Skip to main content
SpringerLink
Log in

Graph mappings and Poincaré duality

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We give a new geometric description for a compact, oriented, pseduo-manifold X of the Poincaré duality map from the integral cohomology of X to the integral homology of X. Our construction takes a multi-valued Lipschitz map on X with values in a sphere S n to its geometric-measure-theoretic graph in X × S n and then to the slice of this graph as an integral cycle on X. This construction is compatible with analogous constructions on algebraic cocycles on projective varieties employed by the authors and others.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abraham R., Robbin J.: Transversal Mappings and Flows. Benjamin, Reading (1967)

    MATH  Google Scholar 

  2. Almgren F. Jr: Homotopy groups of the integral cycle groups. Topology 1, 257–299 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  3. Dold A., Thom R.: Quasifaserungen und unendliche symmetrische Produkte. Ann. Math. 67, 230–281 (1956)

    MathSciNet  Google Scholar 

  4. Federer H.: Geometric Measure Theory. Springer, New York (1969)

    MATH  Google Scholar 

  5. Friedlander E.: Algebraic cocycles on normal, quasiprojective varieties. Compos. Math. 110, 127–162 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Friedlander, E., Gabber, O.: Cycle spaces and intersection theory. Topol. Methods Mod. Math. 325–370 (1993)

  7. Friedlander E., Lawson H.B. Jr: A theory of algebraic cocycles. Ann. Math. 136, 361–428 (1992)

    Article  MathSciNet  Google Scholar 

  8. Friedlander E., Lawson H.B. Jr: Duality for spaces of algebraic cocycles and algebraic cycles. Topology 36, 535–565 (1997)

    Article  MathSciNet  Google Scholar 

  9. Friedlander, Mazur, B.: Filtrations on the homology of algebraic varieties. Amer. Math. Soc. Memoir, A.M.S. 529 (1994)

  10. Friedlander, E., Voevodsky, V.: Bivariant cycle homology. In: Cycles, transfers, and Motivic Homology Theories. Annals of Math Studies, pp. 138–187 (2000)

  11. Gajer P.: Poincaré duality and integral cycles. Compos. Math. 98, 193–203 (1995)

    MATH  MathSciNet  Google Scholar 

  12. Griffiths, P., Morgan, J.: Rational homotopy theory and differential forms. Birkhauser (1981)

  13. Hironaka H.: Triangulation of algebraic sets. Algebraic Geometry Proc. Symp. Pure Math. 29, 165–185 (1975)

    Google Scholar 

  14. Hirsch M.: Differential Topology. Springer, New York (1976)

    MATH  Google Scholar 

  15. Hocking J., Young G.: Topology. Addison-Wesley, Reading (1961)

    MATH  Google Scholar 

  16. Hormander L.: Linear Partial Differential Operators. Springer, New York (1964)

    Google Scholar 

  17. Lima-Filho P.: Lawson homology for quasiprojective varieties. Compos. Math 84, 1–23 (1992)

    MATH  MathSciNet  Google Scholar 

  18. Lima-Filho P.: On the generalized cycle map. J. Differ. Geom. 38, 105–130 (1993)

    MATH  MathSciNet  Google Scholar 

  19. Milgram R.J.: The homology of symmetric products. Trans. Am. Math. Soc. 138, 251–266 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  20. Poincaré, H.: Compléments à l’Analysis situs. In: Poincaré, H. (ed.) Œuvres, vol. VI, pp. 189–192. Gauthier-Villars (1953)

  21. Shiota M., Yokoi M.: Triangulations of subanalytic sets and locally subanalytic manifolds. Trans. Am. Math. Soc. 286, 727–750 (1984)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Southern California, 3620 S. Vermont Ave, Los Angeles, CA, 90089-2532, USA

    Eric M. Friedlander

  2. Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794, USA

    H. Blaine Lawson Jr

Authors
  1. Eric M. Friedlander
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Blaine Lawson Jr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric M. Friedlander.

Additional information

E. M. Friedlander was partially supported by N.S.F. grant # 030005325. H. B. Lawson Jr was partially supported by the N.S.F.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Friedlander, E.M., Lawson, H.B. Graph mappings and Poincaré duality. Math. Ann. 343, 431–461 (2009). https://doi.org/10.1007/s00208-008-0278-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-008-0278-4

Keywords

Search

Navigation