Classifying space

Formalism

A more formal statement takes into account that G may be a topological group (not simply a discrete group), and that group actions of G are taken to be continuous; in the absence of continuous actions the classifying space concept can be dealt with, in homotopy terms, via the Eilenberg–MacLane space construction. In homotopy theory the definition of a topological space BG, the classifying space for principal G-bundles, is given, together with the space EG which is the total space of the universal bundle over BG. That is, what is provided is in fact a continuous mapping

${\displaystyle \pi :EG⟶BG.}$

Assume that the homotopy category of CW complexes is the underlying category, from now on. The classifying property required of BG in fact relates to π. We must be able to say that given any principal G-bundle

${\displaystyle \gamma :Y⟶Z}$

over a space Z, there is a classifying map φ from Z to BG, such that ${\displaystyle \gamma }$ is the pullback of π along φ. In less abstract terms, the construction of ${\displaystyle \gamma }$ by 'twisting' should be reducible via φ to the twisting already expressed by the construction of π.

For this to be a useful concept, there evidently must be some reason to believe such spaces BG exist. The early work on classifying spaces introduced constructions (for example, the bar construction), that gave concrete descriptions of BG as a simplicial complex for an arbitrary discrete group. Such constructions make evident the connection with group cohomology.

Specifically, let EG be the weak simplicial complex whose n- simplices are the ordered (n+1)-tuples ${\displaystyle [g_0,\dots ,g_n]}$ of elements of G. Such an n-simplex attaches to the (n−1) simplices ${\displaystyle [g_0,\dots ,{\hat{g}}_i,\dots ,g_n]}$ in the same way a standard simplex attaches to its faces, where ${\displaystyle {\hat{g}}_i}$ means this vertex is deleted. The complex EG is contractible. The group G acts on EG by left multiplication,

${\displaystyle g\cdot [g_0,\dots ,g_n]=[g{g}_0,\dots ,g{g}_n],}$

and only the identity e takes any simplex to itself. Thus the action of G on EG is a covering space action and the quotient map ${\displaystyle EG\to EG/G}$ is the universal cover of the orbit space ${\displaystyle BG=EG/G}$, and BG is a ${\displaystyle K(G,1)}$.

In abstract terms (which are not those originally used around 1950 when the idea was first introduced) this is a question of whether a certain functor is representable: the contravariant functor from the homotopy category to the category of sets, defined by

h(Z) = set of isomorphism classes of principal G-bundles on Z.

The abstract conditions being known for this (Brown's representability theorem) ensure that the result, as an existence theorem, is affirmative and not too difficult.

Examples

1. The circle ${\displaystyle S^1}$ is a classifying space for the infinite cyclic group ${\displaystyle \mathbb{Z}.}$ The total space is ${\displaystyle E\mathbb{Z}=\mathbb{R}.}$
2. The n-torus ${\displaystyle {\mathbb{T}}^n}$ is a classifying space for ${\displaystyle {\mathbb{Z}}^n}$, the free abelian group of rank n. The total space is ${\displaystyle E{\mathbb{Z}}^n={\mathbb{R}}^n.}$
3. The wedge of n circles is a classifying space for the free group of rank n.
4. A closed (that is, compact and without boundary) connected surface S of genus at least 1 is a classifying space for its fundamental group ${\displaystyle {\pi }_1(S).}$
5. A closed (that is, compact and without boundary) connected hyperbolic manifold M is a classifying space for its fundamental group ${\displaystyle {\pi }_1(M)}$.
6. A finite locally connected CAT(0) cubical complex is a classifying space of its fundamental group.
7. The infinite-dimensional projective space ${\displaystyle {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}}$ (the direct limit of finite-dimensional projective spaces) is a classifying space for the cyclic group ${\displaystyle {\mathbb{Z}}_2=\mathbb{Z}/2\mathbb{Z}.}$ The total space is ${\displaystyle E{\mathbb{Z}}_2=S^{\mathrm{\infty }}}$ (the direct limit of spheres ${\displaystyle S^n.}$ Alternatively, one may use Hilbert space with the origin removed; it is contractible).
8. The space ${\displaystyle B{\mathbb{Z}}_n=S^{\mathrm{\infty }}/{\mathbb{Z}}_n}$ is the classifying space for the cyclic group ${\displaystyle {\mathbb{Z}}_n.}$ Here, ${\displaystyle S^{\mathrm{\infty }}}$ is understood to be a certain subset of the infinite dimensional Hilbert space ${\displaystyle {\mathbb{C}}^{\mathrm{\infty }}}$ with the origin removed; the cyclic group is considered to act on it by multiplication with roots of unity.
9. The unordered configuration space ${\displaystyle {\mathrm{UConf}}_n({\mathbb{R}}^2)}$ is the classifying space of the Artin braid group ${\displaystyle B_n}$, and the ordered configuration space ${\displaystyle {\mathrm{Conf}}_n({\mathbb{R}}^2)}$ is the classifying space for the pure Artin braid group ${\displaystyle P_n.}$
10. The (unordered) configuration space ${\displaystyle {\mathrm{UConf}}_n({\mathbb{R}}^{\mathrm{\infty }})}$ is a classifying space for the symmetric group ${\displaystyle S_n.}$
11. The infinite dimensional complex projective space ${\displaystyle {\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}}$ is the classifying space BS¹ for the circle S¹ thought of as a compact topological group.
12. The Grassmannian ${\displaystyle Gr(n,{\mathbb{R}}^{\mathrm{\infty }})}$ of n-planes in ${\displaystyle {\mathbb{R}}^{\mathrm{\infty }}}$ is the classifying space of the orthogonal group O(n). The total space is ${\displaystyle EO(n)=V(n,{\mathbb{R}}^{\mathrm{\infty }})}$, the Stiefel manifold of n-dimensional orthonormal frames in ${\displaystyle {\mathbb{R}}^{\mathrm{\infty }}.}$

Applications

This still leaves the question of doing effective calculations with BG; for example, the theory of characteristic classes is essentially the same as computing the cohomology groups of BG, at least within the restrictive terms of homotopy theory, for interesting groups G such as Lie groups (H. Cartan's theorem). As was shown by the Bott periodicity theorem, the homotopy groups of BG are also of fundamental interest.

An example of a classifying space is that when G is cyclic of order two; then BG is real projective space of infinite dimension, corresponding to the observation that EG can be taken as the contractible space resulting from removing the origin in an infinite-dimensional Hilbert space, with G acting via v going to −v, and allowing for homotopy equivalence in choosing BG. This example shows that classifying spaces may be complicated.

In relation with differential geometry (Chern–Weil theory) and the theory of Grassmannians, a much more hands-on approach to the theory is possible for cases such as the unitary groups that are of greatest interest. The construction of the Thom complex MG showed that the spaces BG were also implicated in cobordism theory, so that they assumed a central place in geometric considerations coming out of algebraic topology. Since group cohomology can (in many cases) be defined by the use of classifying spaces, they can also be seen as foundational in much homological algebra.

Generalizations include those for classifying foliations, and the classifying toposes for logical theories of the predicate calculus in intuitionistic logic that take the place of a 'space of models'.