The cohomology of an isospectral flow

Fried, David

doi:10.1090/S0002-9939-1986-0854048-6

The cohomology of an isospectral flow
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by David Fried

Proc. Amer. Math. Soc. 98 (1986), 363-368

DOI: https://doi.org/10.1090/S0002-9939-1986-0854048-6

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Abstract:

Building on work of Tomei, we compute the cohomology of the manifold of real symmetric tridiagonal matrices with distinct fixed eigenvalues. The proof uses the global dynamical properties of the Toda flow on this isospectral manifold.

References

Armand Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115–207 (French). MR 51508, DOI 10.2307/1969728
Raoul Bott and Hans Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964–1029. MR 105694, DOI 10.2307/2372843
Michael W. Davis, Some aspherical manifolds, Duke Math. J. 55 (1987), no. 1, 105–139. MR 883666, DOI 10.1215/S0012-7094-87-05507-4

The homology of a space on which a reflection group acts

Marvin J. Greenberg, Lectures on algebraic topology, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0215295
W.-Y. Hsiang, R. S. Palais, and C. L. Terng, Geometry and topology of isoparametric submanifolds in Euclidean spaces, Proc. Nat. Acad. Sci. U.S.A. 82 (1985), no. 15, 4863–4865. MR 799109, DOI 10.1073/pnas.82.15.4863
Jürgen Moser, Finitely many mass points on the line under the influence of an exponential potential–an integrable system, Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974) Lecture Notes in Phys., Vol. 38, Springer, Berlin, 1975, pp. 467–497. MR 0455038
S. Smale, The $\Omega$-stability theorem, Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 289–297. MR 0271971
Carlos Tomei, The topology of isospectral manifolds of tridiagonal matrices, Duke Math. J. 51 (1984), no. 4, 981–996. MR 771391, DOI 10.1215/S0012-7094-84-05144-5