In topology, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres.
as the union of the upper and lower hemispheres
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
: glue the two bundles together on the boundary, with a twist.
: 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).
, that is, a space X, together with two closed subsets A and B whose union is X.
gives a vector bundle on X.
Moreover, we demand that the union of all the sets
(i.e. the collection is an atlas of trivializations
modulo the equivalence relation
By design, the local trivializations
give a fibrewise equivalence between this quotient space and the fibre bundle
modulo the equivalence relation
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.
The relation to the previous bundle is induced from the principal bundle:
The theory of classifying spaces gives us an induced push-forward fibration
This space is a fibration in two different ways: 1) Project onto the first factor:
, which is a contractible space by the definition of a classifying space.
This map is called the classifying map of the fibre bundle
along the classifying map and 2) The bundle
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.
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
) Similar constructions can be found for various instantons, including the Wess–Zumino–Witten model.