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 $$D^n_\pm$$ 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 $$B(\operatorname{Homeo}(F))$$ 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 gluing 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.