# Characteristic classes not defined on vector bundles

If you read the definition on Wikipedia, you'll see that they allow characteristic classes to be defined on general principal $G$-bundles (vector bundles being subsumed in this general case by looking at the frame bundle (or subbundle thereof)). However, all of the 'big name' characteristic classes I am aware of (Stiefel-Whitney, Chern, Pontryagin, Euler, Todd) are defined in the special case of vector bundles (possibly with extra structure).

Are there any 'naturally occurring' examples of characteristic classes defined on principal $G$-bundles that are not frame bundles? (I say 'naturally occurring' because it is easy enough to cook up an example using the Chern-Weil Theorem, but I would prefer the example to actually be useful in practice.)

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It seems that if $G$ admits a faithful linear representation then there should be an equivalence between principal $G$ bundles and $G$-vector bundles of dimension the dimension of (the chosen faithful linear representation of $G$). So I guess you can most of the time frame your problem (!) in terms of frame bundles. But I'll admit this is silly. In any case, there are such characteristic classes when one studies principal $G$ bundles with finite $G$, there is a paper on the topic by Freed and Quinn (Chern Simons theory with finite gauge group). They recover some character formulas. – Olivier Bégassat Jun 19 '14 at 23:03

• Suppose you want to classify, not vector bundles, but vector bundles with flat connection, e.g. local systems. The corresponding classifying space is $BG_{\delta}$, where $G$ is the appropriate general linear group but given the discrete topology. Cohomology of this thing then naturally gives characteristic classes of vector bundles with flat connections.
• When $G$ is finite, the cohomology of $BG$ appears as characteristic classes in relation with a TQFT called Dijkgraaf-Witten theory; roughly speaking the cohomology classes take the role of Lagrangians.
• Arguably the most conceptually important example: suppose you want to classify, not vector bundles, but e.g. smooth fibrations with fiber $F$ for a smooth manifold $F$. Then the corresponding classifying space is $B \text{Diff}(F)$ and classes in cohomology of this classifying space give characteristic classes of $F$-bundles. For example $F$ might be a sphere or a surface. Already sphere bundles naturally appear in various contexts, e.g. they can be built out of vector bundles, they appear in the study of when homotopy types can be realized by manifolds, they are related to the study of the J-homomorphism, etc. Characteristic classes of surface bundles appear in the literature although I'm not sure in exactly what capacity; see, for example, this paper.