some examples of universal bundles
The universal bundle for a topological group G is usually written as π:EG→BG. Any principal G-bundle for which the total space is contractible
is universal
; this will help us to find universal bundles without worrying about Milnor’s construction of EG involving infinite joins.
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•
G=ℤ2: Eℤ2=S∞ and Bℤ2=ℝP∞.
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G=ℤn: Eℤn=S∞ and Bℤn=S∞/ℤn. Here ℤn acts on S∞ (considered as a subset of a separable
complex Hilbert space) via multiplication
with an n-th root of unity
.
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•
G=ℤn: Eℤn=ℝn and Bℤn=Tn.
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More generally, if G is any discrete group then one can take BG to be any Eilenberg-Mac Lane space
K(G,1) and EG to be its universal cover. Indeed EG is simply connected, and it follows from the lifting theorem that πn(EG)=0 for n≥0. This example includes the previous three and many more.
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G=S1: ES1=S∞ and BS1=ℂP∞.
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G=SU(2): ESU(2)=S∞ and BSU(2)=ℍP∞.
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G=O(n), the n-th orthogonal group
: EO(n)=V(∞,n), the manifold of frames of n orthonormal vectors in ℝ∞, and BO(n)=G(∞,n), the Grassmanian of n-planes in ℝ∞. The projection map is taking the subspace spanned by a frame of vectors.
Title | some examples of universal bundles |
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Canonical name | SomeExamplesOfUniversalBundles |
Date of creation | 2013-03-22 13:12:05 |
Last modified on | 2013-03-22 13:12:05 |
Owner | bwebste (988) |
Last modified by | bwebste (988) |
Numerical id | 11 |
Author | bwebste (988) |
Entry type | Example |
Classification | msc 55R15 |
Classification | msc 55R10 |
Synonym | universal family of spaces |
Related topic | CategoryOfQuantumAutomata |
Defines | Hilbert bundle |