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 : https://doi.org/10.5802/aif.2300
Classification : 03C64,  55M20
Mots clés : Structures o-minimales, théorème de point fixe
@article{AIF_2007__57_5_1441_0,
     author = {Edmundo, M\'ario J.},
     title = {A fixed point theorem in o-minimal structures},
     journal = {Annales de l'Institut Fourier},
     pages = {1441--1450},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {5},
     year = {2007},
     doi = {10.5802/aif.2300},
     mrnumber = {2364135},
     zbl = {1127.03034},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2300/}
}
TY  - JOUR
AU  - Edmundo, Mário J.
TI  - A fixed point theorem in o-minimal structures
JO  - Annales de l'Institut Fourier
PY  - 2007
DA  - 2007///
SP  - 1441
EP  - 1450
VL  - 57
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2300/
UR  - https://www.ams.org/mathscinet-getitem?mr=2364135
UR  - https://zbmath.org/?q=an%3A1127.03034
UR  - https://doi.org/10.5802/aif.2300
DO  - 10.5802/aif.2300
LA  - en
ID  - AIF_2007__57_5_1441_0
ER  - 
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 | Article | MR 2000077 | Zbl 1060.03059

[2] Bochnak, J.; Coste, M.; Roy, M-F. Real algebraic geometry, Springer-Verlag, 1998 | MR 1659509 | Zbl 0912.14023

[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 1226249 | Zbl 0791.55003

[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 | Article | MR 667464 | Zbl 0484.14006

[6] Dold, A. Lectures on algebraic topology, Springer Verlag, 1995 | MR 1335915 | Zbl 0872.55001

[7] van den Dries, L. Tame topology and o-minimal structures, Cambridge University Press, 1998 | MR 1633348 | Zbl 0953.03045

[8] Edmundo, M.; Otero, M. Definably compact abelian groups, J. Math. Logic, Volume 4 (2004), pp. 163-180 | Article | MR 2114966 | Zbl 1070.03025

[9] Peterzil, Y.; Steinhorn, C. Definable compacteness and definable subgroups of o-minimal groups, J. London Math. Soc., Volume 59 (1999), pp. 769-786 | Article | MR 1709079 | Zbl 0935.03047

[10] Rotman, J. An introduction to algebraic topology, Springer Verlag, 1988 | MR 957919 | Zbl 0661.55001

[11] Woerheide, A. O-minimal homology (1996) (Ph. D. Thesis)

Cité par Sources :