Skip to main content
SpringerLink
Log in

Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin class

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based on the notion of a universal local formula introduced by the author and on the usage of bistellar moves. We give a brief sketch for the first formula and a rather detailed exposition for the second one. For the second formula, we also succeed to simplify it by providing a new simpler algorithm for decomposing a cycle in the graph of bistellar moves of two-dimensional combinatorial spheres into a linear combination of elementary cycles.

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. A. M. Gabriélov, I. M. Gel’fand, and M. V. Losik, “Combinatorial Computation of Characteristic Classes,” Funkts. Anal. Prilozh. 9(2), 12–28 (1975) [Funct. Anal. Appl. 9, 103–115 (1975)]; Funkts. Anal. Prilozh. 9 (3), 5–26 (1975) [Funct. Anal. Appl. 9, 186–202 (1975)].

    Google Scholar 

  2. A. M. Gabrielov, I. M. Gel’fand, and M. V. Losik, “A Local Combinatorial Formula for the First Class of Pontryagin,” Funkts. Anal. Prilozh. 10(1), 14–17 (1976) [Funct. Anal. Appl. 10, 12–15 (1976)].

    MathSciNet  Google Scholar 

  3. R. MacPherson, “The Combinatorial Formula of Gabrielov, Gelfand and Losik for the First Pontrjagin Class,” in Sèminaire Bourbaki 1976/77 (Springer, Berlin, 1978), Exp. 497, Lect. Notes Math. 677, pp. 105–124.

    Chapter  Google Scholar 

  4. A. M. Gabriélov, “Combinatorial Formulas for Pontryagin Classes and GL-Invariant Chains,” Funkts. Anal. Prilozh. 12(2), 1–7 (1978) [Funct. Anal. Appl. 12, 75–80 (1978)].

    MATH  Google Scholar 

  5. I. M. Gelfand and R. D. MacPherson, “A Combinatorial Formula for the Pontrjagin Classes,” Bull. Am. Math. Soc. 26(2), 304–309 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Cheeger, “Spectral Geometry of Singular Riemannian Spaces,” J. Diff. Geom. 18(4), 575–657 (1983).

    MATH  MathSciNet  Google Scholar 

  7. A. A. Gaifullin, “Local Formulae for Combinatorial Pontryagin Classes,” Izv. Ross. Akad. Nauk, Ser. Mat. 68(5), 13–66 (2004) [Izv. Math. 68, 861–910 (2004)].

    MathSciNet  Google Scholar 

  8. A. A. Gaifullin, “The Construction of Combinatorial Manifolds with Prescribed Sets of Links of Vertices,” Izv. Ross. Akad. Nauk, Ser. Mat. 72(5), 3–62 (2008) [Izv. Math. 72, 845–899 (2008)].

    MathSciNet  Google Scholar 

  9. N. Levitt and C. Rourke, “The Existence of Combinatorial Formulae for Characteristic Classes,” Trans. Am. Math. Soc. 239, 391–397 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  10. A. A. Gaifullin, “Computation of Characteristic Classes of a Manifold from a Triangulation of It,” Usp. Mat. Nauk 60(4), 37–66 (2005) [Russ. Math. Surv. 60, 615–644 (2005)].

    MathSciNet  Google Scholar 

  11. E. Stiefel, “Richtungsfelder und Fernparallelismus in n-dimensionalen Mannigfaltigkeiten,” Comment. Math. Helv. 8, 305–353 (1936).

    Article  MathSciNet  Google Scholar 

  12. H. Whitney, “On the Theory of Sphere Bundles,” Proc. Natl. Acad. Sci. USA 26, 148–153 (1940).

    Article  MATH  MathSciNet  Google Scholar 

  13. S. Halperin and D. Toledo, “Stiefel-Whitney Homology Classes,” Ann. Math., Ser. 2,96, 511–525 (1972).

    Article  MathSciNet  Google Scholar 

  14. J. Cheeger, “A Combinatorial Formula for Stiefel-Whitney Classes,” in Topology of Manifolds: Proc. Univ. Georgia, 1969 (Markham Publ., Chicago, 1970), pp. 470–471.

    Google Scholar 

  15. M. F. Atiyah, V. K. Patodi, and I. M. Singer, “Spectral Asymmetry and Riemannian Geometry,” Bull. London Math. Soc. 5(2), 229–234 (1973).

    Article  MATH  MathSciNet  Google Scholar 

  16. C. P. Rourke and B. J. Sanderson, Introduction to Piecewise-Linear Topology (Springer, Berlin, 1972).

    MATH  Google Scholar 

  17. J. H. C. Whitehead, “Note on Manifolds,” Q. J. Math., Oxford Ser. 12, 26–29 (1941).

  18. S. S. Cairns, “Isotopic Deformations of Geodesic Complexes on the 2-Sphere and on the Plane,” Ann. Math., Ser. 2., 45(2), 207–217 (1944).

    Article  MathSciNet  Google Scholar 

  19. C.-W. Ho, “On Certain Homotopy Properties of Some Spaces of Linear and Piecewise Linear Homeomorphisms. I, II,” Trans. Am. Math. Soc. 181, 213–233, 235–243 (1973).

    Article  MATH  Google Scholar 

  20. U. Pachner, “Konstruktionsmethoden und das kombinatorische Homöomorphieproblem für Triangulationen kompakter semilinearer Mannigfaltigkeiten,” Abh. Math. Semin. Univ. Hamburg 57, 69–86 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  21. U. Pachner, “P.L. Homeomorphic Manifolds Are Equivalent by Elementary Shellings,” Eur. J. Comb. 12(2), 129–145 (1991).

    MATH  MathSciNet  Google Scholar 

  22. M. È. Kazarian, “The Chern-Euler Number of Circle Bundle via Singularity Theory,” Math. Scand. 82, 207–236 (1998).

    MATH  MathSciNet  Google Scholar 

  23. W. Kühnel and T. F. Banchoff, “The 9-Vertex Complex Projective Plane,” Math. Intell. 5(3), 11–22 (1983).

    Article  MATH  Google Scholar 

  24. A. Björner and F. H. Lutz, “Simplicial Manifolds, Bistellar Flips and a 16-Vertex Triangulation of the Poincaré Homology 3-Sphere,” Exp. Math. 9(2), 275–289 (2000).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow State University, Moscow, 119991, Russia

    Alexander A. Gaifullin

  2. Institute for Information Transmission Problems, Russian Academy of Sciences, Bol’shoi Karetnyi per. 19, Moscow, 127994, Russia

    Alexander A. Gaifullin

Authors
  1. Alexander A. Gaifullin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander A. Gaifullin.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2010, Vol. 268, pp. 76–93.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gaifullin, A.A. Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin class. Proc. Steklov Inst. Math. 268, 70–86 (2010). https://doi.org/10.1134/S0081543810010074

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543810010074

Keywords

Search

Navigation