Pontryagin class

Remarks

The 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. Pontryagin classes are natural, so they are characteristic classes, and $p_i(E) = 0$ for $4i > 2n$, that is, for $i > n/2$. The Whitney sum formula holds modulo $2$-torsion: $2\bigl(p(E \oplus E') - p(E)\, p(E')\bigr) = 0$.

Example: for the complex projective space $\C P^m$ with generator $a \in H^2(\C P^m; \Z)$, the tangent bundle has $p(T\C P^m) = (1 + a^2)^{m+1}$. For $m = 2$ this gives $p_1(T\C P^2) = 3a^2$.

Non-example: Pontryagin classes do not detect every nontrivial bundle. The 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.

For a smooth manifold $M$, $p_i(M) = p_i(TM)$.