nLab
Pontryagin class

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

The Pontryagin classes are characteristic classes on the classifying space O(n)\mathcal{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.

Definition

The universal Pontryagin characteristic classes P kP_k on the classifying space BO(n)B O(n) are, up to a sign, the pullbacks of the Chern classes c 2kc_{2k} along the complexification inclusion

BO(n)BU(n). B O(n) \to B U(n) \,.

Properties

As generating universal characteristic classes

The torsion-free quotient of the cohomology ring H (BSO(2n+1),)H^\bullet(B SO(2n+1), \mathbb{Z}) is the polynomial ring on all Pontryagin classes {P i} i=1 n\{P_i\}_{i = 1}^n. The torsion is generated by Bockstein images of H (BSO(2n+1);𝔽 2)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 (BSO(2n),)H^\bullet(B SO(2n), \mathbb{Z}) is the quotient of the polynomial ring on Pontryagin classes P iP_i and the Euler class χ\chi by the relation χ 2=P n\chi^2 = P_n; again the torsion is generated by Bocksteins of monomials in the Stiefel–Whitney classes.

Further relation to Chern classes

Under the other canonical map

j:BU(n)BO(2n) j \;\colon\; B U(n) \to BO(2n)

one has

j *(P k)= a+b=2k(1) a+kc ac b j^\ast(P_k) = \sum_{a + b = 2 k} (-1)^{a+k} c_a c_b

and

j *(χ)=c n. j^\ast(\chi) = c_n \,.

Splitting principle and Chern roots

Under the inclusion

i:U(1) nU(n)O(2n) i \;\colon\; U(1)^n \hookrightarrow U(n) \to O(2n)

of the maximal torus one has that

(Bi) *(P k)=σ k(x 1,,x n) 2 (B i)^\ast(P_k) = \sigma_k(x_1, \cdots, x_n)^2

and

(Bi) *(χ)=σ n(x 1,,x n) (B i)^\ast(\chi) = \sigma_n(x_1, \cdots, x_n)

where the x iH (BU(1) n,)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.

Trivializations and structures

The twisted differential c-structures corresponding to Pontryagin class include

References

Classical textbook references are

With an eye towards mathematical physics:

  • 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

Last revised on November 11, 2019 at 02:50:53. See the history of this page for a list of all contributions to it.