# 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: ${\ displaystyle G}$ ${\ displaystyle \ xi \ colon EG \ to BG}$${\ displaystyle G}$

• For every numerable -principal bundle there is a continuous mapping such that the bundles and are isomorphic .${\ displaystyle G}$${\ displaystyle \ pi \ colon E \ to X}$${\ displaystyle f \ colon X \ to BG}$${\ displaystyle f ^ {*} \ xi}$${\ displaystyle \ pi}$
• For two mappings the bundles are isomorphic if and only if they are homotopic .${\ displaystyle f, g \ colon X \ to BG}$${\ displaystyle f ^ {*} \ xi, g ^ {*} \ xi}$${\ displaystyle f, g}$

So you have a bijection

${\ displaystyle \ left \ {G {\ text {-principal bundle over}} X \ right \} = \ left [X, BG \ right]}$,

where denotes the homotopy classes of images . ${\ displaystyle \ left [X, BG \ right]}$${\ displaystyle X \ to BG}$

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. ${\ displaystyle G}$ ${\ displaystyle BG}$${\ displaystyle G}$${\ displaystyle BG}$

## 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 .) ${\ displaystyle EG = G * G * \ ldots * G * \ ldots}$${\ displaystyle G}$${\ displaystyle \ textstyle \ sum _ {i} t_ {i} g_ {i}}$${\ displaystyle \ textstyle g_ {i} \ in G, t_ {i} \ in \ left [0,1 \ right], \ sum _ {i} t_ {i} = 1}$${\ displaystyle t_ {i} \ not = 0}$${\ displaystyle 0g_ {i} = 0g_ {i} ^ {\ prime}}$${\ displaystyle g_ {i} \ not = g_ {i} ^ {\ prime}}$

The group works through the Milnor room . The quotient is the classifying space of the group , the principal bundle ${\ displaystyle G}$${\ displaystyle EG}$${\ displaystyle \ textstyle (\ sum _ {i} t_ {i} g_ {i}) g = \ sum _ {i} t_ {i} (g_ {i} g)}$${\ displaystyle BG: = EG / G}$${\ displaystyle G}$

${\ displaystyle \ xi \ colon EG \ to BG}$

is the universal bundle.

For various Lie groups, for example and there are simpler realizations of the classifying space using Graßmann manifolds, see below. ${\ displaystyle O (n)}$${\ displaystyle U (n)}$

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. ${\ displaystyle G}$${\ displaystyle E}$${\ displaystyle B = E / G}$${\ displaystyle BG}$${\ displaystyle E \ to B}$${\ displaystyle G}$

## Topology of the classifying space

${\ displaystyle EG}$is contractible . For the homotopy groups of true ${\ displaystyle BG}$

${\ displaystyle \ pi _ {i} (BG) = \ pi _ {i-1} (G)}$.

In particular, the following applies to groups provided with the discrete topology : ${\ displaystyle \ Gamma}$

${\ displaystyle \ pi _ {1} (B \ Gamma) = \ Gamma}$
${\ displaystyle \ pi _ {i} (B \ Gamma) = 0}$for .${\ displaystyle i \ not = 1}$

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 . ${\ displaystyle K \ to G}$${\ displaystyle BK \ to BG}$${\ displaystyle BO (n)}$${\ displaystyle BGL (n, \ mathbb {R})}$

## 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). ${\ displaystyle BG}$${\ displaystyle EG}$${\ displaystyle BG}$${\ displaystyle BG _ {\ delta}}$

• ${\ displaystyle B \ mathbb {Z} _ {n} = L_ {n} ^ {\ infty}}$with total space (in particular )${\ displaystyle S ^ {\ infty}}$${\ displaystyle B \ mathbb {Z} _ {2} = \ mathbb {R} P ^ {\ infty}}$
• ${\ displaystyle B \ mathbb {Z} = S ^ {1}}$ with total space ${\ displaystyle \ mathbb {R}}$
• ${\ displaystyle BS ^ {1} = \ mathbb {C} P ^ {\ infty}}$ with total space ${\ displaystyle S ^ {\ infty}}$
• ${\ displaystyle B (F_ {2}) = S ^ {1} \ vee S ^ {1}}$with total space (infinite tree of degree 4)${\ displaystyle {\ mathcal {T}}}$
• ${\ displaystyle BO (n) = BGL_ {n} (\ mathbb {R}) = G_ {n} (\ mathbb {R} ^ {\ infty})}$ with total space ${\ displaystyle V_ {n} (\ mathbb {R} ^ {\ infty})}$
• ${\ displaystyle B \ mathbb {R} = \ lbrace pt. \ rbrace}$ with total space ${\ displaystyle \ mathbb {R}}$
• ${\ displaystyle B \ langle a_ {1}, b_ {1}, \ ldots, a_ {g}, b_ {g} \; | \; \ prod _ {i = 1} ^ {g} [a_ {i} , b_ {i}] \ rangle = S_ {g}}$with total space (hyperbolic plane)${\ displaystyle {\ mathcal {H}}}$
• ${\ displaystyle B (G_ {1} \ times G_ {2}) = BG_ {1} \ times BG_ {2}}$

## 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 . ${\ displaystyle r}$${\ displaystyle GL (r, \ mathbb {R})}$${\ displaystyle BGL (r, \ mathbb {R})}$${\ displaystyle BO (r) \ simeq BGL (r, \ mathbb {R})}$${\ displaystyle BO (r)}$${\ displaystyle r}$${\ displaystyle BU (r) \ simeq BGL (r, \ mathbb {C})}$${\ displaystyle r}$

The Graßmann manifolds for or are explicit realizations of the classifying spaces or . ${\ displaystyle Gr _ {\ mathbb {K}} (r, \ infty)}$${\ displaystyle {\ mathbb {K}} = \ mathbb {R}}$${\ displaystyle {\ mathbb {K}} = \ mathbb {C}}$${\ displaystyle BO (r)}$${\ displaystyle BU (r)}$

Analogously, oriented vector bundles can be classified from the rank by the universal bundle via the Graßmann manifold of the oriented sub-vector spaces. ${\ displaystyle r}$${\ displaystyle BSO (r) = Gr ^ {+} (r, \ infty)}$

## 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 ${\ displaystyle n}$${\ displaystyle BSO (n)}$${\ displaystyle char (F) \ not = 2}$

${\ displaystyle H ^ {*} (BSO (2n), F) = F \ left [e, p_ {1}, \ ldots, p_ {n} \ right] / (e ^ {2} -p_ {n} )}$
${\ displaystyle H ^ {*} (BSO (2n + 1, F) = F \ left [p_ {1}, \ ldots, p_ {n} \ right]}$,

where denotes the Euler class and the Pontryagin classes . For is ${\ displaystyle e}$${\ displaystyle p_ {i}}$${\ displaystyle char (F) = 2}$

${\ displaystyle H ^ {*} (BSO (n), F) = F \ left [w_ {2}, \ ldots, w_ {n} \ right]}$,

being denoted the Stiefel-Whitney classes . ${\ displaystyle w_ {i}}$

## 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 .

1. 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.${\ displaystyle (U_ {i}) _ {i \ in I}}$ ${\ displaystyle 1 = \ Sigma _ {i \ in I} u_ {i}}$${\ displaystyle supp (u_ {i}) \ subset U_ {i}}$${\ displaystyle U_ {i}}$