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An upper bound for the Lusternik–Schnirelmann category of the symplectic group

Published online by Cambridge University Press:  17 May 2013

E. MACÍAS–VIRGÓS  and
M. J. PEREIRA–SÁEZ
E. MACÍAS–VIRGÓS
Affiliation:
Departamento de Xeometría e Topoloxía, Universidade de Santiago de Compostela, 15782Spain. e-mail: [email protected]
M. J. PEREIRA–SÁEZ
Affiliation:
Departamento de Economía Aplicada II, Universidade da Coruña, 15071Spain. e-mail: [email protected]
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Abstract

We prove that the LS category of the symplectic group Sp(n) is bounded above by $(n+1 \choose 2)$. This is achieved by computing the number of critical levels of a height function.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 2 , September 2013 , pp. 271 - 276
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Fernández–Suárez, L., Gómez–Tato, A., Strom, J. and Tanré, D.The Lusternik–Schnirelmann category of Sp(3). Proc. Amer. Math. Soc. 132 (2004), no. 2, 587–595.CrossRefGoogle Scholar
[2]Frankel, T.Critical submanifolds of the classical groups and Stiefel manifolds. Differ. and Combinat. Topology. Sympos. Marston Morse (Princeton, 1963), 37–53.Google Scholar
[3]Ganea, T.Some problems on numerical homotopy invariants. Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle Wash., 1971), pp. 2330. Lecture Notes in Math. vol. 249 (Springer, Berlin, 1971).Google Scholar
[4]Gómez–Tato, A., Macías–Virgós, E. and Pereira–Sáez, M. J.Trace map, Cayley transform and L–S category of Lie groups. Ann. Global Anal. Geom. 39 (2011), no. 3, 325–335.CrossRefGoogle Scholar
[5]Hunziker, M. and Sepanski, M. R.Distinguished orbits and the L–S category of simply connected compact Lie groups. Topology Appl. 156 (2009), no. 15, 2443–2451.CrossRefGoogle Scholar
[6]Iwase, N. and Mimura, M.L-S categories of simply-connected compact simple Lie groups of low rank. Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001), 199–212, Progr. Math. 215 (Birkhäuser, Basel, 2004).Google Scholar
[7]Macías–Virgós, E. and Pereira–Sáez, M. J.Left eigenvalues of 2 × 2 symplectic matrices. Electron. J. Linear Algebra 18 (2009), 274–280.CrossRefGoogle Scholar
[8]Macías–Virgós, E. and Pereira–Sáez, M. J.Symplectic matrices with predetermined left eigenvalues. Linear Algebra Appl. 432 (2010), no. 1, 347–350.CrossRefGoogle Scholar
[9]Reeken, M.Stability of critical points under small perturbations. Part I. Topological theory, Manuscripta Math. 7 (1972), 387–411.CrossRefGoogle Scholar
[10]Rudyak, Y. and Schlenk, F.Lusternik-Schnirelmann theory for fixed points of maps. Topol. Methods Nonlinear Anal. 21 (2003), no. 1, 171–194.CrossRefGoogle Scholar
[11]Schweitzer, P. A.Secondary cohomology operations induced by the diagonal mapping. Topology 3 (1965), 337–355.CrossRefGoogle Scholar
[12]Singhof, W.On the Lusternik–Schnirelmann category of Lie groups. Math. Z. 145 (1975), no. 2, 111–116.CrossRefGoogle Scholar
[13]Singhof, W.Minimal coverings of manifolds with balls. Manuscripta Math. 29 (1979), 385–415.CrossRefGoogle Scholar