Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Algebraic cycles and algebraic models of smooth manifolds

Author: W. Kucharz
Journal: J. Algebraic Geom. 11 (2002), 101-127
Published electronically: November 16, 2001
MathSciNet review: 1865915
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Abstract | References | Additional Information

Abstract: By Tognoli's theorem, any smooth compact manifold $M$has an algebraic model, that is, there exists a nonsingular real algebraic set $X$ diffeomorphic to $M$. In fact, one can find an uncountable family of pairwise nonisomorphic algebraic models of $M$, assuming that $M$ has a positive dimension. In the present paper we are concerned with the group of homology classes on $X$ (with integer coefficients modulo $2$) that are represented by $d$-dimensional algebraic subsets of $X$. We investigate how this group varies as $X$ runs through the class of all algebraic models of $M$.

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

  • 1. M. Abánades and W. Kucharz, Algebraic equivalence of real algebraic cycles, Ann. Inst. Fourier (Grenoble), 49 6(1999), 1797-1804.
  • 2. S. Akbulut and H. King, The topology of real algebraic sets with isolated singularities, Ann. of Math. 113 (1981), 425-446.
  • 3. S. Akbulut and H. King, The topology of real algebraic sets, Enseign. Math. 29 (1983), 221-261.
  • 4. S. Akbulut and H. King, A resolution theorem for homology cycles of real algebraic varieties, Invent. Math. 79 (1985), 589-601.
  • 5. S. Akbulut and H. King, Topology of Real Algebraic Sets, Math. Sci. Research Institute Publ. 25, Springer 1992.
  • 6. R. Benedetti, On a resolution theorem for homology classes of a real algebraic variety, Boll. Un. Mat. Ital. A (6) 4 (1985), 459-466.
  • 7. R. Benedetti and M. Dedò, Counterexamples to representing homology classes by real algebraic subvarieties up to homeomorphism, Compositio Math. 53 (1984), 143-151.
  • 8. R. Benedetti and A. Tognoli, Théorèmes d'approximation en géométrie algébrique réelle. Sémin. sur la géometrie algébrique réelle, Publ. Math. Univ. Paris VII, 9 (1980), 123-145.
  • 9. R. Benedetti and A. Tognoli, On real algebraic vector bundles, Bull. Sci. Math. (2) 104 (1980), 89-112.
  • 10. R. Benedetti and A. Tognoli, Remarks and counterexamples in the theory of real vector bundles and cycles. Géométrie algébrique réelle et formes quadratiques. Lecture Notes in Math. 959, 198-211, Springer, 1982.
  • 11. J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry, Ergebnisse der Math. und ihrer Grenzgeb. Folge 3, Vol. 36, Berlin Heidelberg, New York, Springer, 1998.
  • 12. J. Bochnak, M. Buchner and W. Kucharz, Vector bundles over real algebraic varieties, K-Theory 3 (1989), 271-298. Erratum, K-Theory 4 (1990), p.103.
  • 13. J. Bochnak and W. Kucharz, K-theory of real algebraic surfaces and threefolds, Math. Proc. Cambridge Phil. Soc. 106 (1989), 471-480.
  • 14. J. Bochnak and W. Kucharz, Algebraic models of smooth manifolds, Invent. Math. 97 (1989), 585-611.
  • 15. J. Bochnak and W. Kucharz, Nonisomorphic algebraic models of a smooth manifold, Math. Ann. 290 (1991), 1-2.
  • 16. J. Bochnak and W. Kucharz, Algebraic cycles and approximation theorems in real algebraic geometry, Trans. Amer. Math. Soc. 337 (1993), 463-472.
  • 17. J. Bochnak and W. Kucharz, On homology classes represented by real algebraic varieties, Banach Center Publications Vol. 44, 21-35, Warsaw, 1998.
  • 18. A. Borel and A. Haefliger, La classe d'homologie fondamentale d'un espace analytique, Bull. Soc. Math. France 89 (1961), 461-513.
  • 19. P. E. Conner, Differentiable Periodic Maps, 2nd Edition, Lecture Notes in Math. 738, Springer, 1979.
  • 20. A. Dold, Lectures on Algebraic Topology, Grundlehren Math. Wiss. Vol. 200, Berlin Heidelberg New York, Springer, 1972.
  • 21. W. Fulton, Intersection Theory, Ergebnisse der Math. und ihrer Grenzgeb. Folge 3, Vol. 2, Berlin Heidelberg New York, Springer, 1984.
  • 22. M. Hirsch, Differential Topology, Graduate Texts in Math. Vol. 33, New York Heidelberg Berlin, Springer, 1976.
  • 23. S. T. Hu, Homotopy Theory, New York, Academic Press, 1959.
  • 24. D. Husemoller, Fibre Bundles, Berlin New York Springer, 1975.
  • 25. W. Kucharz, Algebraic equivalence and homology classes of real algebraic cycles, Math. Nachr. 180 (1996), 135-140.
  • 26. W. Kucharz, Algebraic morphisms into rational real algebraic surfaces, J. Algebraic Geometry 8 (1999), 569-579.
  • 27. J. Milnor and J. Stasheff, Characteristic Classes, Ann. of Math. Studies 76, Princeton Univ. Press, Princeton, New Jersey, 1974.
  • 28. M. Shiota, Equivalence of differentiable functions, rational functions and polynomials, Ann. Inst. Fourier (Grenoble) 32, 4 (1982), 167-204.
  • 29. R. Silhol, A bound on the order of $H^{(a)}_{n-1}(X,Z/2)$ on a real algebraic variety. Lecture Notes in Math. 959, 443-450, Springer, 1982.
  • 30. E. Spanier, Algebraic Topology, New York, McGraw-Hill, 1966.
  • 31. N. Steenrod, The Topology of Fibre Bundles, Princeton Univ. Press, Princeton, New Jersey, 1951.
  • 32. P. Teichner, 6-dimensional manifolds without totally algebraic homology, Proc. Amer. Math. Soc. 123 (1995), 2909-2914.
  • 33. R. Thom, Quelques propriétés globales de variétés différentiables, Comment. Math. Helvetici 28 (1954), 17-86.
  • 34. A. Tognoli, Su una congettura di Nash, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. (3) 27 (1973), 167-185.

Additional Information

W. Kucharz
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
Address at time of publication: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131-1141

Received by editor(s): March 24, 2000
Received by editor(s) in revised form: May 2, 2000
Published electronically: November 16, 2001
Additional Notes: The author was partially supported by NSF Grant DMS-9503138. The paper was completed at the Max-Planck-Institut für Mathematik in Bonn, whose support and hospitality is gratefully acknowledged

American Mathematical Society