In mathematics, the universal bundle in the theory of fiber bundles with structure group a given

topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...

, is a specific bundle over a

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'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...

, such that every bundle with the given structure group over is a

pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...

by means of a

continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...

Existence of a universal bundle

In the CW complex category

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

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...

es, existence theorems for universal bundles arise from

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

For compact Lie groups

We will first prove: :Proposition. Let be a compact Lie group. 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 In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...

for big enough. If we find then we can take to be . The construction of is given in classifying space for . 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 , unique 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 :

\begin
P          & \to                   & P\times EG   & \to & EG \\
\downarrow &                       & \downarrow   &     & \downarrow \pi \\
M          & \to_ & P\times_G EG & \to & BG
\end

Since is a fibration with contractible fibre , sections of exist.A.~Dold -- ''Partitions of Unity in the Theory of Fibrations'', Annals of Mathematics, vol. 78, No 2 (1963) 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: :

\Phi: \left \  \to  P

Now, simply define a section by :

\begin
M \to P\times_G EG \\
x \mapsto \lbrack \Phi(x,u),u \rbrack
\end

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 . These spaces are of interest in their own right, despite typically being

contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...

. For example, in defining the homotopy quotient or homotopy orbit space of a

group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

of , in cases where the

orbit space In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...

pathological Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in th ...

(in the sense of being a non-

Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...

, for example). The idea, if acts on the space , is to consider instead the action on , and corresponding quotient. See

equivariant cohomology In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ordi ...

for more detailed discussion. If is contractible then and are

homotopy equivalent In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...

spaces. But the diagonal action on , i.e. where acts on both and coordinates, may be

well-behaved In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved. Th ...

when the action on is not.

Examples

Classifying space for U(n) In mathematics, the classifying space for the unitary group U(''n'') is a space BU(''n'') together with a universal bundle EU(''n'') such that any hermitian bundle on a paracompact space ''X'' is the pull-back of EU(''n'') by a map ''X'' → BU(' ...

External links

PlanetMath page of universal bundle examples

Notes

{{Manifolds Fiber bundles Homotopy theory