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 D^n_+ and D^n_- 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 f\colon S^{n-1} \to G (called the clutching map), glue the two trivial bundles together via f.

Formally, it is the coequalizer of the inclusions S^{n-1} \times F \to D^n_+ \times F \coprod D^n_- \times F via (x,v) \mapsto (x,v) \in D^n_+ \times F and (x,v) \mapsto (x,f(x)(v)) \in D^n_- \times F: glue the two bundles together on the boundary, with a twist.

Thus we have a map \pi_{n-1} G \to \text{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 \pi_{n-1} O(k) \to \text{Vect}_k(S^n), 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 A \cap B 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 S^{n-1} \to 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} \to G: the gluing is trivial in the base, but not in the fibers.

[edit] References

version 2.0, p. 22.