Spaces of Hermitian operators with simple spectra and their finite-order cohomology

TL;DR

This paper introduces a new spectral sequence for studying the cohomology of spaces of Hermitian operators with simple spectra, connecting topological, combinatorial, and algebraic methods.

Contribution

It constructs a novel spectral sequence based on topological order complexes, generalizing previous combinatorial formulas and providing tools to analyze cohomology of infinite Hermitian operators.

Findings

Spectral sequence degenerates at E_1 term.

Converges to stable cohomology of infinite Hermitian operators.

Defines finite degree cohomology classes for these spaces.

Abstract

The topology of spaces of Hermitian operators in $C^n$ with non-simple spectra was studied by V.Arnold 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 {\em topological order complexes} and resolutions of algebraic varieties, generalizing the combinatorial inclusion-exclusion formula and similar to the construction of finite degree 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 degree 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.