Computing o-minimal topological invariants using differential topology

Peterzil, Ya’acov; Starchenko, Sergei

doi:10.1090/S0002-9947-06-04220-6

Computing o-minimal topological invariants using differential topology
HTML articles powered by AMS MathViewer

by Ya’acov Peterzil and Sergei Starchenko PDF

Trans. Amer. Math. Soc. 359 (2007), 1375-1401 Request permission

Abstract:

We work in an o-minimal expansion of a real closed field. Using piecewise smoothness of definable functions we define the topological degree for definable continuous functions. Using this notion of the degree we obtain a new proof for the existence of torsion points in a definably compact group, and also a new proof of an o-minimal analogue of the Brouwer fixed point theorem.

References

Alessandro Berarducci and Margarita Otero, Intersection theory for o-minimal manifolds, Ann. Pure Appl. Logic 107 (2001), no. 1-3, 87–119. MR 1807841, DOI 10.1016/S0168-0072(00)00027-0
Alessandro Berarducci and Margarita Otero, Transfer methods for o-minimal topology, J. Symbolic Logic 68 (2003), no. 3, 785–794. MR 2000077, DOI 10.2178/jsl/1058448438
Coste, Michel, An introduction to O-minimal geometry, Raag Publications.
Lou van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348, DOI 10.1017/CBO9780511525919
B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern geometry—methods and applications. Part II, Graduate Texts in Mathematics, vol. 104, Springer-Verlag, New York, 1985. The geometry and topology of manifolds; Translated from the Russian by Robert G. Burns. MR 807945, DOI 10.1007/978-1-4612-1100-6
Mário J. Edmundo and Margarita Otero, Definably compact abelian groups, J. Math. Log. 4 (2004), no. 2, 163–180. MR 2114966, DOI 10.1142/S0219061304000358
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR 1336822
Joseph Johns, An open mapping theorem for o-minimal structures, J. Symbolic Logic 66 (2001), no. 4, 1817–1820. MR 1877024, DOI 10.2307/2694977
Ta Lé Loi, The existence of Morse functions on Sets definable in o-minimal structures, preprint.
J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331, DOI 10.1515/9781400881802
Ya’acov Peterzil, Some topological and differentiable invariants in o-minimal structures—a survey of the solution to the torsion point problem, Model theory and applications, Quad. Mat., vol. 11, Aracne, Rome, 2002, pp. 307–323. MR 2159722
Y. Peterzil, A. Pillay, and S. Starchenko, Definably simple groups in o-minimal structures, Trans. Amer. Math. Soc. 352 (2000), no. 10, 4397–4419. MR 1707202, DOI 10.1090/S0002-9947-00-02593-9
Peterzil, Ya’acov, and Starchenko, Sergei, Complex geometry and analytic-geometric categories, preprint (2005).
Y. Peterzil, A. Pillay, and S. Starchenko, Definably simple groups in o-minimal structures, Trans. Amer. Math. Soc. 352 (2000), no. 10, 4397–4419. MR 1707202, DOI 10.1090/S0002-9947-00-02593-9
Ya’acov Peterzil and Charles Steinhorn, Definable compactness and definable subgroups of o-minimal groups, J. London Math. Soc. (2) 59 (1999), no. 3, 769–786. MR 1709079, DOI 10.1112/S0024610799007528
Anand Pillay, On groups and fields definable in $o$-minimal structures, J. Pure Appl. Algebra 53 (1988), no. 3, 239–255. MR 961362, DOI 10.1016/0022-4049(88)90125-9
Vladimir Razenj, One-dimensional groups over an $o$-minimal structure, Ann. Pure Appl. Logic 53 (1991), no. 3, 269–277. MR 1129780, DOI 10.1016/0168-0072(91)90024-G
Masahiro Shiota, Geometry of subanalytic and semialgebraic sets, Progress in Mathematics, vol. 150, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1463945, DOI 10.1007/978-1-4612-2008-4
Adam W. Strzebonski, Euler characteristic in semialgebraic and other $\textrm {o}$-minimal groups, J. Pure Appl. Algebra 96 (1994), no. 2, 173–201. MR 1303545, DOI 10.1016/0022-4049(94)90127-9
Woerheide, A., O-minimal homology Ph.D. Thesis, University of Illinois at Urbana-Champaign, (1996).
Kam-Chau Wong, A fixed point theorem for o-minimal structures, MLQ Math. Log. Q. 49 (2003), no. 6, 598–602. MR 2013720, DOI 10.1002/malq.200310065