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We develop the theory of halving spaces to obtain lower bounds in realenumerative geometry. Halving spaces are topological spaces with an action of aLie group $\Gamma$ with additional cohomological properties. For$\Gamma=\mathbb{Z}_2$ we recover the conjugation spaces of Hausmann, Holm andPuppe. For $\Gamma=\mathrm{U}(1)$ we obtain the circle spaces. We show thatreal even and quaternionic partial flag manifolds are circle spaces leading tonon-trivial lower bounds for even real and quaternionic Schubert problems. Toprove that a given space is a halving space, we generalize results of Borel andHaefliger on the cohomology classes of real subvarieties and theircomplexifications. The novelty is that we are able to obtain results inrational cohomology instead of modulo 2. The equivariant extension of thetheory of circle spaces leads to generalizations of the results of Borel andHaefliger on Thom polynomials.

Halving spaces and lower bounds in real enumerative geometry