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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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