  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 $\Gamma$ with additional cohomological properties. For
$\Gamma=\mathbb{Z}_2$ we recover the conjugation spaces of Hausmann, Holm and
Puppe. For $\Gamma=\mathrm{U}(1)$ we obtain the circle spaces. We show that
real even and quaternionic partial flag manifolds are circle spaces leading to
non-trivial 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.

Algebraic Topology;Algebraic Geometry