Algebraic cycles and algebraic models of smooth manifolds
Author:
W. Kucharz
Journal:
J. Algebraic Geom. 11 (2002), 101-127
DOI:
https://doi.org/10.1090/S1056-3911-01-00292-2
Published electronically:
November 16, 2001
MathSciNet review:
1865915
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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 $M$.
Aban M. Abánades and W. Kucharz, Algebraic equivalence of real algebraic cycles, Ann. Inst. Fourier (Grenoble), 49 6(1999), 1797-1804.
Akbulut S. Akbulut and H. King, The topology of real algebraic sets with isolated singularities, Ann. of Math. 113 (1981), 425-446.
Akbulut1 S. Akbulut and H. King, The topology of real algebraic sets, Enseign. Math. 29 (1983), 221-261.
Akbulut1a S. Akbulut and H. King, A resolution theorem for homology cycles of real algebraic varieties, Invent. Math. 79 (1985), 589-601.
Akbulut2 S. Akbulut and H. King, Topology of Real Algebraic Sets, Math. Sci. Research Institute Publ. 25, Springer 1992.
Bened1 R. Benedetti, On a resolution theorem for homology classes of a real algebraic variety, Boll. Un. Mat. Ital. A (6) 4 (1985), 459-466.
Bened R. Benedetti and M. Dedò, Counterexamples to representing homology classes by real algebraic subvarieties up to homeomorphism, Compositio Math. 53 (1984), 143-151.
Bened2 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.
Bened3 R. Benedetti and A. Tognoli, On real algebraic vector bundles, Bull. Sci. Math. (2) 104 (1980), 89-112.
Bened4 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.
BCR 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.
BBK 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.
BK J. Bochnak and W. Kucharz, K-theory of real algebraic surfaces and threefolds, Math. Proc. Cambridge Phil. Soc. 106 (1989), 471-480.
BK2 J. Bochnak and W. Kucharz, Algebraic models of smooth manifolds, Invent. Math. 97 (1989), 585-611.
BK3 J. Bochnak and W. Kucharz, Nonisomorphic algebraic models of a smooth manifold, Math. Ann. 290 (1991), 1-2.
BK4 J. Bochnak and W. Kucharz, Algebraic cycles and approximation theorems in real algebraic geometry, Trans. Amer. Math. Soc. 337 (1993), 463-472.
BK5 J. Bochnak and W. Kucharz, On homology classes represented by real algebraic varieties, Banach Center Publications Vol. 44, 21-35, Warsaw, 1998.
Borel A. Borel and A. Haefliger, La classe d’homologie fondamentale d’un espace analytique, Bull. Soc. Math. France 89 (1961), 461-513.
Conner P. E. Conner, Differentiable Periodic Maps, 2nd Edition, Lecture Notes in Math. 738, Springer, 1979.
Dold A. Dold, Lectures on Algebraic Topology, Grundlehren Math. Wiss. Vol. 200, Berlin Heidelberg New York, Springer, 1972.
Fulton W. Fulton, Intersection Theory, Ergebnisse der Math. und ihrer Grenzgeb. Folge 3, Vol. 2, Berlin Heidelberg New York, Springer, 1984.
Hirsch M. Hirsch, Differential Topology, Graduate Texts in Math. Vol. 33, New York Heidelberg Berlin, Springer, 1976.
Hu S. T. Hu, Homotopy Theory, New York, Academic Press, 1959.
Husemoller D. Husemoller, Fibre Bundles, Berlin New York Springer, 1975.
Kucharz W. Kucharz, Algebraic equivalence and homology classes of real algebraic cycles, Math. Nachr. 180 (1996), 135-140.
Kucharz2 W. Kucharz, Algebraic morphisms into rational real algebraic surfaces, J. Algebraic Geometry 8 (1999), 569-579.
Milnor J. Milnor and J. Stasheff, Characteristic Classes, Ann. of Math. Studies 76, Princeton Univ. Press, Princeton, New Jersey, 1974.
Shiota M. Shiota, Equivalence of differentiable functions, rational functions and polynomials, Ann. Inst. Fourier (Grenoble) 32, 4 (1982), 167-204.
Silhol 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.
Spanier E. Spanier, Algebraic Topology, New York, McGraw-Hill, 1966.
Steenrod N. Steenrod, The Topology of Fibre Bundles, Princeton Univ. Press, Princeton, New Jersey, 1951.
Teichner P. Teichner, 6-dimensional manifolds without totally algebraic homology, Proc. Amer. Math. Soc. 123 (1995), 2909-2914.
