(Equivariant) Cohomology of Homogeneous Spaces Associated with Composition Algebras

Abstract

We study homogeneous spaces arising from embeddings of associative composition algebras of dimensions 2 and 4, with particular emphasis on the Hamiltonian quaternion algebra. For inclusions D \hookrightarrow C, we analyze the (equivariant) geometry and topology of the quotients GL(n,C)/GL(n,D). In the quaternionic case, we prove that these spaces are equivariantly diffeomorphic to homogeneous vector bundles over compact symmetric spaces, and hence admit equivariant deformation retractions onto compact homogeneous models. This description determines their homotopy type and reduces the computation of their cohomological invariants to the compact setting. Using this reduction together with Weyl group invariant theory, we compute the rational equivariant cohomology rings with respect to maximal tori and obtain explicit formulas for the Poincar\'e polynomials of the quaternionic homogeneous spaces. We further extend the analysis to noncompact symmetric spaces associated with Clifford groups. In this setting, we establish equivariant deformation retractions onto compact homogeneous spaces and derive explicit expressions for their Betti numbers.