Nice things that can be proved easily with characteristic classes

In this blog post you'll find a computation of the cohomology ring of a hypersurface of degree $d$ in $\mathbb{CP}^3$ using characteristic classes. This turns out to be a weirdly good exercise in using characteristic classes: the computation invokes, in order, Euler classes, Chern classes, Pontryagin classes, Wu classes, and Stiefel-Whitney classes, and doing it is what made me comfortable with characteristic class computations.

I like this example. The Spheres $S^{2n}$ cannot be complex manifolds unless $n=0,1,3$. One proves that $TS^{2n}$ does not have the structure of a complex vector bundle in these cases. If $TS^{2n}$ were a complex vector bundle, then $c_{n}(TS^{2n}) = e (TS^{2n})$, so $c_n (TS^{2n})$ is twice a generator of $H^{2n} (S^{2n}; \mathbb{Z})$.

Case 1: $n=2m$ even. Then $p_m (TS^{2n}) = \pm c_{2m} (TS^{2n} \otimes \mathbb{C}) = c_{2m} (TS^{2n} \oplus \overline{TS^{2n}})$. By the product formula for Chern classes, this is $= \pm c_{2m}(TS^{2n}) + c_{2m} \overline{TS^{2n}}) =\pm 2 c_{2m }(TS^{2n}) \neq 0$. This is a contradiction since $TS^{2n} \oplus \mathbb{R}$ is trivial.

Case 2: $n \geq 4$. This is more difficult and relies on the Bott periodicity theorem, one of whose corollaries states that a the top Chern class $c_n (V)$ of a complex vector bundle $V\to S^{2n}$ is divisible by $(n-1)!$

Secondary characteristic classes allow us to see the large scale behaviour of leaves of foliations of 3-manifolds by surfaces; see

MR1040572 (91h:57015) Reviewed
Ghys, Étienne(F-ENSLY)
L'invariant de Godbillon-Vey. (French) [The Godbillon-Vey invariant]
Séminaire Bourbaki, Vol. 1988/89.
Astérisque No. 177-178 (1989), Exp. No. 706, 155–181.
57R30 (58F18)