Conjugation Equation for Quaternionic Conjugation Spaces

There is a considerable collection of examples of spaces $$X$$ equipped with an involution $$\tau$$ such that the mod 2-cohomology rings $$H^{2*}(X)$$ and $$H^*(X^\tau)$$ are isomorphic. In [4], it was shown that such an isomorphism is a part of a certain structure on the equivariant cohomology of $$X$$ and $$X^\tau$$ , which is called an $$H$$ -frame. An important part of the $$H$$ -frame structure in [4] was the so-called conjugation equation. In [3], the coefficients of the conjugation equation were calculated in terms of the Steenrod squares. Later, another proofs were obtained, [9, 10]. In this paper, we develop a similar notion of a $$Q$$ -framing, which occurs in the situation when a space $$X$$ is equipped with two commuting involutions $$\tau_1,\tau_2$$ and the mod 2-cohomology rings $$H^{4*}(X)$$ and $$H^*(X^{\tau_1,\tau_2})$$ are isomorphic. Basic examples are the quaternionic Grassmannians and the quaternionic flag manifolds equipped with two complex involutions. Our main result is the establishment of the quaternionic conjugation equations and identifying their coefficients in terms of Steenrod operations. DOI 10.1134/S1061920825600096