# Classifying space

In mathematics , with the help of the **classifying space** and the **universal bundle of** a topological group G, the principal bundles with G are classified as a structure group. The classifying space and the universal bundle are characterized by a universal property , an explicit construction goes back to John Milnor . Bundles and their classification play an important role in mathematics and theoretical physics.

## Universal bundle

A - principal bundle is *universal bundle* when all (numerierbaren) -Prinzipalbündel by retracting can win the universal bundle; formal: if it has the following universal property for numerable G-principal bundles:

- For every numerable -principal bundle there is a continuous mapping such that the bundles and are isomorphic .
- For two mappings the bundles are isomorphic if and only if they are homotopic .

So you have a bijection

- ,

where denotes the homotopy classes of images .

The basis of a universal bundle is called the *classifying space of* the topological group . Using general nonsense one can easily show that (if a universal bundle exists) it is uniquely determined except for homotopy equivalence . The following construction, which goes back to Milnor, also proves the existence of classifying space.

## Milnor construction

The infinite union of countably many copies of the topological group is called Milnor space. The elements are of the form with and only finitely many . (Note also for .)

The group works through the Milnor room . The quotient is the classifying space of the group , the principal bundle

is the universal bundle.

For various Lie groups, for example and there are simpler realizations of the classifying space using Graßmann manifolds, see below.

In general, every free effect of gives a quotient on a contractible space , which is a classifying space (and thus in particular homotopy-equivalent to the above construction). The quotient mapping is then a universal -principal bundle.

## Topology of the classifying space

is contractible . For the homotopy groups of true

- .

In particular, the following applies to groups provided with the discrete topology :

- for .

The classifying space of a discrete group is therefore an Eilenberg-MacLane space .

If there is a homotopy equivalence , then there is also a homotopy equivalence. In particular, is homotopy equivalent to .

## Examples of classifying spaces

The following list gives examples of classifying spaces with associated total space (of the universal bundle) . Note that for topological groups generally does not match (the classifying space for the same group with the discrete topology).

- with total space (in particular )

- with total space

- with total space

- with total space (infinite tree of degree 4)

- with total space

- with total space

- with total space (hyperbolic plane)

## Vector bundle

For a real vector bundle of rank one has the frame bundle as a bundle over the same base. In particular , and because of the homotopy also a classifying space for real vector bundle of rank . Correspondingly, a classifying space for complex vector bundles is of rank .

The Graßmann manifolds for or are explicit realizations of the classifying spaces or .

Analogously, oriented vector bundles can be classified from the rank by the universal bundle via the Graßmann manifold of the oriented sub-vector spaces.

## Characteristic classes

Cohomology classes of a classifying space are used to define characteristic classes.

For example, one obtains characteristic classes of oriented vector bundles of rank from the cohomology of . For a field F with applies

- ,

where denotes the Euler class and the Pontryagin classes . For is

- ,

being denoted the Stiefel-Whitney classes .

## literature

- John Milnor:
*Construction of universal bundles.*Part I In:*Ann. of Math.*(2) 63 (1956), pp. 272-284. pdf ; Part II In:*Ann. of Math.*(2) 63 (1956), pp. 430-436. pdf - Dale Husemoller:
*Fiber bundles.*McGraw-Hill Book Co., New York / London / Sydney 1966, OCLC 909937420 . - Tammo tom Dieck:
*Topology.*(= de Gruyter textbook). Walter de Gruyter, Berlin 1991, ISBN 3-11-012463-7 .

## Web links

- Classifying space (Encyclopedia of Mathematics)
- BG (literature guide to algebraic topology)
- Mitchell: Notes on principal bundles and classifying spaces (Chapters 7-11)
- May: A concise course in algebraic topology (Chapter 23: "Characteristic classes of vector bundles")
- Stasheff: Cohomology of groups and classifying spaces

## Individual evidence

- ↑ An open covering of a topological space is called numerable if there is a locally finite decomposition of the one with . A principal bundle is called numerable if there is a numerable cover so that the bundle's restrictions on the can be trivialized . See Husemoller, op.cit., Section I.4.9.