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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.

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    $\begingroup$ 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. $\endgroup$
    – Will Sawin
    Commented Jan 23, 2016 at 22:37
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    $\begingroup$ 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 $B$.) $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jan 23, 2016 at 23:08
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    $\begingroup$ @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. $\endgroup$
    – Will Sawin
    Commented Jan 24, 2016 at 1:56
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    $\begingroup$ 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. $\endgroup$
    – Qiaochu Yuan
    Commented Jan 24, 2016 at 6:32
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    $\begingroup$ $E$ admits a spin$^c$ structure iff $E$ is orientable and the third integral Stiefel-Whitney class vanishes. $\endgroup$
    – José Figueroa-O'Farrill
    Commented Jan 24, 2016 at 20:45

1 Answer 1

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The following classes are of a slightly different flavour because they depend on the additional choice of a connection.

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).

Note that the Kamber-Tondeur classes can be interpreted as the imaginary parts of the Cheeger-Simons differential characters.

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    $\begingroup$ Just to make sure, does "parallel trivilization" mean the same thing in this context as "parallel global section of the frame bundle"? (Which is equivalent to flat + no monodromy). $\endgroup$
    – Saal Hardali
    Commented Jan 27, 2016 at 15:34
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    $\begingroup$ @SaalHardali Indeed. Differential characters are able to see (part of) the monodromy of a flat connection. They are truely global invariants (in contrast to Chern-Weil forms, say). They even see some holonomy information if the connection is not flat. $\endgroup$
    – Sebastian Goette
    Commented Jan 27, 2016 at 17:19
  • $\begingroup$ Can you please suggest some reference for differential characters.. my background is Chern-Weil homomorphism.. $\endgroup$
    – Praphulla Koushik
    Commented Oct 10, 2019 at 13:36
  • $\begingroup$ I suggest to read the original article by Cheeger and Simons, or the paper by Simons and Sullivan about axioms for differential characters. The latter paper also lists some other references that you may try. $\endgroup$
    – Sebastian Goette
    Commented Oct 10, 2019 at 13:39

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