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  "body": "\n## Summary\n\nThe Stiefel–Whitney classes $w_i(E) \\in H^i(B; \\Z/2\\Z)$ are mod $2$ [[characteristic classes|7HACJC0JE2]] of real vector bundles $E \\to B$.\nFour axioms determine them: naturality, the [[Whitney sum|BE9NJRVDR3]] formula, a normalization, and vanishing above the rank.\nThe first class detects orientability and the second detects the existence of a spin structure.\nStiefel–Whitney numbers of manifolds decide when two closed manifolds are cobordant.\n\n## Explanations\n\n### L1 human\n\nA real vector bundle can be twisted in ways that ordinary integer invariants do not see.\nStiefel–Whitney classes count such twists modulo $2$, as cohomology classes with values $0$ or $1$.\nThe first one answers a yes-or-no question: can the fibers be oriented consistently along every loop?\nFor the Möbius band over the circle the answer is no, and $w_1$ is nonzero; for the cylinder it is yes, and $w_1 = 0$.\n",
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