Contents

Idea

The Pontryagin classes are characteristic classes on the classifying space $ℬO(n)$ 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 on the classifying space $BO(n)$ are, up to a sign, the pullbacks of the Chern classes along the inclusion

Trivializations and structures

The twisted differential c-structures corresponding to Pontryagin class include

• twisted differential string structure for the first fractional Pontryagin class $1/2p_1:B\mathrm{Spin}\to B^{3}U(1)$;

• twisted differential fivebrane structure for the second fractional Pontryagin class $\frac{1}{6}{p}_2:B\mathrm{String}\to B^{7}U(1)$.

Related concepts

• Pontryagin class

• Chern class

• Stiefel-Whitney class

• Chern character

References

Classical textbook references are

• John Milnor, Jim Stasheff, Characteristic classes, Princeton Univ. Press

• Werner Greub, Stephen Halperin, Ray Vanstone, Connections, Curvature, and Cohomology Academic Press (1973)

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

• Peter May, A concise course in algebraic topology (pdf)