Conjugation spaces

Abstract

There are classical examples of spaces X with an involution whose mod2-comhomology ring resembles that of their fixed point set X : there is a ring isomorphism : H 2 * (X) H * (X ). 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 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 conjugateequivariant 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.