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)



Comparison theorems for o-minimal singular (co)homology

Authors: Mário J. Edmundo and Arthur Woerheide
Journal: Trans. Amer. Math. Soc. 360 (2008), 4889-4912
MSC (2000): Primary 03C64, 55N10
Published electronically: April 9, 2008
MathSciNet review: 2403708
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

  • [bo] A.Berarducci and M.Otero O-minimal fundamental group, homology and manifolds J. London Math. Soc. 65 (2) (2002) 257-270. MR 1883182 (2003b:03048)
  • [BCR] J.Bochnak, M. Coste and M-F. Roy Real algebraic geometry Springer-Verlag 1998. MR 1659509 (2000a:14067)
  • [D1] H.Delfs Kohomologie affiner semialgebraischer Räume Diss. Univ. Regensburg 1980.
  • [D2] H.Delfs Homology of locally semialgebraic spaces Lecture Notes in Mathematics 1484 Springer-Verlag 1991. MR 1176311 (94c:14047)
  • [dk] H.Delfs and M.Knebusch On the homology of algebraic varieties over real closed fields J. reine angew. Math. 335 (1982) 122-163. MR 667464 (84b:14010)
  • [dp] C.Dodson and P.Parker A user's guide to algebraic topology Kluwer Academic Publishers 1997. MR 1430097 (97j:55001)
  • [d] A.Dold Lectures on algebraic topology Springer-Verlag 1980. MR 606196 (82c:55001)
  • [vdd] L. van den Dries Tame topology and o-minimal structures Cambridge University Press 1998. MR 1633348 (99j:03001)
  • [dd] J.Denef and L.van den Dries $ p$-adic and real subanalytic sets Ann. Math. 128 (1988) 79-138. MR 951508 (89k:03034)
  • [dm] L. van den Dries and C.Miller On the real exponential field with restricted analytic functions Israel J. Math. 85 (1994) 19-56. MR 1264338 (95e:03099)
  • [ds1] L. van den Dries and P. Speissegger The real field with convergent generalized power series Trans. Amer. Math. Soc. 350 (11) (1998) 4377-4421. MR 1458313 (99a:03036)
  • [ds2] L. van den Dries and P. Speissegger The field of reals with multisummable series and the exponential function Proc. London Math. Soc. 81 (3) (2000) 513-565. MR 1781147 (2002k:03057)
  • [e] 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.
  • [es] S.Eilenberg and N.Steenrod Foundations of algebraic topology Princeton University Press 1952. MR 0050886 (14:398b)
  • [k] M.Knebusch Semi-algebraic topology in the last ten years in Real Algebraic Geometry, Lecture Notes in Mathematics 1524, Springer-Verlag 1992. MR 1226239 (95d:14055)
  • [KPS] J.Knight, A.Pillay and C. Steinhorn Definable sets in ordered structures II Trans. Amer. Math. Soc. 295 (1986) 593-605. MR 833698 (88b:03050b)
  • [ps] Y.Peterzil and C.Steinhorn Definable compacteness and definable subgroups of o-minimal groups J. London Math. Soc. 59 (2) (1999) 769-786. MR 1709079 (2000i:03055)
  • [Sh] S.Shelah Classification theory and the number of non isomorphic models 2nd ed. revised, North-Holland 1990. MR 1083551 (91k:03085)
  • [sp] E. H. Spanier Algebraic topology McGraw-Hill 1966. MR 0210112 (35:1007)
  • [Wo] A.Woerheide O-minimal homology Ph.D. Thesis (1996), University of Illinois at Urbana-Champaign.
  • [w] A.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) 1051-1094. MR 1398816 (98j:03052)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03C64, 55N10

Retrieve articles in all journals with MSC (2000): 03C64, 55N10

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

Keywords: O-minimal singular homology and cohomology.
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
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society