A fixed point theorem in o-minimal structures
[Théorème de point fixe dans les structures o-minimal.]
Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1441-1450.

Dans cet article nous montrons un théorème de point fixe o-minimal pour les applications définissables continues sur les ensembles définissables et définissablement compacts, qui généralise la version de Brumfiel du théorème de point fixe de Hopf pour les applications semi-algébriques.

Here we prove an o-minimal fixed point theorem for definable continuous maps on definably compact definable sets, generalizing Brumfiel’s version of the Hopf fixed point theorem for semi-algebraic maps.

DOI : 10.5802/aif.2300
Classification : 03C64, 55M20
Keywords: O-minimal structures, fixed point theorems
Mot clés : Structures o-minimales, théorème de point fixe
Edmundo, Mário J. 1

1 Universidade de Lisboa CMAF Av. Prof. Gama Pinto 2 1649-003 Lisboa (Portugal)
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Edmundo, Mário J. A fixed point theorem in o-minimal structures. Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1441-1450. doi : 10.5802/aif.2300. http://archive.numdam.org/articles/10.5802/aif.2300/

[1] Berarducci, A.; Otero, M. Transfer methods for o-minimal topology, J. Symbolic Logic, Volume 68 (2003), pp. 785-794 | DOI | MR | Zbl

[2] Bochnak, J.; Coste, M.; Roy, M-F. Real algebraic geometry, Springer-Verlag, 1998 | MR | Zbl

[3] Brumfiel, G. W. A Hopf fixed point theorem for semi-algebraic maps, Lecture Notes in Math. 1524, Springer Verlag, Berlin, 1992 Real algebraic geometry (Rennes, 1991) | MR | Zbl

[4] Coste, M. An introduction to o-minimal geometry Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa (2000). Available in RAAG preprint server 2000, http://ihp-raag.org/

[5] Delfs, H.; Knebusch, M. On the homology of algebraic varieties over real closed fields, J. reine u.angew. Math., Volume 335 (1982), pp. 122-163 | DOI | MR | Zbl

[6] Dold, A. Lectures on algebraic topology, Springer Verlag, 1995 | MR | Zbl

[7] van den Dries, L. Tame topology and o-minimal structures, Cambridge University Press, 1998 | MR | Zbl

[8] Edmundo, M.; Otero, M. Definably compact abelian groups, J. Math. Logic, Volume 4 (2004), pp. 163-180 | DOI | MR | Zbl

[9] Peterzil, Y.; Steinhorn, C. Definable compacteness and definable subgroups of o-minimal groups, J. London Math. Soc., Volume 59 (1999), pp. 769-786 | DOI | MR | Zbl

[10] Rotman, J. An introduction to algebraic topology, Springer Verlag, 1988 | MR | Zbl

[11] Woerheide, A. O-minimal homology, University of Illinois at Urbana-Champaign (1996) (Ph. D. Thesis PhD. Thesis)

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