In mathematics, the Pontryagin classes are certain characteristic classes. The real Pontryagin class lies in cohomology groups with index a multiple of four. It applies to real vector bundles. There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure.
These classes have a meaning in real differential geometry - unlike the Chern class, which assumes a complex vector bundle at the outset. This theory relates directly to curvature, as was shown by Shiing-shen Chern and André Weil around 1948. The initial introduction by Pontryagin was in relation with the bordism problem. of finding necessary conditions on a manifold to be a boundary. On the other hand the current definition through algebraic topology depends on taking a real vector bundle and complexifying it to , which has a Chern class.
For a vector bundle E over a 2n-dimensional differentiable manifold M, its Pontryagin class is (up to a normalization factor) with n copies of F where for any auxiliary connection A over E. It turns out the value is independent of the connection.
See also Chern-Simons.
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