Cohomology of spaces of Hopf equivariant maps of spheres

TL;DR

This paper computes the rational cohomology rings of spaces of Hopf equivariant maps between spheres, revealing their isomorphism to cohomology rings of certain Stiefel manifolds, and discusses the properties of natural cohomology maps.

Contribution

It establishes the isomorphism of rational cohomology rings between equivariant map spaces and Stiefel manifolds, and analyzes the surjectivity and non-injectivity of natural cohomology maps.

Findings

Cohomology rings of equivariant map spaces are isomorphic to those of Stiefel manifolds.

Natural cohomology maps are surjective but not injective.

Results apply to maps equivariant under circle and quaternionic Hopf actions.

Abstract

For any natural numbers $k \leq n$, the rational cohomology ring of the space of continuous maps $S^{2k-1} \to S^{2n-1}$ (respectively, $S^{4k-1} \to S^{4n-1}$) equivariant under the Hopf action of the circle (respectively, of the group $S^3$ of unit quaternions) is naturally isomorphic to that of the Stiefel manifold $V_k({\mathbb C}^n)$ (respectively, $V_k({\mathbb H}^n)$). The natural maps of integral cohomology groups of these spaces of equivariant maps to cohomology of Stiefel manifolds are surjective but not injective.