Classifying space

This article covers the classifying space of a topological group. Generalizations of this concept are the classifying space of a family of subgroups and the classifying space of a category .

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

${\displaystyle \left\{G{\text{-Prinzipalbündel über }}X\right\}=\left[X,BG\right]}$,

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 .

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

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