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universal bundle
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(Definition)
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Let  be a topological group. A universal bundle for  is a principal bundle
 such that for any principal bundle
 , with  a CW-complex, there is a map
 , unique up to homotopy, such that the pullback bundle
 is equivalent to  , that is such that there is a bundle map  .
with
 , such that any bundle map of any bundle over  extending  factors uniquely through  .
As is obvious from the universal property, the universal bundle for a group  is unique up to unique homotopy equivalence.
The base space  is often called a classifying space of  , since homotopy classes of maps to it from a given space classify  -bundles over that space.
There is a useful criterion for universality: a bundle is universal if and only if all the homotopy groups of  , its total space, are trivial. This allows us to construct the universal bundle any subgroup from that of a larger group. Assume  and that
 is a universal bundle for  . Then  also acts freely on  which is contractable so
 must be a universal bundle for  .
In 1956, John Milnor gave a general construction of the universal bundle for any topological group  (see Annals of Mathematics, Second Series, Volume 63 Issue 2 and Issue 3 for details). His construction uses the infinite join of the group  with itself to define the total space of the universal bundle.
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"universal bundle" is owned by mps. [ full author list (3) | owner history (1) ]
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(view preamble)
| Also defines: |
classifying space |
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Cross-references: infinite join, volume, series, subgroup, homotopy groups, universal, universality, classes, base, homotopy equivalence, group, universal property, obvious, factors, bundle map, equivalent, pullback bundle, homotopy, map, CW-complex, principal bundle, topological group
There is 1 reference to this entry.
This is version 11 of universal bundle, born on 2002-11-01, modified 2004-03-28.
Object id is 3556, canonical name is UniversalBundle.
Accessed 5262 times total.
Classification:
| AMS MSC: | 55R10 (Algebraic topology :: Fiber spaces and bundles :: Fiber bundles) | | | 55R15 (Algebraic topology :: Fiber spaces and bundles :: Classification) |
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Pending Errata and Addenda
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![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{E\ar[d]^{\pi}\ar[r]^{\varphi '(E)}&EG\ar[d]^p\ B\ar[r]^{\varphi '(B)}&BG} } \end{xy}$](http://images.planetmath.org:8080/cache/objects/3556/l2h/img10.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{E\ar[d]^{\pi}\ar[r]^{\varphi '(E)}&EG\ar[d]^p\ B\ar[r]^{\varphi '(B)}&BG} } \end{xy}$")