Communications in Mathematical Physics

, Volume 178, Issue 1, pp 225–236 | Cite as

Čech cocycles for characteristic classes

  • J. -L. Brylinski
  • D. A. McLaughlin


We give general formulae for explicit Čech cocycles representing characteristic classes of real and complex vector bundles, as well as for cocycles representing Chern-Simons classes of bundles with arbitrary connections. Our formulae involve integrating differential forms over moving simplices inside homogeneous spaces. An important feature of our cocycles is that they take integer values (as opposed to real or rational values). We find in particular a formula for the instanton number of a connection over a closed four-manifold with arbitrary structure group. For flat connections, our formulae recover and generalize those of Cheeger and Simons. The methods of this paper apply also to the purely geometric construction of the Quillen line bundle with its metric.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Vector Bundle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beilinson, A.: Higher regulators and values of L-functions. J. Sov. Math.30, 2036–2070 (1985)Google Scholar
  2. 2.
    Beresin, F.A., Retakh, V.S.: A method of computing characteristic classes of vector bundles. Rep. Math. Phys.18, 363–378 (1980)Google Scholar
  3. 3.
    Borel, A.: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. Math. (2)57, 115–207 (1953)Google Scholar
  4. 4.
    Borel, A., Hirzebruch, F.: Characteristic classes and homogeneous spaces 2. Am. J. Math.81, 315–382 (1959)Google Scholar
  5. 5.
    Bott, R., Tu, L.: “Differential forms in algebraic topology”. Berlin, Heidelberg, New York: Springer, 1982Google Scholar
  6. 6.
    Brylinski, J.-L., McLaughlin, D.A.: A geometric construction of the first Pontryagin class. In: Quantum Topology, Series on Knots and Everything. L. Kauffman, R. Baadhio, (eds.) Singapore: World Scientific, 1993, pp. 209–220Google Scholar
  7. 7.
    Brylinski, J.-L., McLaughlin, D.A.: Holomorphic quantization and unitary representations of the Teichmüller group. In: Lie Theory and Geometry: In honor of Bertram Kostant, Progress in Math., Birkhaüser, vol.123, 1994, pp. 21–64Google Scholar
  8. 8.
    Brylinski, J.-L., McLaughlin, D.A.: The geometry of degree 4 characteristic classes and of line bundles on loop spaces I. Duke Math. J.75, 603–638 (1994)Google Scholar
  9. 9.
    Brylinski, J.-L., McLaughlin, D.A.: The geometry of degree 4 characteristic classes and of line bundles on loop spaces II. Preprint (1995). To appear in Duke Math. J.Google Scholar
  10. 10.
    Cheeger, J.: Spectral geometry of singular riemannian spaces. J. Diff. Geom.18, 575–657 (1983)Google Scholar
  11. 11.
    Cheeger, J., Simons, J.: Differential characters and geometric invariants. Lecture Notes in Math. vol.1167, Berlin, Heidelberg, New York: Springer, 1985, pp. 50–80Google Scholar
  12. 12.
    Chern, S.S., Simons, J.: Characteristic forms and geometric invariants. Ann. Math.99, 48–69 (1974)Google Scholar
  13. 13.
    Dupont, J.: The dilogarithm as a characteristic class for flat bundles. J. Pure Appl. Alg.44, 137–164 (1987)Google Scholar
  14. 14.
    Dupont, J.: Characteristic classes for flat bundles and their formulas. Topology33, 575–590 (1994)Google Scholar
  15. 15.
    Esnault, H.: Characteristic classes of flat bundles. Topology27, 323–352 (1987)Google Scholar
  16. 16.
    Gabrielov, A., Gel'fand, I.M., Losik, M.V.: Combinatorial calculation of characteristic classes. Funct. Anal. Appl.9, 48–50, 103–115, 186–202 (1975)Google Scholar
  17. 17.
    Gel'fand, I.M., MacPherson, R.: A combinatorial formula for the Pontryagin classes. Bull. A. M. S.26, no. 2, 304–309 (1992)Google Scholar
  18. 18.
    Goncharov, A.B.: Explicit construction of characteristic classes. Adv. Soviet Math.16, Part I 169–210 (1993)Google Scholar
  19. 19.
    Laursen, M.L., Schierholz, G., Wiese, U-J.: 2 and 3-cochains in 4-dimensionalSU(2)-gauge theory. Commun. Math. Phys.103, 693–699 (1986)Google Scholar
  20. 20.
    MacPherson, R.: The combinatorial formula of Gabrielov, Gel'fand and Losik for the first Pontryagin class. Séminaire Bourbaki, Exposés 498–506, Lecture Notes in Math. vol.677, Berlin, Heidelberg, New York, 1977, pp. 105–124Google Scholar
  21. 21.
    Narasimhan, M.S., Ramanan, S.: Existence of universal connections. Am. J. Math.83, 563–572 (1961)85, 223–231 (1963)Google Scholar
  22. 22.
    Weil, A.: Sur les théorèmes de de Rham. Commun. Math. Helv.26, 119–145 (1952)Google Scholar
  23. 23.
    Zucker, S.: The Cheeger-Simons invariant as a Chern class. In: “Algebraic analysis, Geometry and Number Theory, Proc. JAMI Inaugural Conference”, JHU Press, 1989, pp. 397–417Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • J. -L. Brylinski
    • 1
  • D. A. McLaughlin
    • 2
  1. 1.Mathematics DepartmentPennsylvania State UniversityUniversity ParkUSA
  2. 2.Mathematics Department, Fine HallPrinceton UniversityPrincetonUSA

Personalised recommendations