# Clutching construction

In topology, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres.

## Definition

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.

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).

### 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.

### Classifying map construction

1) Project onto the first factor: $M_{p}\times _{G}EG\to M_{p}/G=N$ . The fibre in this case is $EG$ , which is a contractible space by the definition of a classifying space.

Thus we have a fibration $M_{p}\to N\simeq M_{p}\times _{G}EG\to BG$ . This map is called the classifying map of the fibre bundle $p:M\to N$ since 1) the principal bundle $G\to M_{p}\to N$ is the pull-back of the bundle $G\to EG\to BG$ along the classifying map and 2) The bundle $p$ is induced from the principal bundle as above.

### Contrast with twisted spheres

{{#invoke:see also|seealso}} Twisted spheres are sometimes referred to as a "clutching-type" construction, but this is misleading: the clutching construction is properly about fiber bundles.