Clutching construction
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In topology, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres.
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[edit] Definition
Consider the sphere Sn as the union of the upper and lower hemispheres and along their intersection, the equator, an Sn − 1.
Given trivialized fiber bundles with fiber F and structure group G over the two disks, 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).
[edit] Generalization
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 gives a vector bundle on X.
[edit] 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.
- 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 : 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 : the gluing is trivial in the base, but not in the fibers.
[edit] References
- Allan Hatcher's book-in-progress Vector Bundles & K-Theory
version 2.0, p. 22.