simple Lie groups.
Analyzed homogeneous spaces G/K where G and K share the same rank.
Focused on examples from Cartan’s list of symmetric spaces, specifically EIII, EVI, and EVIII.
Applied general methods relevant for classical groups and exceptional groups.
Examined three symmetric spaces EIII, EVI, and EVIII, revealing their topological characteristics.
Identified the dimensions of these spaces as 32, 64, and 128, respectively.
Established that these spaces correspond to Rosenfeld projective planes.

Abstract

We study the homogeneous spaces of a simply connected, compact, simple Lie group G through the lens of K -theory. Our general methods apply equally well to the case where G is in one of the four infinite families of classical groups, or one of the five exceptional groups. In this paper we focus on the case of homogeneous spaces G/K where G and K have the same rank. The main examples we study in detail are the three symmetric spaces EIII, EVI, EVIII in Cartan’s list of symmetric spaces. These are, respectively, homogeneous spaces for E₆, E₇, E₈ with dimensions 32, 64, 128 and known as Rosenfeld projective planes.

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