Principal fibre bundle
A 
-fibration 
such that the group 
acts freely and perfectly on the space 
. The significance of principal fibre bundles lies in the fact that they make it possible to construct associated fibre bundles with fibre 
if a representation of 
in the group of homeomorphisms 
is given. Differentiable principal fibre bundles with Lie groups play an important role in the theory of connections and holonomy groups. For instance, let 
be a topological group with 
as a closed subgroup and let 
be the homogeneous space of left cosets of 
with respect to 
; the fibre bundle 
will then be principal. Further, let 
be a Milnor construction, i.e. the join of an infinite number of copies of 
, each point of which has the form
 |
where 
, 
, and where only finitely many 
are non-zero. The action of 
on 
defined by the formula 
is free, and the fibre bundle 
is a numerable principal fibre bundle.
Each fibre of a principal fibre bundle is homeomorphic to 
.
A morphism of principal fibre bundles is a morphism of the fibre bundles 
for which the mapping of the fibres 
induces a homomorphism of groups:
 |
where 
, 
. In particular, a morphism is called equivariant if 
is independent of 
, so that 
for any 
, 
. If 
and 
, an equivariant morphism is called a 
-morphism. Any 
-morphism (i.e. a 
-morphism over 
) is called a 
-isomorphism.
For any mapping 
and principal fibre bundle 
the induced fibre bundle 
is principal with the same group 
; moreover, the mapping 
is a 
-morphism which unambiguously determines the action of 
on the space 
. For instance, if the principal fibre bundle 
is trivial, it is isomorphic to the principal fibre bundle 
, where 
is the 
-bundle over a single point and 
is the constant mapping. The converse is also true, and for this reason principal fibre bundles with a section are trivial. For each numerable principal fibre bundle 
there exists a mapping 
such that 
is 
-isomorphic to 
, and for the principal fibre bundles 
and 
to be isomorphic, it is necessary and sufficient that 
and 
be homotopic (cf. Homotopy). This is the principal theorem on the homotopy classification of principal fibre bundles, which expresses the universality of the principal fibre bundle 
(obtained by Milnor's construction), with respect to the classifying mapping 
.
References
| [1] | R.L. Bishop, R.J. Crittenden, "Geometry of manifolds" , Acad. Press (1964) |
| [2] | K. Nomizu, "Lie groups and differential geometry" , Math. Soc. Japan (1956) |
| [3] | S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) |
| [4] | , Fibre spaces and their applications , Moscow (1958) (In Russian; translated from English) |
| [5] | N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951) |
| [6] | D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) |
Comments
Let 
be a principal fibre bundle. It is called numerable if there is a sequence 
of continuous mappings 
such that the open sets 
form an open covering (cf. Covering (of a set)) of 
and 
is trivializable over each 
(i.e. the restricted bundles 
are trivial, cf. Fibre space).
References
| [a1] | J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989) |
Principal fibre bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_fibre_bundle&oldid=48289