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  "body": "\n## Definition\n\nLet $G$ be a topological group, let $R$ be an [[abelian group|082GHNPJ53]] and let $k \\ge 0$ be an integer.\nWrite $H^k(X; R)$ for [[singular cohomology|6N7BC70CX0]] with coefficients in $R$.\nA *characteristic class* of degree $k$ for principal $G$-bundles is a rule $c$ that assigns to each [[principal $G$-bundle|8E37E74XVP]] $P \\to X$ a class $c(P) \\in H^k(X; R)$, such that:\n\n1. If $P \\to X$ and $P' \\to X$ are isomorphic, then $c(P) = c(P')$.\n2. For every continuous map $f \\colon Y \\to X$, $c(f^* P) = f^* c(P)$, where $f^* P$ is the pullback bundle and $f^*$ on the right is the induced map in cohomology.\n\n## Remarks\n\nThe same definition applies to real or complex vector bundles of a fixed rank, which correspond to principal $GL_n(\\R)$- or $GL_n(\\C)$-bundles.\nOver CW complexes, characteristic classes of degree $k$ correspond bijectively to elements of $H^k(BG; R)$: the class $u$ gives $c(P) = f_P^* u$, where $f_P \\colon X \\to BG$ classifies $P$.\n\nExample: for real line bundles, the first Stiefel–Whitney class $w_1 \\in H^1(X; \\Z/2\\Z)$ is a characteristic class.\nFor the Möbius line bundle $M$ over $S^1$, $w_1(M)$ is the generator of $H^1(S^1; \\Z/2\\Z)$, while $w_1$ of the trivial bundle is $0$.\n\nNon-example: for line bundles over spaces $X$, the rule $c(P) = 1 \\in H^0(X; \\Z)$ if $X$ is compact and $c(P) = 0$ otherwise fails clause 2.\nFor the inclusion $f \\colon [0, 1] \\to \\R$ and the trivial bundle $P$ over $\\R$, $c(f^* P) = 1$ but $f^* c(P) = 0$.\n\nClause 2 implies that a trivial bundle $X \\times G$ has $c(X \\times G) = 0$ when $k \u003e 0$, since it is pulled back from a point.\n",
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