In topology, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres.

Definition

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 hemispheres, then given a map (called the clutching map), glue the two trivial bundles together via f. Formally, it is the coequalizer of the inclusions via and : glue the two bundles together on the boundary, with a twist. Thus we have a map : clutching information on the equator yields a fiber bundle on the total space. In the case of vector bundles, this yields, and indeed this map is an isomorphism (under connect sum of spheres on the right).

Generalization

The above can be generalized by replacing D^n_\pm and S^n 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 \cap B gives a vector bundle on X.

Classifying map construction

Let be a fibre bundle with fibre F. Let \mathcal U be a collection of pairs (U_i,q_i) such that is a local trivialization of p over. Moreover, we demand that the union of all the sets U_i is N (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 q_i give a fibrewise equivalence between this quotient space and the fibre bundle p. Consider the space modulo the equivalence relation is equivalent to if and only if and consider to be a map then we demand that. That is, in our re-construction of p we are replacing the fibre F 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 N 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 G-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 EG, which is a contractible space by the definition of a classifying space.
2. Project onto the second factor: . The fibre in this case is M_p. 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 p is induced from the principal bundle as above.

Contrast with twisted spheres

Twisted spheres are sometimes referred to as a "clutching-type" construction, but this is misleading: the clutching construction is properly about fiber bundles.

Examples

The clutching construction is used to form the chiral anomaly, by gluing together a pair of self-dual curvature forms. Such forms are locally exact on each hemisphere, as they are differentials of the Chern–Simons 3-form; by gluing them together, the curvature form is no longer globally exact (and so has a non-trivial homotopy group \pi_3.) Similar constructions can be found for various instantons, including the Wess–Zumino–Witten model.