
Spaces of Hermitian operators with simple spectra and their finiteorder cohomology
V. A. Vassiliev^{ab} ^{a} Steklov Mathematical Institute, Russian Academy of Sciences
^{b} Independent University of Moscow
Abstract:
V. I. Arnold studied the topology of spaces of Hermitian operators with nonsimple 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, which generalizes the combinatorial inclusionexclusion formula, and is similar to the construction of finiteorder 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 all terms $E_r^{p,q}$ are finitely generated. This allows us to define the finiteorder cohomology classes of this space and apply wellknown 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.
Key words and phrases:
Hermitian operator, simple spectrum, simplicial resolution, continuous order complex, finite type cohomology, stable filtration.
DOI:
https://doi.org/10.17323/1609451420033311451165
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MSC: 06B35, 06F30, 15A57, 81V70 Received: May 28, 2002
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Citation:
V. A. Vassiliev, “Spaces of Hermitian operators with simple spectra and their finiteorder cohomology”, Mosc. Math. J., 3:3 (2003), 1145–1165
Citation in format AMSBIB
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\by V.~A.~Vassiliev
\paper Spaces of Hermitian operators with simple spectra and their finiteorder cohomology
\jour Mosc. Math.~J.
\yr 2003
\vol 3
\issue 3
\pages 11451165
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\crossref{https://doi.org/10.17323/1609451420033311451165}
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