The cohomology of real Grassmannians via Schubert stratifications

TL;DR

This paper derives a closed-form formula for the cohomology of real Grassmannians by employing stratified space theory and differential topology to compute differentials in a chain complex.

Contribution

It introduces a novel approach using stratified space theory to explicitly compute the cohomology of real Grassmannians, including the additive structure with arbitrary coefficients.

Findings

Closed formula for cohomology of real Grassmannians

Identification of the chain complex's isomorphism type

Application of stratified space theory to differential computations

Abstract

In this paper, we present a closed formula for the cohomology of real Grassmannians. To achieve this, we use a theory of stratified spaces to compute the differentials in a chain complex that computes the cohomology. Specifically, we organize Schubert cells as a conically smooth stratified space in the sense of Ayala, Francis, Tanaka; the links therein yield the sought differentials, using methods in differential topology. Further, we identify the isomorphism type of this chain complex and we use this result to provide a closed formula for the additive structure of the cohomology of real Grassmannians with arbitrary coefficients.