Thom R. Thom, Quelques propriétés globales de variétés différentiables, Comment. Math. Helvetici 28 (1954), 17-86.
Tognoli A. Tognoli, Su una congettura di Nash, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. (3) 27 (1973), 167-185.
Aban M. Abánades and W. Kucharz, Algebraic equivalence of real algebraic cycles, Ann. Inst. Fourier (Grenoble), 49 6(1999), 1797-1804.
Akbulut S. Akbulut and H. King, The topology of real algebraic sets with isolated singularities, Ann. of Math. 113 (1981), 425-446.
Akbulut1 S. Akbulut and H. King, The topology of real algebraic sets, Enseign. Math. 29 (1983), 221-261.
Akbulut1a S. Akbulut and H. King, A resolution theorem for homology cycles of real algebraic varieties, Invent. Math. 79 (1985), 589-601.
Akbulut2 S. Akbulut and H. King, Topology of Real Algebraic Sets, Math. Sci. Research Institute Publ. 25, Springer 1992.
Bened1 R. Benedetti, On a resolution theorem for homology classes of a real algebraic variety, Boll. Un. Mat. Ital. A (6) 4 (1985), 459-466.
Bened R. Benedetti and M. Dedò, Counterexamples to representing homology classes by real algebraic subvarieties up to homeomorphism, Compositio Math. 53 (1984), 143-151.
Bened2 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.
Bened3 R. Benedetti and A. Tognoli, On real algebraic vector bundles, Bull. Sci. Math. (2) 104 (1980), 89-112.
Bened4 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.
BCR 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.
BBK 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.
BK J. Bochnak and W. Kucharz, K-theory of real algebraic surfaces and threefolds, Math. Proc. Cambridge Phil. Soc. 106 (1989), 471-480.
BK2 J. Bochnak and W. Kucharz, Algebraic models of smooth manifolds, Invent. Math. 97 (1989), 585-611.
BK3 J. Bochnak and W. Kucharz, Nonisomorphic algebraic models of a smooth manifold, Math. Ann. 290 (1991), 1-2.
BK4 J. Bochnak and W. Kucharz, Algebraic cycles and approximation theorems in real algebraic geometry, Trans. Amer. Math. Soc. 337 (1993), 463-472.
BK5 J. Bochnak and W. Kucharz, On homology classes represented by real algebraic varieties, Banach Center Publications Vol. 44, 21-35, Warsaw, 1998.
Borel A. Borel and A. Haefliger, La classe d’homologie fondamentale d’un espace analytique, Bull. Soc. Math. France 89 (1961), 461-513.
Conner P. E. Conner, Differentiable Periodic Maps, 2nd Edition, Lecture Notes in Math. 738, Springer, 1979.
Dold A. Dold, Lectures on Algebraic Topology, Grundlehren Math. Wiss. Vol. 200, Berlin Heidelberg New York, Springer, 1972.
Fulton W. Fulton, Intersection Theory, Ergebnisse der Math. und ihrer Grenzgeb. Folge 3, Vol. 2, Berlin Heidelberg New York, Springer, 1984.
Hirsch M. Hirsch, Differential Topology, Graduate Texts in Math. Vol. 33, New York Heidelberg Berlin, Springer, 1976.
Hu S. T. Hu, Homotopy Theory, New York, Academic Press, 1959.
Husemoller D. Husemoller, Fibre Bundles, Berlin New York Springer, 1975.
Kucharz W. Kucharz, Algebraic equivalence and homology classes of real algebraic cycles, Math. Nachr. 180 (1996), 135-140.
Kucharz2 W. Kucharz, Algebraic morphisms into rational real algebraic surfaces, J. Algebraic Geometry 8 (1999), 569-579.
Milnor J. Milnor and J. Stasheff, Characteristic Classes, Ann. of Math. Studies 76, Princeton Univ. Press, Princeton, New Jersey, 1974.
Shiota M. Shiota, Equivalence of differentiable functions, rational functions and polynomials, Ann. Inst. Fourier (Grenoble) 32, 4 (1982), 167-204.
Silhol 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.
Spanier E. Spanier, Algebraic Topology, New York, McGraw-Hill, 1966.
Steenrod N. Steenrod, The Topology of Fibre Bundles, Princeton Univ. Press, Princeton, New Jersey, 1951.
Teichner P. Teichner, 6-dimensional manifolds without totally algebraic homology, Proc. Amer. Math. Soc. 123 (1995), 2909-2914.
Thom R. Thom, Quelques propriétés globales de variétés différentiables, Comment. Math. Helvetici 28 (1954), 17-86.
Tognoli 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
Email:
kucharz@math.unm.edu
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
