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  "body": "\n## Definition\n\nLet $E \\to B$ be a real vector bundle of rank $n$ over a paracompact space $B$.\nLet $E \\otimes \\C$ be its complexification, the complex vector bundle of rank $n$ with fibers $E_b \\otimes_\\R \\C$.\nFor $i \\ge 0$, the $i$-th *Pontryagin class* of $E$ is\n$$p_i(E) = (-1)^i c_{2i}(E \\otimes \\C) \\in H^{4i}(B; \\Z),$$\nwhere $c_{2i}$ is the $2i$-th Chern class.\nThe total Pontryagin class is $p(E) = 1 + p_1(E) + p_2(E) + \\cdots$.\n\n## Remarks\n\nThe odd Chern classes of $E \\otimes \\C$ satisfy $2 c_{2i+1}(E \\otimes \\C) = 0$, because $E \\otimes \\C$ is isomorphic to its complex conjugate; this is why only the even ones are used.\nPontryagin classes are natural, so they are [[characteristic classes|7HACJC0JE2]], and $p_i(E) = 0$ for $4i \u003e 2n$, that is, for $i \u003e n/2$.\nThe Whitney sum formula holds modulo $2$-torsion: $2\\bigl(p(E \\oplus E') - p(E)\\, p(E')\\bigr) = 0$.\n\nExample: for the complex projective space $\\C P^m$ with generator $a \\in H^2(\\C P^m; \\Z)$, the tangent bundle has $p(T\\C P^m) = (1 + a^2)^{m+1}$.\nFor $m = 2$ this gives $p_1(T\\C P^2) = 3a^2$.\n\nNon-example: Pontryagin classes do not detect every nontrivial bundle.\nThe Möbius line bundle over $S^1$ is not trivial, but $p_1$ lies in $H^4(S^1; \\Z) = 0$, so all its Pontryagin classes beyond $p_0$ vanish.\n\nFor a smooth manifold $M$, $p_i(M) = p_i(TM)$.\n",
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