Spaces of Hermitian operators with simple spectra and their finite-order cohomology.
V.I.Arnold studied the topology of spaces of Hermitian operators in $\C^n$
with non-simple spectra in a relation with the theory of adiabatic
connections and the quantum Hall effect. The natural filtration of these
spaces by the sets of operators with fixed numbers of eigenvalues defines
the spectral sequence, providing interesting combinatorial and homological
information on this stratification.
We construct a different spectral sequence, also counting the homology
groups of these spaces and based on the universal techniques of topological
order complexes and conical resolutions of algebraic varieties, generalizing
the combinatorial inclusion-exclusion formula and similar to the
construction of finite-order knot invariants.
This spectral sequence degenerates at the term $E_1$, is (conjecturally)
multiplicative, and as $n$ grows then it converges to a stable spectral
sequence counting the cohomology of the space of infinite Hermitian
operators without multiple eigenvalues, all whose terms
$E^{p,q}_r$ are finitely generated. It allows us to define the finite-order
cohomology classes of this space, and to apply the well-known facts and
methods of the topological theory of flag manifolds to the problems of
geometrical combinatorics, especially concerning the continuous partially
ordered sets of subspaces and flags.