On the cohomology rings of real flag manifolds: Schubert cycles

TL;DR

This paper presents an algorithm for computing the integer cohomology of real flag manifolds, explores torsion properties, and develops Schubert calculus techniques relevant for real enumerative geometry.

Contribution

It introduces a new algorithm for cohomology computation, proves torsion classes have order 2 in even flag manifolds, and advances Schubert calculus for real flag manifolds.

Findings

Integer cohomology groups computed via incidence coefficients

Torsion classes in even flag manifolds have order 2

Schubert calculus developed for real flag manifolds and Grassmannians

Abstract

We give an algorithm to compute the integer cohomology groups of any real partial flag manifold, by computing the incidence coefficients of the Schubert cells. For even flag manifolds we determine the integer cohomology groups, by proving that any torsion class has order 2 (generalizing a result of Ehresmann). We conjecture this to hold for any real flag manifold. We obtain results concerning which Schubert varieties represent integer cohomology classes, their structure constants and how to express them in terms of characteristic classes. For even flag manifolds and Grassmannians we also describe Schubert calculus. The Schubert calculus can be used to obtain lower bounds for certain real enumerative geometry problems (Schubert problems).