I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians.

In an effort to get a hold of characteristic classes I had the idea of compiling a short dictionary relating characteristic classes to their obstructions. Unfortunately I didn't find anything of this sort on the web. It could be a nice thing If we could compile such a list here.

Let $E \to B$ be a real vector bundle over a compact manifold (for simplicity):

**Euler class**($E$ orientable)**:**$E \to B$ has a nowhere vanishing section $ \implies e(E)=0 $.**Stiefel-Whitney classes:**- $w_1(E)=w_1(\det E)=0 \iff E$ orientable.
- $w_1(E) =w_2(E) = 0 \iff E$ has spin structure.
- $E$ has a trivial subbundle of rank $m$ $\implies$ $w_k=0$ for all $k>rank(E)-m$.
- $E$ orientable $\implies$ $w_{top} (E) = e(E) \text{ mod 2}$

**Pontryagin classes:**- For $E$ spin vector bundle: $\frac{1}{2} p_1(E)=0 \iff E$ has string structure.
- For $E$ string vector bundle: $\frac{1}{6}p_2(E)= 0 \iff E$ has 5-brane structure.
- If $rank(E)$ is even: $e(E) \cup e(E) = p_{top}(E)$

**Chern classes:**Suppose $E \to B$ is now a complex vector bundle.- $E$ has a trivial complex subbundle (or is it quotient bundle here?) of rank $m$ $\implies$ $c_k=0$ for all $k>rank(E)-m$.
- $c_i(E)=w_{2i}(E_{\mathbb{R}}) \text{ mod 2}$.
- $c_1(E) = c_1(\wedge^{top} E) = 0 \iff E$ has reduction of structure group to $SU$. I read in several places that this has something to do with the possible number of linearly independent parallel spinors - notice $w_2( E_{ \mathbb{R}}) = c_1(E) = 0$ so $E$ is spin in particular.
- $c_{top}(E)=e(E_{\mathbb{R}})$

**Todd class:**?**Chern character:**?**Wu class:**?

Additions and corrections are welcome.

integralStiefel-Whitney class vanishes. $\endgroup$7more comments