Consider the sphere S n as the union of the upper and lower hemispheres D + n and D − n along their intersection, the equator, an S n − 1 .
Given trivialized fiber bundles with fiber F and structure group G over the two disks, then given a map f : S n − 1 → G (called the clutching map), glue the two trivial bundles together via f.
Formally, it is the coequalizer of the inclusions S n − 1 × F → D + n × F ∐ D − n × F via ( x , v ) ↦ ( x , v ) ∈ D + n × F and ( x , v ) ↦ ( x , f ( x ) ( v ) ) ∈ D − n × F : glue the two bundles together on the boundary, with a twist.
Thus we have a map π n − 1 G → Fib F ( S n ) : clutching information on the equator yields a fiber bundle on the total space.
In the case of vector bundles, this yields π n − 1 O ( k ) → Vect k ( S n ) , and indeed this map is an isomorphism (under connect sum of spheres on the right).
The above can be generalized by replacing the disks and sphere with any closed triad ( X ; A , B ) , that is, a space X, together with two closed subsets A and B whose union is X. Then a clutching map on A ∩ B gives a vector bundle on X.
Let p : M → N be a fibre bundle with fibre F . Let U be a collection of pairs ( U i , q i ) such that q i : p − 1 ( U i ) → N × F is a local trivialization of p over U i ⊂ N . Moreover, we demand that the union of all the sets U i is N (i.e. the collection is an atlas of trivializations ∐ i U i = N ).
Consider the space ∐ i U i × F modulo the equivalence relation ( u i , f i ) ∈ U i × F is equivalent to ( u j , f j ) ∈ U j × F if and only if U i ∩ U j ≠ ϕ and q i ∘ q j − 1 ( u j , f j ) = ( u i , f i ) . By design, the local trivializations q i give a fibrewise equivalence between this quotient space and the fibre bundle p .
Consider the space ∐ i U i × H o m e o ( F ) modulo the equivalence relation ( u i , h i ) ∈ U i × H o m e o ( F ) is equivalent to ( u j , h j ) ∈ U j × H o m e o ( F ) if and only if U i ∩ U j ≠ ϕ and consider q i ∘ q j − 1 to be a map q i ∘ q j − 1 : U i ∩ U j → H o m e o ( F ) then we demand that q i ∘ q j − 1 ( u j ) ( h j ) = h i . Ie: in our re-construction of p we are replacing the fibre F by the topological group of homeomorphisms of the fibre, H o m e o ( F ) . If the structure group of the bundle is known to reduce, you could replace H o m e o ( F ) with the reduced structure group. This is a bundle over B with fibre H o m e o ( F ) and is a principal bundle. Denote it by p : M p → N . The relation to the previous bundle is induced from the principal bundle: ( M p × F ) / H o m e o ( F ) = M .
So we have a principal bundle H o m e o ( F ) → M p → N . The theory of classifying spaces gives us an induced push-forward fibration M p → N → B ( H o m e o ( F ) ) where B ( H o m e o ( F ) ) is the classifying space of H o m e o ( F ) . Here is an outline:
Given a G -principal bundle G → M p → N , consider the space M p × G E G . This space is a fibration in two different ways:
1) Project onto the first factor: M p × G E G → M p / G = N . The fibre in this case is E G , which is a contractible space by the definition of a classifying space.
2) Project onto the second factor: M p × G E G → E G / G = B G . The fibre in this case is M p .
Thus we have a fibration M p → N ≃ M p × G E G → B G . This map is called the classifying map of the fibre bundle p : M → N since 1) the principal bundle G → M p → N is the pull-back of the bundle G → E G → B G along the classifying map and 2) The bundle p is induced from the principal bundle as above.
Twisted spheres are sometimes referred to as a "clutching-type" construction, but this is misleading: the clutching construction is properly about fiber bundles.
In twisted spheres, you glue two disks along their boundary. The disks are a priori identified (with the standard disk), and points on the boundary sphere do not in general go to their corresponding points on the other boundary sphere. This is a map S n − 1 → S n − 1 : the gluing is non-trivial in the base.In the clutching construction, you glue two bundles together over the boundary of their base disks. The boundary spheres are glued together via the standard identification: each point goes to the corresponding one, but each fiber has a twist. This is a map S n − 1 → G : the gluing is trivial in the base, but not in the fibers.