Cohomology of spaces of Hopf equivariant maps of spheres

For any natural numbers $k \leqslant n$, the rational cohomology ring of the space of continuous maps $S^{2k-1} \to S^{2n-1}$ (respectively, $S^{4k-1} \to S^{4n-1}$, which are 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.

Keywords

equivariant map, configuration space, Stiefel manifold, spectral sequence

2010 Mathematics Subject Classification

55P91, 55R91, 57R91

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This work was supported by the Absorption Center in Science of the Ministry of Immigration and Absorption of the State of Israel.

Received 23 August 2023

Received revised 22 November 2023

Accepted 22 January 2024

Published 2 October 2024