nLab
characteristic class

Idea

In the general context of cohomology, as described there, a cocycle representing a cohomology class on an object $X$ with coefficients in an object $A$ is a morphism $c:X\to A$ in a given ambient (∞,1)-topos $H$.

The same applies with the object $A$ taken as the domain object: for $B$ yet another object, the $B$-valued cohomology of $A$ is similarly $H(A,B)={\pi }_{0}H(A,B)$. For $[k]\in H(A,B)$ any cohomology class in there, we obtain an ∞-functor

$$[k(-)]:H(X,A)\to H(X,B)$$[k(-)] : \mathbf{H}(X,A) \to H(X,B)

from the $A$-valued cohomology of $X$ to its $B$-valued cohomology, simply from the composition operation

$$H(X,A)\times H(A,B)\to H(X,B).$$\mathbf{H}(X,A) \times \mathbf{H}(A,B) \to \mathbf{H}(X,B) \,.

Quite generally, for $[c]\in H(X,A)$ an $A$-cohomology class, its image $[k(c)]\in H(X,B)$ is the corresponding characteristic class.

From the nPOV, where cocycles are elements in an (∞,1)-categorical hom-space, forming characteristic classes is nothing but the composition of cocycles.

In practice one is interested in this notion for particularly simple objects $B$, notably for $B$ an Eilenberg-MacLane object $B^{n}K$ for some component $K$ of a spectrum object. This serves to characterize cohomology with coefficients in a complicated object $A$ by a collection of cohomology classes with simpler coefficients. Therefore the name characteristic class .

Zoran: While the discussion of the name ‘characteristic class’ is plausible, it is, I think, unfortunately not historically true. The continuous map into the classifying space, by which the pullback of a universal class gives the characteristic class of a manifold is traditionally called the characteristic map. because that map characterizes that cohomology class. It is not the cohomology theory which is characterized by that map, but the very class. So characteristic classes are those which can be characterized by the maps to given classifying space.

Then with the usual notation $H^n(X,K):=H(X,B^{n}K)$ a given characteristic class in degree $n$ assigns

$$[k(-)]:H(X,A)\to H^n(X,K).$$[k(-)] : \mathbf{H}(X,A) \to H^n(X,K) \,.

Moreover, recall from the discussion at cohomology that to every cocycle $c:X\to A$ is associated the object $P\to X$ that it classifies – its homotopy fiber – which may be thought of as an $A$-principal ∞-bundle over $X$ with classifying map $X\to A$. One typically thinks of the characteristic class $[k(c)]$ as characterizing this principal ∞-bundle $P$.

Examples

Integral characteristic classes of principal bundles

This is the archetypical example: let $H=$ Top and let $G$ be a topological group and $ℬG$ its classifying space.

The integral cohomology of this classifying space is of course

$$H^n(ℬG,\mathbb{Z})={\pi }_0\mathrm{Top}(ℬG,{ℬ}^n\mathbb{Z}).$$H^n(\mathcal{B}G, \mathbb{Z}) = \pi_0 Top(\mathcal{B}G, \mathcal{B}^n\mathbb{Z}) \,.

A $G$-principal bundle $P\to X$ is classified by some map $c:X\to ℬG$. For any $k\in H^n(ℬG,\mathbb{Z})$ a degree $n$ cohomology class of the classifying space, the corresponding composite map $X\stackrel{c}{\to }ℬG\stackrel{k}{\to }{ℬ}^n\mathbb{Z}$ represents a class $[k(c)]\in H^n(X,\mathbb{Z})$. This is the corresponding characteristic class of the bundle.

Notable families of examples include:

• for $G=O$ the orthogonal group: Pontryagin class?es;

• for $G=U$ the unitary group: Chern class?es;

Chern character

The Chern character is a natural characteristic class with values in real cohomology. See there for more details.

Literature

• J. Milnor, J. Stasheff, Characteristic classes, Princeton Univ. Press

• Jean-Pierre Schneiders, Introduction to characteristic classes and index theory (book), Lisboa (Lisabon) 2000

• Johan L. Dupont, Fibre bundles and Chern-Weil theory, Lecture Notes Series 69, Dept. of Math., University of Aarhus, 2003, 115 pp. pdf

• R. Bott, L. W. Tu, Differential forms in algebraic topology, GTM 82, Springer 1982.