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Journal of Topology Advance Access published online on October 31, 2007
Journal of Topology, doi:10.1112/jtopol/jtm006
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© 2007 London Mathematical Society

Axiomatic characterization of ordinary differential cohomology

James Simons

Renaissance Technologies LLC, 800 Third Avenue, New York, NY 10022, USA jim{at}rentec.com

Dennis Sullivan

Mathematics Department, SUNY – Stonybrook, Stonybrook, NY 11794, USA dennis{at}math.sunysb.edu

The Cheeger–Simons differential characters, the Deligne cohomology in the smooth category, the Hopkins–Singer construction of ordinary differential cohomology, and the recent Harvey–Lawson constructions are each in two distinct ways abelian group extensions of known functors. In one description, these objects are extensions of integral cohomology by the quotient space of all differential forms by the subspace of closed forms with integral periods. In the other, they are extensions of closed differential forms with integral periods by the cohomology with coefficients in the circle. These two series of short-exact sequences mesh with two interlocking long-exact sequences (the Bockstein sequence and the de Rham sequence) to form a commutative DNA-like array of functors called the Character Diagram. Our first theorem shows that on the category of smooth manifolds and smooth maps, any package consisting of a functor into graded abelian groups together with four natural transformations that fit together so as to form a Character Diagram as mentioned earlier is unique up to a unique natural equivalence. Our second theorem shows that natural product structure on differential characters is uniquely characterized by its compatibility with the product structures on the known functors in the Character Diagram. The proof of our first theorem couples the naturality with results about approximating smooth singular cycles and homologies by embedded pseudomanifolds.

Received February 5, 2007.

2000 Mathematics Subject Classification 57R19



References

  1. Brylinski J.-L. Loop spaces, characteristic classes, and geometric quantization (1993) Boston, MA: Birkhäuser.
  2. Cheeger J. ‘Multiplication of differential characters’. (1973) Proceedings of the Instituto Nazionale di Alta Mathematica, Symposia Mathematica XI: London, UK. Academic Press. 441–445.
  3. Cheeger J., Simons J. ‘Differential characters and geometric invariants’ Notes of Stanford Conference 1973, Lecture Notes in Mathematics 1167. (1985) New York: Springer. 50–90.
  4. Chern S. S., Simons J. ‘Characteristic forms and geometric invariants’. Ann. Math. (1974) 99:48–69.[CrossRef]
  5. Gillet H., Soule C. ‘Arithmetic Chow groups and differential characters’. In: Algebraic K-theory: Connections with geometry and topology—Jardine J. F., Snaith V., eds. (1989) Norwell, MA: Kluwer Academic. 30–68.
  6. Harris B. ‘Differential characters and the Abel–Jacobi map’. In: Algebraic K-theory: Connections with geometry and topology—Jardine J. F., Snaith V., eds. (1989) Norwell, MA: Kluwer Academic. 69–86.
  7. Harvey R., Lawson B. ‘From Sparks to Grundles–differential characters’. Commun. Anal. Geom. (2006) 14:25–58.
  8. Harvey R., Lawson B., Zweck J. ‘The de Rham–Federer theory of differential characters and character duality’. Amer. J. Math. (2003) 125:791–847.[CrossRef]
  9. Hopkins M., Singer I. ‘Quadratic functions in geometry, topology, and M-theory’, Preprint. (2002) arXiv.org/pdf/math/0211216.
  10. Hurewicz W., Wallman H. Dimension theory (1941) Princeton, NJ: Princeton University Press.
  11. Munkres J. R. ‘Elementary differential topology’. In: Annals of Mathematics Studies (1969) 54. Princeton, NJ: Princeton University Press. 79.
  12. Poincaré H. ‘Complement a l’Analysis Situs'. In: OEuvres completes (1954) tome VI. Paris, France: Gauthier-Villars. 291.
  13. Simons J. ‘Characteristic forms and transgression: characters associated to a connection’, Preprint. (1972) Stony Brook University.
  14. Sullivan D. ‘Singularities in spaces’. (1971) Proceedings of the Liverpool Conference on Singularities Symposium II, Liverpool, England, Lecture Notes in Mathematics 209: Berlin, Germany. Springer. 196–206.

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This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
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Right arrow Articles by Simons, J.
Right arrow Articles by Sullivan, D.
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What's this?