  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.
