# A dictionary of Characteristic classes and obstructions

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: ?

• trivial complex subbundles and trivial complex quotient bundles are equivalent here because you can choose a Hermitian form by partitions of unity and use it to split any inclusion. – Will Sawin Jan 23 '16 at 22:37
• Note that the Wu class is not the class of a bundle. Rather, its only input is the base $B$. (Maybe one can attempt to define $v(E)$ such that $\operatorname{Sq}(v(E)) = w(E)$, the total Stiefel–Whitney class. I don't know whether this always exists, and I don't think one classically considers these. The classical case would then correspond to $E = TB$, the tangent bundle of $B$.) – R. van Dobben de Bruyn Jan 23 '16 at 23:08
• @SaalHardali Complex vector bundles are not holomorphic vector bundles. In the case of holomorphic vector bundles, the obstruction you gave is an obstruction to having trivial quotients and also an obstruction to subbundles - you could put either and the statement would still be true. – Will Sawin Jan 24 '16 at 1:56
• The Todd class and Chern character are not supposed to be obstructions to anything. They're supposed to appear, for example, in the Grothendieck-Riemann-Roch theorem. – Qiaochu Yuan Jan 24 '16 at 6:32
• $E$ admits a spin$^c$ structure iff $E$ is orientable and the third integral Stiefel-Whitney class vanishes. – José Figueroa-O'Farrill Jan 24 '16 at 20:45

Assume that $E\to B$ carries a flat connection $\nabla$. Then the Kamber-Tondeur classes are obstructions against the existence of a $\nabla$-parallel metric on $E$. In the case of a complex line bundle, the first Kamber-Tondeur class is the only obstruction.
The Cheeger-Simons differential characters of a vector bundle $E\to B$ with connection $\nabla$ are obstructions against a parallel trivialisation. For a complex line bundle, the first Cheeger-Simons class is the only obstruction (in fact, this class classifies complex line bundles with connections).