Characteristic Class with arbitrary coefficient

Question

I'm looking for some "natural definition" for characteristic class with arbitrary coefficient. For example, the chern class satisfies four conditions and Stiefel-Whitney class satisfies three conditions. Are there any axiomatic approach to define a characteristic class for Abelian groups $G$ other than $\mathbb{Z}$(Chern class) or $\mathbb{Z}/2\mathbb{Z}$(Stiefel-Whitney). Thanks!

Answer 1

I don't think this is really what you had in mind, but here's another perspective on characteristic classes which lends itself to some generality.

Let $E,F$ be two multiplicative cohomology theories and $T \colon E \to F$ a natural transformation. Let $\Xi$ be some class of vector bundles which are naturally $E$- and $F$-oriented, in the following sense: each $\xi \in \Xi$, say of dimension $n$ over a space $X$, has Thom classes $u_\xi \in E^n(\xi, \xi-0)$ and $v_\xi \in F^n(\xi, \xi-0)$ which are natural in bundle maps, so that we have natural Thom isomorphisms $$ \phi_\xi \colon E^*(X) \to E^{*+n}(\xi,\xi-0) \quad\text{and}\quad \psi_\xi \colon F^*(X) \to F^{*+n}(\xi,\xi-0), $$ given by multiplication by $u_\xi$ and $v_\xi$, respectively. Then we can define a characteristic class $$ c_T(\xi) \mathrel{:=} (\psi_\xi^{-1} \circ T \circ \phi_\xi)(1) = T(u_\xi)/v_\xi \in F^*(X). $$ By naturality of the Thom classes and $T$ this will be natural in bundle maps. If we require our Thom classes to also be multiplicative, and $T$ to respect the multiplicative structures on $E$ and $F$, and the characteristic class will also be multiplicative.

Examples:

1. Take $E = F = \mathrm{H}\mathbb{Z}/2$, ordinary cohomology with $\mathbb{Z}/2$ coefficients, and $T = \operatorname{Sq} = \sum \operatorname{Sq^i}$ the total Steenrod square. All vector bundles are naturally $\mathrm{H}\mathbb{Z}/2$-oriented (this is the same thing as an orientation in the classical sense with $\mathbb{Z}/2$ coefficients). Then $c_{\operatorname{Sq}}$ is just the total Stiefel-Whitney class $w$ (this is immediate from the axiomatic definition of $w$), and taking the graded pieces we get $c_{\operatorname{Sq^i}} = w_i$.

2. I'm guessing the same story runs with $\mathbb{Z}$ coefficients to give Chern classes? But right now I don't know how cohomology operations work with $\mathbb{Z}$ coefficients so I'm not actually claiming this with any certainty; perhaps someone else can chime in. Note that these will only be defined for complex vector bundles because you need natural multiplicative $\mathrm{H}\mathbb{Z}$-orientations.

3. Take $E = F = \mathrm{K}$, complex K-theory, and $T = \psi^i$ an Adams operation. The resulting classes $c_{\psi^i}$ are called the cannibalistic classes, and were used by Adams to solve the vector fields on spheres problem (which is where I read about this stuff).

So I guess all I'm saying is that, from this perspective, characteristic classes in $\mathrm{H}R$, ordinary cohomology with coefficients in a ring $R$, come from Thom isomorphisms and natural transformations from some cohomology theory to $\mathrm{H}R$, e.g. cohomology operations on $\mathrm{H}R$.