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 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 on the classifying space are, up to a sign, the pullbacks of the Chern classes along the complexification inclusion into the classifying space
The torsion-free quotient of the cohomology ring is the polynomial ring on all Pontryagin classes . The torsion is generated by Bockstein images of , which is generated by the Stiefel-Whitney classes.
The torsion-free quotient of the cohomology ring is the quotient of the polynomial ring on Pontryagin classes and the Euler class by the relation ; 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 are the “Chern roots”.
See at Chern class - Properties – Splitting principle and Chern roots and at splitting principle - Examples - Real vector bundles for more.
We spell out the formulas for the images under the Chern-Weil homomorphism of the Chern classes, Pontrjagin classes and Euler classes as characteristic forms over smooth manifolds.
Let be a smooth manifold.
Write
for the commutative algebra over the real numbers of even-degree differential forms on , under the wedge product of differential forms. This is naturally a graded commutative algebra, graded by form degree, but since we consider only forms in even degree it is actually a plain commutative algebra, too, after forgetting the grading.
Let be a semisimple Lie algebra (such as or ) with Lie algebra representation over the complex numbers of finite dimension (for instance the adjoint representation or the fundamental representation), hence a homomorphism of Lie algebras
to the linear endomorphism ring , regarded here through its commutator as the endomorphism Lie algebra of .
When regarded as an associative ring this is isomorphic to the matrix algebra of square matrices
The tensor product of the -algebras (1) and (2)
is equivalently the matrix algebra with coefficients in the complexification of even-degree differential forms:
The multiplicative unit
in this algebra is the smooth function (differential 0-forms) which is constant on the identity matrix and independent of .
Given a connection on a -principal bundle, we regard its -valued curvature form as an element of this algebra
The total Chern form is the determinant of the sum of the unit (3) with the curvature form (4), and its component in degree , for , is the th Chern form :
By the relation between determinant and trace, this is equal to the exponential of the trace of the logarithm of , this being the exponential series in the trace of the Mercator series in :
Setting in these expressions (5) yields the total Pontrjagin form with degree=-components the Pontrjagin forms :
Hence the first couple of Pontrjagin forms are
(See also, e.g., Nakahara 2003, Exp. 11.5)
For and with the curvature form again regarded as a 2-form valued -square matrix
the Euler form is its Pfaffian of this matrix, hence the following sum over permutations with summands signed by the the signature :
The first of these is, using the Einstein summation convention and the Levi-Civita symbol:
(See also, e.g., Nakahara 2003, Exp. 11.7)
The twisted differential c-structures corresponding to Pontryagin class include
twisted differential string structure for the first fractional Pontryagin class ;
twisted differential fivebrane structure for the second fractional Pontryagin class .
The original definition is in
Л. С. Понтрягин, Характеристические циклы многообразий, ДАН, XXXV, № 2 (1942), 35–39.
English translation: Characteristic Cycles of Manifolds. L. S. Pontryagin Selected Works. Volume 1. Selected Research Papers. Edited by R. V. Gamkrelidze. CRC Press, 1986. 283–287. doi.
Л. С. Понтрягин, Характеристические циклы дифференцируемых многообразий, Матем. сб., 21(63):2 (1947), 233–284. MathNet.Ru PDF.
English translation by A. A. Brown: Lev Pontrjagin, Characteristic cycles on differentiable manifolds, Mat. Sbornik N. S. 21(63) (1947), 233-284; A.M.S. Translation 32 (1950). PDF. English translation by P. S. V. Naidu: Characteristic Cycles of Differentiable Manifolds. L. S. Pontryagin Selected Works. Volume 1. Selected Research Papers. Edited by R. V. Gamkrelidze. CRC Press, 1986. 375–433. doi.
Early 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 1974 (ISBN:9780691081229, doi:10.1515/9781400881826, pdf)
Werner Greub, Stephen Halperin, Ray Vanstone, Connections, Curvature, and Cohomology Academic Press (1973)
With an eye towards mathematical physics:
Mikio Nakahara, Section 11.4.1 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)
Gerd Rudolph, Matthias Schmidt, around Def. 4.2.19 in: Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields, Theoretical and Mathematical Physics series, Springer 2017 (doi:10.1007/978-94-024-0959-8)
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 March 4, 2024 at 21:33:42. See the history of this page for a list of all contributions to it.