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

*(English)*Zbl 1047.47052Summary: V. I. Arnold [Sel. Math., New Ser. 1, 1–19 (1995; Zbl 0841.58005)] studied the topology of spaces of Hermitian operators with non-simple spectra in \(\mathbb C^n\) in relation to the theory of adiabatic connections and the quantum Hall effect. (Important physical motivations of this problem were also suggested by S. P. Novikov.) The natural stratification of these spaces into the sets of operators with fixed numbers of eigenvalues defines a spectral sequence providing interesting combinatorial and homological information on this stratification.

We construct a different spectral sequence, also converging to homology groups of these spaces; it is based on the universal techniques of topological order complexes and conical resolutions of algebraic varieties, generalizes the combinatorial inclusion-exclusion formula, and is similar to the construction of finite-order knot invariants.

This spectral sequence stabilizes at the term \(E_1\), is (conjecturally) multiplicative, and it converges as \(n\to\infty\) to a stable spectral sequence calculating the cohomology of the space of infinite Hermitian operators without multiple eigenvalues whose terms \(E_r^{p,q}\) are all finitely generated. This allows us to define the finite-order cohomology classes of this space and apply well-known facts and methods of the topological theory of flag manifolds to problems of geometric combinatorics, especially to those concerning continuous partially ordered sets of subspaces and flags.

We construct a different spectral sequence, also converging to homology groups of these spaces; it is based on the universal techniques of topological order complexes and conical resolutions of algebraic varieties, generalizes the combinatorial inclusion-exclusion formula, and is similar to the construction of finite-order knot invariants.

This spectral sequence stabilizes at the term \(E_1\), is (conjecturally) multiplicative, and it converges as \(n\to\infty\) to a stable spectral sequence calculating the cohomology of the space of infinite Hermitian operators without multiple eigenvalues whose terms \(E_r^{p,q}\) are all finitely generated. This allows us to define the finite-order cohomology classes of this space and apply well-known facts and methods of the topological theory of flag manifolds to problems of geometric combinatorics, especially to those concerning continuous partially ordered sets of subspaces and flags.

##### MSC:

47L05 | Linear spaces of operators |

55R80 | Discriminantal varieties and configuration spaces in algebraic topology |

15B57 | Hermitian, skew-Hermitian, and related matrices |

18G40 | Spectral sequences, hypercohomology |

47B15 | Hermitian and normal operators (spectral measures, functional calculus, etc.) |

06B35 | Continuous lattices and posets, applications |

06F30 | Ordered topological structures |