Quaternionic conjugation spaces

There is a considerable amount 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. Hausmann, Holm, and Puppe have shown that such an isomorphism is a part of a certain structure on equivariant cohomology of $X$ and $X^\tau$, which is called an $H$-frame. The simplest examples are complex Grassmannians and flag manifolds with complex conjugation. We develop a similar notion of $Q$-frame which appears 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. Motivating examples are quaternionic Grassmannians and quaternionic flag manifolds equipped with two complex involutions. We show naturality and uniqueness of $Q$-framing. We prove that $Q$-framing can be defined for direct limits, products, etc. of $Q$-framed spaces. This list of operations contains glueing a disk in $\HH^n$ with complex involutions $\tau_1$ and $\tau_2$ to a $Q$-framed space by an equivariant map of boundary sphere.
An important part of $H$-frame structure in paper by H.–H.–P. was so called conjugation equation. Franz and Puppe calculated the coefficients of the conjugation equation in terms of the Steenrod squares. As a part of a $Q$-framing we introduce corresponding structure equation and express its coefficients by explicit formula in terms of the Steenrod operations.