Conjugation spaces

Abstract

There are classical examples of spaces $X$ with an involution $\tau$ whose mod 2 cohomology ring resembles that of their fixed point set $X^{\tau }$: there is a ring isomorphism $\kappa :H^{2\ast }\left(X\right)\approx H^{\ast }\left(X^{\tau }\right)$. 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 $H^{\ast }$–frame. An $H^{\ast }$–frame, if it exists, is natural and unique. A space with involution admitting an $H^{\ast }$–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.