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Splitting definably compact groups in o-minimal structures

Published online by Cambridge University Press:  12 March 2014

Marcello Mamino
Marcello Mamino*
Affiliation:
Classe di Scienze - Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy, E-mail: m.mamino@sns.it
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Abstract

An argument of A. Borel [Bor-61, Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.

Keywords

03C6455S40Definable groupso-minimalityfibre bundles

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 3 , September 2011 , pp. 973 - 986
Copyright
Copyright © Association for Symbolic Logic 2011

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References

REFERENCES

[Bar-09]Baro, Elías, On the o-minimal LS-category, to appear in the Israel Journal of Mathematics arXiv:0905.1391vl, 2009.Google Scholar
[BO-10]Baro, Elíias and Otero, Margarita, On o-minimal homotopy groups, The Quarterly Journal of Mathematics, vol. 61 (2010), no. 3, pp. 275–289.CrossRefGoogle Scholar
[BMO-10]Berarducci, Alessandro, Mamino, Marcello, and Otero, Margarita, Higher homotopy of groups definable in o-minimal structures, Israel Journal of Mathematics, vol. 180 (2010), pp. 143–161.CrossRefGoogle Scholar
[Bor-61]Borel, Armand, Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes, The Tôhoku Mathematical Journal. Second Series, vol. 13 (1961), no. 2, pp. 216–240.Google Scholar
[HM-98]Hofmann, Karl H. and Morris, Sidney A., The structure of compact groups, de Gruyter Studies in Mathematics, vol. 25, Walter de Gruyter & Co., Berlin, 1998.Google Scholar
[HPP-08]Hrushovski, Ehud, Peterzil, Ya'acov, and Pillay, Anand, On central extensions and definably compact groups in o-minimal structures, arXiv:0811.0089vl, 2008.Google Scholar
[MS-92]Madden, James J. and Stanton, Charles M., One-dimensional Nash groups, Pacific Journal of Mathematics, vol. 154 (1992), no. 2, pp. 331–344.CrossRefGoogle Scholar
[Ote-08]Otero, Margarita, A survey on groups definable in o-minimal structures, Model theory with applications to algebra and analysis, Vol. 2, London Mathematical Society Lecture Note Series, vol. 350, Cambridge University Press, Cambridge, 2008, pp. 177–206.Google Scholar
[PS-99]Peterzil, Ya'acov and Steinhorn, Charles, Definable compactness and definable subgroups of o-minimal groups, Journal of the London Mathematical Society. Second Series, vol. 59 (1999), no. 3, pp. 769–786.CrossRefGoogle Scholar
[Pil-88]Pillay, Anand, On groups and fields definable in o-minimal structures, Journal of Pure and Applied Algebra, vol. 53 (1988), no. 3, pp. 239–255.CrossRefGoogle Scholar
[Ste-51]Steenrod, Norman, The topology of fibre bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N.J., 1951.CrossRefGoogle Scholar
[Str-94]Strzebonski, Adam W., Euler characteristic in semialgebraic and other o-minimal groups, Journal ofPure and Applied Algebra, vol. 96 (1994), no. 2, pp. 173–201.CrossRefGoogle Scholar
[vdD-98]van den Dries, Lou, Tame topology and o-minimal structures, vol. 248, Cambridge University Press, Cambridge, 1998.CrossRefGoogle Scholar