Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Computing o-minimal topological invariants using differential topology


Authors: Ya'acov Peterzil and Sergei Starchenko
Journal: Trans. Amer. Math. Soc. 359 (2007), 1375-1401
MSC (2000): Primary 03C64, 03C98, 57R99
DOI: https://doi.org/10.1090/S0002-9947-06-04220-6
Published electronically: October 24, 2006
MathSciNet review: 2262855
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. Berarducci, Alessandro, and Otero, Margarita, Intersection theory for o-minimal manifolds, Ann. Pure Appl. Logic, 107, (2001), no. 1-3, 87-119. MR 1807841 (2001m:03074)
  • 2. Berarducci, Alessandro, and Otero, Margarita, Transfer methods for o-minimal topology, J. Symbolic Logic, 68, (2003), no. 3, 785-794. MR 2000077 (2004h:03082)
  • 3. Coste, Michel, An introduction to O-minimal geometry, Raag Publications.
  • 4. van den Dries, Lou, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, 248, Cambridge University Press, Cambridge, 1998. MR 1633348 (99j:03001)
  • 5. Dubrovin, B. A., Fomenko, A. T., and Novikov, S. P., Modern geometry--methods and applications. Part II, Graduate Texts in Mathematics, 104, The geometry and topology of manifolds; Translated from the Russian by Robert G. Burns, Springer-Verlag, New York, 1985. MR 807945 (86m:53001)
  • 6. Edmundo, Mário J., Otero, Margarita, Definably compact abelian groups, J. Math. Log., 4, (2004), no. 2, 163-180. MR 2114966
  • 7. Hirsch, Morris W., Differential topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR 1336822 (96c:57001)
  • 8. Johns, Joseph, An open mapping theorem for o-minimal structures, J. Symbolic Logic, 66, (2001), no. 4, 1817-1820. MR 1877024 (2003f:03051)
  • 9. Ta Lé Loi, The existence of Morse functions on Sets definable in o-minimal structures, preprint.
  • 10. Milnor, J., Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR 0163331 (29:634)
  • 11. Peterzil, Ya'acov, Some topological and differential invariants--a survey of the solution to the torsion-point problem, in Proceedings of the 2001 Ravello meeting in honor of A. Macintyre 60's birthday. MR 2159722 (2006d:03062)
  • 12. Peterzil, Y., Pillay, A., and Starchenko, S., Definably simple groups in o-minimal structures, Trans. Amer. Math. Soc., 352, (2000), no. 10, 4397-4419. MR 1707202 (2001b:03036)
  • 13. Peterzil, Ya'acov, and Starchenko, Sergei, Complex geometry and analytic-geometric categories, preprint (2005).
  • 14. Peterzil, Y., Pillay, A., Starchenko, S., Definably simple groups in o-minimal structures, Trans. Amer. Math. Soc., 352, (2000), no, 10, 4397-4419. MR 1707202 (2001b:03036)
  • 15. Peterzil, Ya'acov, Steinhorn, Charles, Definable compactness and definable subgroups of o-minimal groups, J. London Math. Soc. (2), 59, (1999), no. 3, 769-786. MR 1709079 (2000i:03055)
  • 16. Pillay, Anand, On groups and fields definable in $ o$-minimal structures, J. Pure Appl. Algebra, 53, (1988), no. 3, 239-255. MR 961362 (89i:03069)
  • 17. Razenj, Vladimir, One-dimensional groups over an $ o$-minimal structure, Ann. Pure Appl. Logic, 53, (1991), no. 3, 269-277, MR 1129780 (93b:03053a)
  • 18. Shiota, Masahiro, Geometry of subanalytic and semialgebraic sets, Progress in Mathematics, 150, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1463945 (99b:14061)
  • 19. Strzebonski, Adam W., Euler characteristic in semialgebraic and other $ {\rm o}$-minimal groups, J. Pure Appl. Algebra, 96, (1994), no. 2, 173-201. MR 1303545 (95j:03067)
  • 20. Woerheide, A., O-minimal homology Ph.D. Thesis, University of Illinois at Urbana-Champaign, (1996).
  • 21. Wong, Kam-Chau, A fixed point theorem for o-minimal structures, MLQ Math. Log. Q., 49, (2003), no. 6, 598-602. MR 2013720 (2004m:03143)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03C64, 03C98, 57R99

Retrieve articles in all journals with MSC (2000): 03C64, 03C98, 57R99


Additional Information

Ya'acov Peterzil
Affiliation: Department of Mathematics, University of Haifa, Haifa, Israel
Email: kobi@math.haifa.ac.il

Sergei Starchenko
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: starchenko.1@nd.edu

DOI: https://doi.org/10.1090/S0002-9947-06-04220-6
Keywords: O-minimality
Received by editor(s): June 20, 2005
Published electronically: October 24, 2006
Additional Notes: The second author was supported in part by NSF Grant #0400163
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society