We develop the theory of
halving spaces to obtain lower bounds in real enumerative
geometry. Halving spaces are topological spaces with an action of a Lie group
with additional
cohomological properties. For
we recover the conjugation spaces of Hausmann, Holm and Puppe. For
we
obtain the
circle spaces. We show that real even and quaternionic partial flag
manifolds are circle spaces, leading to nontrivial lower bounds for even real and
quaternionic Schubert problems. To prove that a given space is a halving space, we
generalize results of Borel and Haefliger on the cohomology classes of real subvarieties
and their complexifications. The novelty is that we are able to obtain results in
rational cohomology instead of modulo 2. The equivariant extension of the theory of
circle spaces leads to generalizations of the results of Borel and Haefliger on Thom
polynomials.
Keywords
conjugation spaces, equivariant cohomology, circle actions,
real flag manifolds, real enumerative geometry
