Characteristic class

Remarks

The 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. Over 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 $P$.

Example: for real line bundles, the first Stiefel–Whitney class $w_1 \in H^1(X; \Z/2\Z)$ is a characteristic class. For 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 $0$.

Non-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. For the inclusion $f \colon [0, 1] \to \R$ and the trivial bundle $P$ over $\R$, $c(f^* P) = 1$ but $f^* c(P) = 0$.

Clause 2 implies that a trivial bundle $X \times G$ has $c(X \times G) = 0$ when $k > 0$, since it is pulled back from a point.