group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The Pontryagin classes are characteristic classes on the classifying space $B O(n)$ of the orthogonal group and, by pullback, on the base of any bundle with structural group the orthogonal group. The latter is where they were originally defined.
The analogs for the unitary group are the Chern classes.
The universal Pontryagin characteristic classes $P_k$ on the classifying space $B O(n)$ are, up to a sign, the pullbacks of the Chern classes $c_{2k}$ along the complexification inclusion
The torsion-free quotient of the cohomology ring $H^\bullet(B SO(2n+1), \mathbb{Z})$ is the polynomial ring on all Pontryagin classes $\{P_i\}_{i = 1}^n$. The torsion is generated by Bockstein images of $H^\bullet(BSO(2n+1);\mathbb{F}_2)$, which is generated by the Stiefel-Whitney classes.
The torsion-free quotient of the cohomology ring $H^\bullet(B SO(2n), \mathbb{Z})$ is the quotient of the polynomial ring on Pontryagin classes $P_i$ and the Euler class $\chi$ by the relation $\chi^2 = P_n$; again the torsion is generated by Bocksteins of monomials in the Stiefel–Whitney classes.
Under the other canonical map
one has
and
Under the inclusion
of the maximal torus one has that
and
where the $x_i \in H^\bullet(B U(1)^n, \mathbb{Z})$ are the “Chern roots”.
See at Chern class - Properties – Splitting principle and Chern roots and at splitting principle - Examples - Real vector bundles for more.
The twisted differential c-structures corresponding to Pontryagin class include
twisted differential string structure for the first fractional Pontryagin class $\frac{p_1}{2} : B Spin \to B^3 U(1)$;
twisted differential fivebrane structure for the second fractional Pontryagin class $\frac{1}{6}p_2 : B String \to B^7 U(1)$.
Original accounts:
Classical textbook references are
Shoshichi Kobayashi, Katsumi Nomizu, Section XII.4 in: Foundations of Differential Geometry, Volume 1, Wiley 1963 (web, ISBN:9780471157335, Wikipedia)
John Milnor, Jim Stasheff, Characteristic classes, Princeton Univ. Press
Werner Greub, Stephen Halperin, Ray Vanstone, Connections, Curvature, and Cohomology Academic Press (1973)
With an eye towards mathematical physics:
See also
Paul Bressler, The first Pontryagin class, math.AT/0509563
Ivan Panin, Charles Walter, Quaternionic Grassmannians and Pontryagin classes in algebraic geometry, arxiv/1011.0649
A brief introduction is in chapter 23, section 7
First Pontrjagin class as counting gravitational instanton number:
Tohru Eguchi, Peter Freund, Quantum Gravity and World Topology, Phys. Rev. Lett. 37, 1251 (1967) (doi:10.1103/PhysRevLett.37.1251)
Alexander Belavin, D. Burlankov, The renormalisable theory of gravitation and the Einstein equations, Physics Letters A Volume 58, Issue 1, 26 July 1976, Pages 7-8 (doi:10.1016/0375-9601(76)90530-2)
Serdar Nergiz, Cihan Saclioglum, equation (27) in: A Quasiperiodic Gibbons–Hawking Metric and Spacetime Foam, Phys. Rev. D 53, 2240 (1996) (arXiv:hep-th/9505141)
Last revised on November 25, 2020 at 07:41:59. See the history of this page for a list of all contributions to it.