Cohomology of real Grassmann manifold and KP flow

TL;DR

This paper links the cohomology of real Grassmann manifolds to KP flows, providing explicit constructions of incidence graphs, Poincare polynomials, and point counts over finite fields, revealing deep geometric and algebraic connections.

Contribution

It introduces a novel flow-based realization of Gr(k,n) that simplifies the construction of its cohomology and relates it to finite field point counts.

Findings

Explicit incidence graphs for cohomology of Gr(k,n) derived from KP flow

Formulas for Poincare polynomials of Gr(k,n)

Number of F_q points computed via singularities along KP flow

Abstract

We consider a realization of the real Grassmann manifold Gr(k,n) based on a particular flow defined by the corresponding (singular) solution of the KP equation. Then we show that the KP flow can provide an explicit and simple construction of the incidence graph for the integral cohomology of Gr(k,n). It turns out that there are two types of graphs, one for the trivial coefficients and other for the twisted coefficients, and they correspond to the homology groups of the orientable and non-orientable cases of Gr(k,n) via the Poincare-Lefschetz duality. We also derive an explicit formula of the Poincare polynomial for Gr(k,n) and show that the Poincare polynomial is also related to the number of points on a suitable version of Gr(k,n) over a finite field $\F_q$ with q being a power of a prime. In particular, we find that the number of $\F_q$ points on Gr(k,n) can be computed by counting the number of singularities along the KP flow.