Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Beilinson, A.: Higher regulators and values of L-functions. J. Sov. Math.30, 2036–2070 (1985)
Beresin, F.A., Retakh, V.S.: A method of computing characteristic classes of vector bundles. Rep. Math. Phys.18, 363–378 (1980)
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)
Borel, A., Hirzebruch, F.: Characteristic classes and homogeneous spaces 2. Am. J. Math.81, 315–382 (1959)
Bott, R., Tu, L.: “Differential forms in algebraic topology”. Berlin, Heidelberg, New York: Springer, 1982
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. 209–220
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. 21–64
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)
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.
Cheeger, J.: Spectral geometry of singular riemannian spaces. J. Diff. Geom.18, 575–657 (1983)
Cheeger, J., Simons, J.: Differential characters and geometric invariants. Lecture Notes in Math. vol.1167, Berlin, Heidelberg, New York: Springer, 1985, pp. 50–80
Chern, S.S., Simons, J.: Characteristic forms and geometric invariants. Ann. Math.99, 48–69 (1974)
Dupont, J.: The dilogarithm as a characteristic class for flat bundles. J. Pure Appl. Alg.44, 137–164 (1987)
Dupont, J.: Characteristic classes for flat bundles and their formulas. Topology33, 575–590 (1994)
Esnault, H.: Characteristic classes of flat bundles. Topology27, 323–352 (1987)
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)
Gel'fand, I.M., MacPherson, R.: A combinatorial formula for the Pontryagin classes. Bull. A. M. S.26, no. 2, 304–309 (1992)
Goncharov, A.B.: Explicit construction of characteristic classes. Adv. Soviet Math.16, Part I 169–210 (1993)
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)
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. 105–124
Narasimhan, M.S., Ramanan, S.: Existence of universal connections. Am. J. Math.83, 563–572 (1961)85, 223–231 (1963)
Weil, A.: Sur les théorèmes de de Rham. Commun. Math. Helv.26, 119–145 (1952)
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. 397–417
Author information
Authors and Affiliations
Additional information
Communicated by H. Araki
The first author was supported in part by N.S.F. grant DMS-9203517.
The second author was supported in part by N.S.F. grant DMS-9310433.
Rights and permissions
About this article
Cite this article
Brylinski, J.L., McLaughlin, D.A. Čech cocycles for characteristic classes. Commun.Math. Phys. 178, 225–236 (1996). https://doi.org/10.1007/BF02104916
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02104916