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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.

Qfwfq
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Saal Hardali
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    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
  • @WillSawin Isn't there an issue with non-holomorphic partitions? – Saal Hardali Jan 23 '16 at 22:41
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    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
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    @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 01:56
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    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 06:32
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    Regarding the Euler class, the "only if" statement is not true without some restrictions (such as the rank of the vector bundle equaling the dimension of the base). See here http://mathoverflow.net/questions/31376/vanishing-of-euler-class – Mark Grant Jan 24 '16 at 13:39
  • For a string structure, one needs $p_1/2=0$ as far as I know, where $p_1/2$ is a natural characteristic class for spin bundles such that $2p_1/2=p_1$. I would guess that one needs a similar class instead of $p_2$ (maybe $p_2/6$ is a natural class of string vector bundles?). – Sebastian Goette Jan 24 '16 at 18:57
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    @MarkGrant Indeed. The best you can say if $e(E)=0$ is that for a generic section, the zero set is nullhomologous, that is, a boundary in singular homology. – Sebastian Goette Jan 24 '16 at 19:01
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    $E$ 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
  • @QiaochuYuan I just noticed that in a strange way, the Todd class is an obstruction: against commutativity of Chern character and Thom isomorphism. See the original proof of Atiyah-Singer or this answer. – Sebastian Goette Feb 09 '16 at 11:18
  • @SebastianGoette That's interesting, On a different note: I've read somewhere that vanishing Pontryagin classes implies flatness, is that correct? – Saal Hardali Feb 09 '16 at 16:10
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    @SaalHardali I doubt that. Take a nontrivial complex line bundle on a surface. It must have curvature if $c_1\ne0$, but there is no room for nontrivial Pontryagin classes. – Sebastian Goette Feb 09 '16 at 16:51

1 Answers1

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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.

Sebastian Goette
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    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). – Saal Hardali Jan 27 '16 at 15:34
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    @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. – Sebastian Goette Jan 27 '16 at 17:19
  • Can you please suggest some reference for differential characters.. my background is Chern-Weil homomorphism.. – Praphulla Koushik Oct 10 '19 at 13:36
  • 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. – Sebastian Goette Oct 10 '19 at 13:39