Clutching construction
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 formula_1 as the union of the upper and lower hemispheres formula_2 and formula_3 along their intersection, the equator, an formula_4. Given trivialized fiber bundles with fiber formula_5 and structure group formula_6 over the two hemispheres, then given a map formula_7 (called the "clutching map"), glue the two trivial bundles together via "f". Formally, it is the coequalizer of the inclusions formula_8 via formula_9 and formula_10: glue the two bundles together on the boundary , with a twist. Thus we have a map formula_11: clutching information on the equator yields a fiber bundle on the total space. In the case of vector bundles, this yields formula_12, and indeed this map is an isomorphism (under connect sum of spheres on the right). Generalization. The above can be generalized by replacing formula_13 and formula_1 with any closed triad formula_15, that is, a space "X", together with two closed subsets "A" and "B" whose union is "X". Then a clutching map on formula_16 gives a vector bundle on "X". Classifying map construction. Let formula_17 be a fibre bundle  with fibre formula_5. Let formula_19 be a collection of pairs formula_20 such that formula_21 is a local trivialization of formula_22 over formula_23. Moreover, we demand that the union of all the sets formula_24 is formula_25 (i.e. the collection is an atlas of trivializations formula_26). Consider the space formula_27 modulo the equivalence relation formula_28 is equivalent to formula_29 if and only if formula_30 and formula_31. By design, the local trivializations formula_32 give a fibrewise equivalence between this quotient space and the fibre bundle formula_22. Consider the space formula_34 modulo the equivalence relation formula_35 is equivalent to formula_36 if and only if  formula_30 and consider formula_38 to be a map formula_39 then we demand that formula_40. That is, in our re-construction of formula_22 we are replacing the fibre formula_5 by the topological group of homeomorphisms of the fibre, formula_43. If the structure group of the bundle is known to reduce, you could replace formula_43 with the reduced structure group. This is a bundle over formula_25 with fibre formula_43 and is a principal bundle. Denote it by formula_47. The relation to the previous bundle is induced from the principal bundle: formula_48. So we have a principal bundle formula_49. The theory of classifying  spaces gives us an induced push-forward fibration formula_50 where formula_51 is the classifying space of formula_43. Here is an outline: Given a formula_6-principal bundle formula_54, consider the space formula_55. This space is a fibration in two different ways: 1) Project onto the first factor: formula_56. The fibre in this case is formula_57, which is a contractible space by the definition of a classifying space. 2) Project onto the second factor: formula_58. The fibre in this case is formula_59. Thus we have a fibration formula_60. This map is called the classifying map of the fibre bundle formula_17 since 1 ) the principal bundle formula_54 is the pull-back of the bundle formula_63 along the classifying map and 2) The bundle formula_22 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 glueing 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 formula_67) Similar constructions can be found for various instantons, including the Wess–Zumino–Witten model.