There are classical examples of spaces with an involution whose mod 2 cohomology ring resembles that of their fixed point set : there is a ring isomorphism . Such examples include complex Grassmannians, toric manifolds, polygon spaces. In this paper, we show that the ring isomorphism is part of an interesting structure in equivariant cohomology called an –frame. An –frame, if it exists, is natural and unique. A space with involution admitting an –frame is called a conjugation space. Many examples of conjugation spaces are constructed, for instance by successive adjunctions of cells homeomorphic to a disk in with the complex conjugation. A compact symplectic manifold, with an anti-symplectic involution compatible with a Hamiltonian action of a torus , is a conjugation space, provided is itself a conjugation space. This includes the co-adjoint orbits of any semi-simple compact Lie group, equipped with the Chevalley involution. We also study conjugate-equivariant complex vector bundles (“real bundles” in the sense of Atiyah) over a conjugation space and show that the isomorphism maps the Chern classes onto the Stiefel-Whitney classes of the fixed bundle.
"Conjugation spaces." Algebr. Geom. Topol. 5 (3) 923 - 964, 2005. https://doi.org/10.2140/agt.2005.5.923