nLab
characteristic class

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Cohomology

cohomology

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Special notions

Variants

Extra structure

Operations

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Contents

Idea

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

The same applies with the object AA taken as the domain object: for BB yet another object, the BB-valued cohomology of AA is similarly H(A,B)=π 0H(A,B)H(A,B) = \pi_0 \mathbf{H}(A,B). For [k]H(A,B)[k] \in H(A,B) any cohomology class in there, we obtain an ∞-functor

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

from the AA-valued cohomology of XX to its BB-valued cohomology, simply from the composition operation

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

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

Notice that if A=BGA = \mathbf{B}G is connected, an AA-cocycle on XX is a GG-principal ∞-bundle. Hence characteristic classes are equivalently characteristic classes of principal \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 BB, notably for BB an Eilenberg-MacLane object B nK\mathbf{B}^n K for some component KK of a spectrum object. This serves to characterize cohomology with coefficients in a complicated object AA 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 nK)H^n(X,K) := H(X, \mathbf{B}^n K) a given characteristic class in degree nn assigns

[k()]:H(X,A)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:XAc : X \to A is associated the object PXP \to X that it classifies – its homotopy fiber – which may be thought of as an AA-principal ∞-bundle over XX with classifying map XAX \to A. One typically thinks of the characteristic class [k(c)][k(c)] as characterizing this principal ∞-bundle PP.

Examples

Characteristic classes of principal bundles

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

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

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

Generally,

H n(BG,)=π 0H(BG,B nA) H^n(B G, \mathbb{Z}) = \pi_0 \mathbf{H}(B G, B^n A)

is the cohomology of BGB G with coefficients in AA. Every cocycle c:BGB nAc : B G \to B^n A represents a characteristic class [c][c] on BGB G with coefficients in AA.

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

Notable families of examples include:

Of line bundles

Chern character

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

kk-Invariants

Level in \infty-Chern-Simons theory

Of Lagrangian submanifolds

A characteristic class of Lagrangian submanifolds is the 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\mathcal{T} of “spaces” and a category 𝒮\mathcal{S} of spaces with a structure (for example, space together with a vector bundle on it), this category should be a category over 𝒯\mathcal{T}, i.e. at least equipped with a functor U:𝒮𝒯U : \mathcal{S}\to\mathcal{T}.

A morphism of categories with structures is a morphism in the overcategory Cat/𝒯/\mathcal{T}, i.e. a morphism UUU\to U' is a functor F:dom(U)dom(U)F: dom(U)\to dom(U') such that UF=UU' F = U.

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

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

Discussion

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

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

Therefore morphisms of fibered categories over 𝒯\mathcal{T}

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

are equivalently morphisms of (pre)stacks

c:F 𝒮H. c : F_{\mathcal{S}} \to H \,.

In either picture, these are morphism in a 2-topos over the site 𝒯\mathcal{T}.

So, as before, for X𝒯X \in \mathcal{T} some space, a 𝒮\mathcal{S}-structure on XX (for instance a vector bundle) is a moprhism in the topos

g:XF 𝒮 g : X \to F_{\mathcal{S}}

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

c(g):XgF 𝒮cH. c(g) : X \stackrel{g}{\to} F_{\mathcal{S}} \stackrel{c}{\to} H \,.

References

A standard textbook is

A concise introduction is in chapter 23

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.

Revised on November 10, 2013 12:40:02 by Urs Schreiber (82.113.121.220)