Pontryagin class




Special and general types

Special notions


Extra structure





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.


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) \,.


As generating universal characteristic classes

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 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.

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


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


(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


Classical textbook references are

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

Revised on January 5, 2016 22:17:27 by Urs Schreiber (