  We describe a class of real Banach manifolds, which classify $K^{-1}$. These
manifolds are Grassmannians of (hermitian) lagrangian subspaces in a complex
Hilbert space. Certain finite codimensional real subvarieties described by
incidence relations define geometric representatives for the generators of the
cohomology rings of these classifying spaces. Any family of self-adjoint,
Fredholm operators parametrized by a closed manifold comes with a map to one of
these spaces. We use these Schubert varieties to describe the Poincare duals of
the pull-backs to the parameter space of the cohomology ring generators. The
class corresponding to the first generator is the spectral flow.
