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.

Notice that if $A=BG$ is connected, an $A$-cocycle on $X$ is a $G$-principal ∞-bundle. Hence characteristic classes are equivalently characteristic classes of principal $\mathrm{\infty }$-bundles.

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. Historically the name characteristic class came a little different way about, however (see also historical note on characteristic classes).

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

Characteristic classes of principal bundles

This is the archetypical example: let $H=$ Top $\simeq$ ∞Grpd, the canonical (∞,1)-topos of discrete ∞-groupoids, or more generally let $H=$ ETop∞Grpd, the cohesive (∞,1)-topos of Euclidean-topological ∞-groupoids.

For $G$ topological group write $BG$ for its classifying space: the (geometric realization of its) delooping.

For $A$ any other abelian topological group, similarly write $B^{n}A$ for its $n$-fold delooping. If $A$ is a discrete group then this is the Eilenberg-MacLane space $K(A,n)$.

Generally,

$$H^n(BG,\mathbb{Z})={\pi }_{0}H(BG,B^{n}A)$$H^n(B G, \mathbb{Z}) = \pi_0 \mathbf{H}(B G, B^n A)

is the cohomology of $BG$ with coefficients in $A$. Every cocycle $c:BG\to B^{n}A$ represents a characteristic class $[c]$ on $BG$ with coefficients in $A$.

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 $X\stackrel{c}{\to }BG\stackrel{k}{\to }{ℬ}^{n}A$ 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 classes, Stiefel-Whitney classes, Wu classes;

• for $G=\mathrm{SO}$ the special orthogonal group:

  Euler classes;

• for $G=U$ the unitary group:

  Chern classes;

  Todd class

• for $G=BU(1)$ the circle 2-group:

  the Dixmier-Douady class.

Of line bundles

• canonical characteristic class, Theta characteristic

Chern character

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

$k$-Invariants

• k-invariants

Level in $\mathrm{\infty }$-Chern-Simons theory

• level (Chern-Simons theory)

Of subspaces

• Maslov index

Classes in the sense of Fuks

In (Fuks (1987), section 7) an axiomatization of characteristic classes is proposed. We review the definition and discuss how it is a special case of the one given above.

Fuks’s definition

Fuks considers a base category $𝒯$ of “spaces” and a category $𝒮$ of spaces with a structure (for example, space together with a vector bundle on it), this category should be a category over $𝒯$, i.e. at least equipped with a functor $U:𝒮\to 𝒯$.

A morphism of categories with structures is a morphism in the overcategory Cat$/𝒯$, i.e. a morphism $U\to U\prime$ is a functor $F:\mathrm{dom}(U)\to \mathrm{dom}(U\prime )$ such that $U\prime F=U$.

Suppose now the category $𝒯$ is equipped with a cohomology theory which is, for purposes of this definition, a functor of the form $H:{𝒯}^{\mathrm{op}}\to A$ where $A$ is some concrete category, typically category of T-algebras for some algebraic theory in Set, e.g. the category of abelian groups. Define $\mathscr{H}={\mathscr{H}}_H$ as a category whose objects are pairs $(X,a)$ where $X$ is a space (= object in $𝒯$) and $a\in H(X)$. This makes sense as $A$ is a concrete category. A morphism $(X,a)\to (Y,b)$ is a morphism $f:X\to Y$ such that $H(f)(b)=a$. We also denote $f^{*}=H(f)$, hence $f^{*}(b)=a$.

A characteristic class of structures of type $𝒮$ with values in $H$ in the sense of (Fuks) is a morphism of structures $h:𝒮\to {\mathscr{H}}_H$ over $𝒯$. In other words, to each structure $S$ of the type $𝒮$ over a space $X$ in $𝒯$ it assigns an element $h(S)$ in $H(X)$ such that for a morphism $t:S\to T$ in $𝒮$ the homomorphism $(U(t){)}^{*}:H(Y)\to H(X)$, where $Y=U(T)$, sends $h(S)$ to $h(T)$.

Discussion

Notice that ${\mathscr{H}}_H\to 𝒯$ in the above is nothing but the fibered category that under the Grothendieck construction is an equivalent incarnation of the presheaf $H$. In fact, since $A$ in the above is assume to be just a 1-category of sets with structure, ${\mathscr{H}}_H$ is just its category of elements of $H$.

Similarly in all applications that arise in practice (for instance for the structure of vector bundles) that was mentioned, the functor $𝒮\to 𝒯$ is a fibered category, too, corresponding under the inverse of the Grothendieck construction to a prestack $F_{𝒮}$.

Therefore morphisms of fibered categories over $𝒯$

$$c:𝒮\to {\mathscr{H}}_H$$c : \mathcal{S} \to \mathcal{H}_H

are equivalently morphisms of (pre)stacks

$$c:F_{𝒮}\to H.$$c : F_{\mathcal{S}} \to H \,.

In either picture, these are morphism in a 2-topos over the site $𝒯$.

So, as before, for $X\in 𝒯$ some space, a $𝒮$-structure on $X$ (for instance a vector bundle) is a moprhism in the topos

$$g:X\to F_{𝒮}$$g : X \to F_{\mathcal{S}}

(in this setup simply by the 2-Yoneda lemma) and the characteristic class $[c(g)]$ of that bundle is the bullback of that universal class $c$, hence the class represented by the composite

$$c(g):X\stackrel{g}{\to }{F}_{𝒮}\stackrel{c}{\to }H.$$c(g) : X \stackrel{g}{\to} F_{\mathcal{S}} \stackrel{c}{\to} H \,.

Related concepts

• universal characteristic class, universal characteristic map;

• differential characteristic class

References

A standard textbook is

• John Milnor, Jim Stasheff, Characteristic classes, Princeton Univ. Press (1974)

A concise introduction is in chapter 23

• Peter May, A concise course in algebraic topology (pdf)

Further texts include

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

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

• Shigeyuki Morita, Geometry of characteristic classes, Transl. Math. Mon. 199, AMS 2001

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

• D. B. Fuks, Непреривные когомологии топологических групп и характеристические классы , appendix to the Russian translation of K. S. Brown, Cohomology of groups, Moskva, Mir 1987.