Clutching Construction - Definition - Classifying Map Construction

Classifying Map Construction

Let be a fibre bundle with fibre . Let be a collection of pairs such that is a local trivialization of over . Moreover, we demand that the union of all the sets is (i.e. the collection is an atlas of trivializations ).

Consider the space modulo the equivalence relation is equivalent to if and only if and . By design, the local trivializations give a fibrewise equivalence between this quotient space and the fibre bundle .

Consider the space modulo the equivalence relation is equivalent to if and only if and consider to be a map then we demand that . Ie: in our re-construction of we are replacing the fibre by the topological group of homeomorphisms of the fibre, . If the structure group of the bundle is known to reduce, you could replace with the reduced structure group. This is a bundle over with fibre and is a principal bundle. Denote it by . The relation to the previous bundle is induced from the principal bundle: .

So we have a principal bundle . The theory of classifying spaces gives us an induced push-forward fibration where is the classifying space of . Here is an outline:

Given a -principal bundle, consider the space . This space is a fibration in two different ways:

1) Project onto the first factor: . The fibre in this case is, which is a contractible space by the definition of a classifying space.

2) Project onto the second factor: . The fibre in this case is .

Thus we have a fibration . This map is called the classifying map of the fibre bundle since 1) the principal bundle is the pull-back of the bundle along the classifying map and 2) The bundle is induced from the principal bundle as above.