Clutching construction

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

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 hemispheres, 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).

Generalization

The above can be generalized by replacing $D_{\pm }^{n}$ and $S^{n}$ 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

Let $p\colon M\to N$ be a fibre bundle with fibre $F$ . Let ${\mathcal {U}}$ be a collection of pairs $(U_{i},q_{i})$ such that $q_{i}\colon p^{-1}(U_{i})\to N\times F$ is a local trivialization of $p$ over $U_{i}\subset N$ . Moreover, we demand that the union of all the sets $U_{i}$ is $N$ (i.e. the collection is an atlas of trivializations $\coprod _{i}U_{i}=N$ ).

Consider the space $\coprod _{i}U_{i}\times F$ modulo the equivalence relation $(u_{i},f_{i})\in U_{i}\times F$ is equivalent to $(u_{j},f_{j})\in U_{j}\times F$ if and only if $U_{i}\cap U_{j}\neq \phi$ and $q_{i}\circ q_{j}^{-1}(u_{j},f_{j})=(u_{i},f_{i})$ . By design, the local trivializations $q_{i}$ give a fibrewise equivalence between this quotient space and the fibre bundle $p$ .

Consider the space $\coprod _{i}U_{i}\times \operatorname {Homeo} (F)$ modulo the equivalence relation $(u_{i},h_{i})\in U_{i}\times \operatorname {Homeo} (F)$ is equivalent to $(u_{j},h_{j})\in U_{j}\times \operatorname {Homeo} (F)$ if and only if $U_{i}\cap U_{j}\neq \phi$ and consider $q_{i}\circ q_{j}^{-1}$ to be a map $q_{i}\circ q_{j}^{-1}:U_{i}\cap U_{j}\to \operatorname {Homeo} (F)$ then we demand that $q_{i}\circ q_{j}^{-1}(u_{j})(h_{j})=h_{i}$ . That is, in our re-construction of $p$ we are replacing the fibre $F$ by the topological group of homeomorphisms of the fibre, $\operatorname {Homeo} (F)$ . If the structure group of the bundle is known to reduce, you could replace $\operatorname {Homeo} (F)$ with the reduced structure group. This is a bundle over $N$ with fibre $\operatorname {Homeo} (F)$ and is a principal bundle. Denote it by $p\colon M_{p}\to N$ . The relation to the previous bundle is induced from the principal bundle: $(M_{p}\times F)/\operatorname {Homeo} (F)=M$ .

So we have a principal bundle $\operatorname {Homeo} (F)\to M_{p}\to N$ . The theory of classifying spaces gives us an induced push-forward fibration $M_{p}\to N\to B(\operatorname {Homeo} (F))$ where $B(Homeo(F))$ is the classifying space of $\operatorname {Homeo} (F)$ . Here is an outline:

Given a $G$ -principal bundle $G\to M_{p}\to N$ , consider the space $M_{p}\times _{G}EG$ . This space is a fibration in two different ways:

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.

2) Project onto the second factor: $M_{p}\times _{G}EG\to EG/G=BG$ . The fibre in this case is $M_{p}$ .

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 $p\colon M\to N$ 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

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 halves along their boundary. The halves are a priori identified (with the standard ball), 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 hemispheres. 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.

Examples

The clutching construction is used to form the chiral anomaly, by glueing 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 $\pi _{3}.$ )

Similar constructions can be found for various instantons, including the Wess–Zumino–Witten model.

References

Allen Hatcher's book-in-progress Vector Bundles & K-Theory version 2.0, p. 22.