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Halving spaces and lower bounds in real enumerative geometry

László M Fehér and Ákos K Matszangosz

Algebraic & Geometric Topology 22 (2022) 433–472
Abstract

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 Γ with additional cohomological properties. For Γ = 2 we recover the conjugation spaces of Hausmann, Holm and Puppe. For Γ = U (1) 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.

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Keywords
conjugation spaces, equivariant cohomology, circle actions, real flag manifolds, real enumerative geometry
Mathematical Subject Classification
Primary: 14N10, 55N91
Secondary: 14M15, 14P25, 57N80, 57R91
References
Publication
Received: 14 September 2020
Revised: 1 March 2021
Accepted: 15 March 2021
Published: 26 April 2022
Authors
László M Fehér
Department of Analysis
Eötvös Loránd University
Budapest
Hungary
Ákos K Matszangosz
Alfréd Rényi Institute of Mathematics
Budapest
Hungary