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  "body": "\n## Summary\n\nThe Pontryagin classes $p_i(E) \\in H^{4i}(B; \\Z)$ are [[characteristic classes|7HACJC0JE2]] of real vector bundles $E \\to B$ with integer coefficients.\nThey are defined from the Chern classes of the complexification $E \\otimes \\C$.\nPontryagin classes of the tangent bundle of a manifold give the Pontryagin numbers, which are invariants of oriented cobordism.\nThey enter the Hirzebruch signature theorem and Milnor's construction of exotic spheres.\n\n## Explanations\n\n### L1 human\n\nA real vector bundle can be turned into a complex one by allowing complex scalars in each fiber.\nComplex bundles have Chern classes, which are integer cohomology classes.\nThe Pontryagin classes are the Chern classes of this complexified bundle in degrees divisible by four, with a sign.\nFor the complex projective plane, the first Pontryagin class is three times the square of the generator, and this three is the reason its signature is one.\n",
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