nLab Pontrjagin character

Idea

The Pontrjagin character is the analog for KO--theory of what the Chern character is for KU--theory. Like the latter may be expressed as a series of Chern classes, so the Pontrjagin character is expressed as a series of Pontrjagin classes.

Equivalently, the Pontrjagin character is the Chern-Dold character on KO.

References

• Friedrich Hirzebruch, Chapter 1, Section 4 of: Neue topologische Methoden in der Algebraischen Geometrie, Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge, Springer 1956 (doi:10.1007/978-3-662-41083-7)

• Pierre Conner, Edwin Floyd, p. 53 in: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)

• Werner Greub, Stephen Halperin, Ray Vanstone, Pontrjagin, Pfaffian, and Chern Classes, Section 9.6 in: Lie groups, principal bundles and characteristic classes, Volume 2 of: Connections, Curvature, and Cohomology, Pure and Applied Mathematics Volume 47, Part B, 1972, Pages 420-476 (doi:doi.org/10.1016/S0079-8169(08)62879-2)

• Mitsunori Imaoka, Kouji Kuwana, Stably extendible vector bundles over the quaternionic projective spaces, Hiroshima Math. J. 29 (1999), 273-279 (euclid:hmj/1206125008)

• Kiyoshi Igusa, Pontrjagin classes and higher torsion of sphere bundles, pp. 21-29 in: R. Penner et al. (eds.) Groups of Diffeomorphisms , Advanced Studies in Pure Mathematics 52 2008 (euclid:aspm/1543447476)

• Manuel Krannich, Jens Reinhold, Section 2.4 of: Characteristic numbers of manifold bundles over surfaces with highly connected fibers, Journal of the London Mathematical Society, June 2020. (arXiv:1807.11539, doi:10.1112/jlms.12344)

• Daniel Grady, Hisham Sati, Section 2.1 of: Differential KO-theory: constructions, computations, and applications (arXiv:1809.07059)

See also:

• eom, Pontryagin character.