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.
Keywords
cohomology rings, equivariant cohomology, spaces with
involution, real spaces