The Floer homotopy type of height functions on complex Grassmann manifolds

Hurtubise, David

doi:10.1090/S0002-9947-97-01848-5

The Floer homotopy type of height functions on complex Grassmann manifolds
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by David E. Hurtubise PDF

Trans. Amer. Math. Soc. 349 (1997), 2493-2505 Request permission

Abstract:

A family of Floer functions on the infinite dimensional complex Grassmann manifold is defined by taking direct limits of height functions on adjoint orbits of unitary groups. The Floer cohomology of a generic function in the family is computed using the Schubert calculus. The Floer homotopy type of this function is computed and the Floer cohomology which was computed algebraically is recovered from the Floer homotopy type. Certain non-generic elements of this family of Floer functions were shown to be related to the symplectic action functional on the universal cover of the loop space of a finite dimensional complex Grassmann manifold in the author’s preprint The Floer homotopy type of complex Grassmann manifolds.

References

Michèle Audin and Jacques Lafontaine (eds.), Holomorphic curves in symplectic geometry, Progress in Mathematics, vol. 117, Birkhäuser Verlag, Basel, 1994. MR 1274923, DOI 10.1007/978-3-0348-8508-9
D. M. Austin and P. J. Braam, Morse-Bott theory and equivariant cohomology, The Floer memorial volume, Progr. Math., vol. 133, Birkhäuser, Basel, 1995, pp. 123–183. MR 1362827
S. Berceanu and A. Gheorghe, On the construction of perfect Morse functions on compact manifolds of coherent states, J. Math. Phys. 28 (1987), no. 12, 2899–2907. MR 917647, DOI 10.1063/1.527691
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1985. MR 781344, DOI 10.1007/978-3-662-12918-0
Ralph L. Cohen, Morse theory via moduli spaces, Mat. Contemp. 2 (1992), 19–66. Workshop on the Geometry and Topology of Gauge Fields (Campinas, 1991). MR 1303157
R.L. Cohen, J.D.S. Jones, and G.B. Segal. Morse theory and classifying spaces. Preprint, 1992.
R. L. Cohen, J. D. S. Jones, and G. B. Segal, Floer’s infinite-dimensional Morse theory and homotopy theory, The Floer memorial volume, Progr. Math., vol. 133, Birkhäuser, Basel, 1995, pp. 297–325. MR 1362832
Andreas Floer, An instanton-invariant for $3$-manifolds, Comm. Math. Phys. 118 (1988), no. 2, 215–240. MR 956166, DOI 10.1007/BF01218578
Andreas Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), no. 3, 513–547. MR 965228
Andreas Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), no. 4, 575–611. MR 987770, DOI 10.1007/BF01260388
Theodore Frankel, Fixed points and torsion on Kähler manifolds, Ann. of Math. (2) 70 (1959), 1–8. MR 131883, DOI 10.2307/1969889
D. Hurtubise. The Floer homotopy type of complex Grassmann manifolds. Preprint, 1996.
J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331, DOI 10.1515/9781400881802
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554, DOI 10.1515/9781400881826
Matthias Schwarz, Morse homology, Progress in Mathematics, vol. 111, Birkhäuser Verlag, Basel, 1993. MR 1239174, DOI 10.1007/978-3-0348-8577-5
Richard P. Stanley, Some combinatorial aspects of the Schubert calculus, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976) Lecture Notes in Math., Vol. 579, Springer, Berlin, 1977, pp. 217–251. MR 0465880
Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition. MR 722297, DOI 10.1007/978-1-4757-1799-0