Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-14T23:28:04.346Z Has data issue: false hasContentIssue false
  • English
  • Français

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: quique.macias@usc.es
M. J. PEREIRA–SÁEZ
Affiliation:
Departamento de Economía Aplicada II, Universidade da Coruña, 15071Spain. e-mail: maria.jose.pereira@udc.es
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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