Conjugation spaces

Abstract

There are classical examples of spaces X with an involution tau whose mod 2-comhomology ring resembles that of their fixed point set X^tau: there is a ring isomorphism kappa: H^2*(X) --> H^*(X^tau). Such examples include complex Grassmannians, toric manifolds, polygon spaces. In this paper, we show that the ring isomorphism kappa is part of an interesting structure in equivariant cohomology called an H^*-frame. An H^*-frame, if it exists, is natural and unique. A space with involution admitting an H^*-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 C^k with the complex conjugation. A compact symplectic manifold, with an anti-symplectic involution compatible with a Hamiltonian action of a torus T, is a conjugation space, provided X^T 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 kappa maps the Chern classes onto the Stiefel-Whitney classes of the fixed bundle.