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
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] Classifying map construction

Let $p : M \to N$ be a fibre bundle with fibre $F$ . Let $\mathcal U$ be a collection of pairs $(U i, f i)$ such that $q_i : p^{-1}(U_i) \to N \times F$ is a local trivialization of $p$ over $U_i \subset N$ . Moreover, we demand that the union of all the sets $U i$ is $N$ (ie: 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 $\coprod_i U_i\times Homeo(F)$ modulo the equivalence relation $(u_i,h_i)\in U_i \times Homeo(F)$ is equivalent to $(u_j,h_j)\in U_j \times Homeo(F)$ if and only if $U_i \cap U_j \neq \phi$ and consider $q_i \circ q_j^{-1}$ to be a map $q_i \circ q_j^{-1} : U_i \cap U_j \to Homeo(F)$ then we demand that $q_i \circ q_j^{-1}(u_j)(h_j)=h_i$ . Ie: in our re-construction of $p$ we are replacing the fibre $F$ by the topological group of homeomorphisms of the fibre, $H o m e o (F)$ . If the structure group of the bundle is known to reduce, you could replace $H o m e o (F)$ with the reduced structure group. This is a bundle over $B$ with fibre $H o m e o (F)$ and is a principal bundle. Denote it by $p : M_p \to N$ . The relation to the previous bundle is induced from the principal bundle: $(M_p \times F)/Homeo(F) = M$ .

So we have a principal bundle $Homeo(F) \to M_p \to N$ . The theory of classifying spaces gives us an induced push-forward fibration $M_p \to N \to B(Homeo(F))$ where $B (H o m e o (F))$ is the classifying space of $H o m e o (F)$ . 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 $E G$ , 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 $p : M \to N$ 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.

[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.