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Transfer methods for o-minimal topology

Published online by Cambridge University Press:  12 March 2014

Alessandro Berarducci  and
Margarita Otero
Alessandro Berarducci
Affiliation:
Dipartimento di Matematica, Università di Pisa, Via Buonarroti 2, 56127 Pisa, Italy, E-mail: [email protected]
Margarita Otero
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain, E-mail: [email protected]
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Abstract

Let M be an o-minimal expansion of an ordered field. Let φ be a formula in the language of ordered domains. In this note we establish some topological properties which are transferred from φM to φR and vice versa. Then, we apply these transfer results to give a new proof of a result of M. Edmundo—based on the work of A. Strzebonski—showing the existence of torsion points in any definably compact group defined in an o-minimal expansion of an ordered field.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 3 , September 2003 , pp. 785 - 794
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[1] Berarducci, A. and Otero, M., Some examples of transfer methods for o-minimal structures, The Proceedings of the Meeting “Spring Stage on Logic, Algebra and Geometry”, Caserta, Italy, 21–24 03 2000, preprint.Google Scholar
[2] Berarducci, A. and Otero, M., Intersection theory for o-minimal manifolds, Annals of Pure and Applied Logic, vol. 107 (2001), pp. 87–119.CrossRefGoogle Scholar
[3] Berarducci, A. and Otero, M., O-minimal fundamental group, homology and manifolds, Journal of the London Mathematical Society, vol. 65 (2002), pp. 257–270.Google Scholar
[4] Brown, R., Elements of modern topology, McGraw-Hill, London, 1968.Google Scholar
[5] Brumfield, G. W., A Hopf fixed point theorem for semialgebraic maps, Real algebraic geometry, Proceedings of Rennes 1991, Lecture Notes in Mathematics, vol. 1524, Springer-Verlag, 1992.Google Scholar
[6] Delfs, H. and Knebusch, M., On the homology of algebraic varieties over real closed fields, Journal für die Reine und Angewandte Mathematik, vol. 335 (1982), pp. 122–163.Google Scholar
[7] Edmundo, M. J., O-minimal cohomology and definably compact definable groups, preprint, 2000 (revised version 2001).Google Scholar
[8] Galewski, D. E. and Stern, R. J., Classification of simplicial triangulations of topological manifolds, Annals of Mathematics, vol. 111 (1980), pp. 1–34.Google Scholar
[9] Hocking, J. G. and Young, G. S., Topology, Dover Publications, New York, 1988.Google Scholar
[10] Johns, J., An open mapping for o-minimal structures, this Journal, vol. 66 (2001), pp. 1817–1820.Google Scholar
[11] Peterzil, Y. Otero, M. and Pillay, A., Groups and rings definable in o-minimal expansions of real closed fields, Bulletin of the London Mathematical Society, vol. 28 (1996), pp. 7–14.Google Scholar
[12] Milnor, J. W. and Stasheff, J. D., Characteristic classes, Annals of Mathematics Studies, Princeton University Press, Princeton, 1974.CrossRefGoogle Scholar
[13] Peterzil, Y. and Steinhorn, C., Definable compactness and definable subgroups of o-minimal groups, Journal of the London Mathematical Society, vol. 59 (1999), pp. 769–786.Google Scholar
[14] Pillay, A., On groups and fields definable in o-minimal structures. Journal of Pure and Applied Algebra, vol. 53 (1988), pp. 239–255.CrossRefGoogle Scholar
[15] Shiota, M., Geometry of subanalytic and semialgebraic sets, Progress in Mathematics, Birkhäuser, Boston, 1997.CrossRefGoogle Scholar
[16] Strzebonski, A., Euler characteristic in semialgebraic and other o-minimal structures. Journal of Pure and Applied Algebra, vol. 96 (1994), pp. 173–201.Google Scholar
[17] Thurston, W. P., Three-dimensional geometry and topology, Princeton University Press, 1997.Google Scholar
[18] van den Dries, L., Tame topology and o-minimal structures, London Mathematical Society Lecture Notes Series, vol. 248, Cambridge University Press, 1998.CrossRefGoogle Scholar
[19] Vick, J. W., Homology theory, Springer-Verlag, 1994.CrossRefGoogle Scholar
[20] Woerheide, A., O-minimal homology, Ph.D. thesis , University of Illinois at Urbana-Champaign, 1996.Google Scholar