In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the universal bundle in the theory of fiber bundle

Fiber bundle

In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...

s with structure group a given topological group

Topological group

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...

 G, is a specific bundle over a classifying space

Classifying space

In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...

 BG, such that every bundle with the given structure group G over M is a pullback

Pullback bundle

In mathematics, a pullback bundle or induced bundle is a useful construction in the theory of fiber bundles. Given a fiber bundle π : E → B and a continuous map f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′...

 by means of a continuous map

M → BG.

In the CW complex category

When the definition of the classifying space takes place within the homotopy category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

 of CW complex

CW complex

In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...

es, existence theorems for universal bundles arise from Brown's representability theorem

Brown's representability theorem

In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions on a contravariant functor F on the homotopy category Hot of pointed CW complexes, to the category of sets Set, to be a representable functor...

.

For compact Lie groups

We will first prove:

Proposition

Let be a compact Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

.
There exists a contractible space on which acts freely. The projection is a
-principal fibre bundle.

Proof
There exists an injection of into a unitary group

Unitary group

In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...

  for big enough.
If we find then we can take to be .

The construction of EU(n) is given in classifying space for U(n)

Classifying space for U(n)

In mathematics, the classifying space for the unitary group U is a space B together with a universal bundle E such that any hermitian bundle on a paracompact space X is the pull-back of E by a map X → B unique up to homotopy.This space with its universal fibration may be constructed as either# the...

.


The following Theorem is a corollary of the above Proposition.

Theorem

If is a paracompact manifold and is a principal -bundle, then there exists a map
, well defined up to homotopy, such that is isomorphic to , the pull-back
of the -bundle by .

Proof
On one hand, the pull-back of the bundle by the natural projection is the bundle . On the other hand, the pull-back of the principal -bundle by the projection
is also




Since is a fibration with contractible fibre ,
sections of exist. To such a section
we associate the composition with the projection . The map we get is the we were
looking for.

For the uniqueness up to homotopy, notice that there exists a one to one correspondence between maps
such that is isomorphic to and sections of . We have just seen
how to associate a to a section. Inversely, assume that is given. Let be an isomorphism
between and

.

Now, simply define a section by



Because all sections of are homotopic, the homotopy class of is unique.

Use in the study of group actions

The total space of a universal bundle is usually written EG. These spaces are of interest in their own right, despite typically being contractible. For example in defining the homotopy quotient or homotopy orbit space of a group action

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

 of G, in cases where the orbit space is pathological

Pathological (mathematics)

In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is well-behaved....

 (in the sense of being a non-Hausdorff space

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

, for example). The idea, if G acts on the space X, is to consider instead the action on

Y = X×EG,

and corresponding quotient. See equivariant cohomology

Equivariant cohomology

In mathematics, equivariant cohomology is a theory from algebraic topology which applies to spaces with a group action. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory....

 for more detailed discussion.

If EG is contractible then X and Y are homotopy equivalent spaces. But the diagonal action on Y, i.e. where G acts on both X and EG coordinates, may be well-behaved

Well-behaved

Mathematicians very frequently speak of whether a mathematical object — a function, a set, a space of one sort or another — is "well-behaved" or not. The term has no fixed formal definition, and is dependent on mathematical interests, fashion, and taste...

when the action on X is not.

External links

• PlanetMath page of universal bundle examples