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  "body": "\n## Definition\n\nConsider real vector bundles $E \\to B$ over paracompact spaces $B$.\nThe *Stiefel–Whitney classes* are the [[characteristic classes|7HACJC0JE2]] $w_i(E) \\in H^i(B; \\Z/2\\Z)$, $i \\ge 0$, in [[singular cohomology|6N7BC70CX0]], that satisfy the following axioms.\nWrite $w(E) = w_0(E) + w_1(E) + w_2(E) + \\cdots$ for the total class.\n\n1. $w_0(E) = 1$, and $w_i(E) = 0$ for $i$ greater than the rank of $E$.\n2. Naturality: $w_i(f^* E) = f^* w_i(E)$ for every continuous map $f \\colon B' \\to B$, and isomorphic bundles have equal classes.\n3. Whitney product formula: $w(E \\oplus E') = w(E) \\smile w(E')$ for the [[Whitney sum|BE9NJRVDR3]] of two bundles over the same base, where $\\smile$ is the [[cup product|JX47MS1R1H]].\n4. Normalization: for the tautological line bundle $\\gamma^1_1$ over $\\R P^1$, $w_1(\\gamma^1_1)$ is the generator of $H^1(\\R P^1; \\Z/2\\Z) \\cong \\Z/2\\Z$.\n\nThere is exactly one family of classes with these properties.\n\n## Remarks\n\nExample: a trivial bundle $B \\times \\R^n$ is pulled back from a point, so by axiom 2 its total class is $w = 1$.\nFor the tangent bundle of $\\R P^n$, one has $w(T\\R P^n) = (1 + a)^{n+1}$, where $a$ generates $H^1(\\R P^n; \\Z/2\\Z)$; for $n = 2$ this gives $w = 1 + a + a^2$.\n\nNon-example: the rule $w(E) = 1$ for every bundle satisfies axioms 1 to 3 but fails axiom 4.\n\nA bundle $E$ is orientable exactly when $w_1(E) = 0$.\nAn orientable bundle admits a spin structure exactly when also $w_2(E) = 0$.\nFor a smooth manifold $M$, one writes $w_i(M) = w_i(TM)$.\n",
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