Comparison theorems for o-minimal singular (co)homology
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- by Mário J. Edmundo and Arthur Woerheide PDF
- Trans. Amer. Math. Soc. 360 (2008), 4889-4912 Request permission
Abstract:
Here we show the existence of the o-minimal simplicial and singular (co)homology in o-minimal expansions of real closed fields and prove several comparison theorems for o-minimal (co)homology theories.References
- Alessandro Berarducci and Margarita Otero, o-minimal fundamental group, homology and manifolds, J. London Math. Soc. (2) 65 (2002), no. 2, 257–270. MR 1883182, DOI 10.1112/S0024610702003502
- Jacek Bochnak, Michel Coste, and Marie-Françoise Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36, Springer-Verlag, Berlin, 1998. Translated from the 1987 French original; Revised by the authors. MR 1659509, DOI 10.1007/978-3-662-03718-8
- H.Delfs Kohomologie affiner semialgebraischer Räume Diss. Univ. Regensburg 1980.
- Hans Delfs, Homology of locally semialgebraic spaces, Lecture Notes in Mathematics, vol. 1484, Springer-Verlag, Berlin, 1991. MR 1176311, DOI 10.1007/BFb0093939
- Hans Delfs and Manfred Knebusch, On the homology of algebraic varieties over real closed fields, J. Reine Angew. Math. 335 (1982), 122–163. MR 667464, DOI 10.1515/crll.1982.335.122
- C. T. J. Dodson and Phillip E. Parker, A user’s guide to algebraic topology, Mathematics and its Applications, vol. 387, Kluwer Academic Publishers Group, Dordrecht, 1997. MR 1430097, DOI 10.1007/978-1-4615-6309-9
- Albrecht Dold, Lectures on algebraic topology, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 200, Springer-Verlag, Berlin-New York, 1980. MR 606196
- 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
- J. Denef and L. van den Dries, $p$-adic and real subanalytic sets, Ann. of Math. (2) 128 (1988), no. 1, 79–138. MR 951508, DOI 10.2307/1971463
- Lou van den Dries and Chris Miller, On the real exponential field with restricted analytic functions, Israel J. Math. 85 (1994), no. 1-3, 19–56. MR 1264338, DOI 10.1007/BF02758635
- Lou van den Dries and Patrick Speissegger, The real field with convergent generalized power series, Trans. Amer. Math. Soc. 350 (1998), no. 11, 4377–4421. MR 1458313, DOI 10.1090/S0002-9947-98-02105-9
- Lou van den Dries and Patrick Speissegger, The field of reals with multisummable series and the exponential function, Proc. London Math. Soc. (3) 81 (2000), no. 3, 513–565. MR 1781147, DOI 10.1112/S0024611500012648
- M.Edmundo Invariance of o-minimal singular (co)homology in elementary extensions Séminaire de Structure Algébriques Ordennées n. 76 (2004) (ed., F. Delon et al.) Publications de la Université de Paris VII.
- Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, N.J., 1952. MR 0050886
- Manfred Knebusch, Semialgebraic topology in the last ten years, Real algebraic geometry (Rennes, 1991) Lecture Notes in Math., vol. 1524, Springer, Berlin, 1992, pp. 1–36. MR 1226239, DOI 10.1007/BFb0084606
- Anand Pillay and Charles Steinhorn, Definable sets in ordered structures. I, Trans. Amer. Math. Soc. 295 (1986), no. 2, 565–592. MR 833697, DOI 10.1090/S0002-9947-1986-0833697-X
- 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
- S. Shelah, Classification theory and the number of nonisomorphic models, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990. MR 1083551
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
- A.Woerheide O-minimal homology Ph.D. Thesis (1996), University of Illinois at Urbana-Champaign.
- A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), no. 4, 1051–1094. MR 1398816, DOI 10.1090/S0894-0347-96-00216-0
Additional Information
- Mário J. Edmundo
- Affiliation: CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
- Arthur Woerheide
- Affiliation: 6203 S. Evans Avenue, Chicago, Illinois 60637
- Received by editor(s): November 13, 2003
- Received by editor(s) in revised form: December 16, 2004, November 3, 2005, July 17, 2006, and September 8, 2006
- Published electronically: April 9, 2008
- Additional Notes: This research was supported by the FCT grant SFRH/BPD/6015/2001 and partially by the European Research and Training Network HPRN-CT-2001-00271 RAAG
- © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 4889-4912
- MSC (2000): Primary 03C64, 55N10
- DOI: https://doi.org/10.1090/S0002-9947-08-04403-6
- MathSciNet review: 2403708