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.

1 Definition
- 1.1 Generalization
- 1.2 Contrast with twisted spheres
2 References

[edit] 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 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.