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 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).

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

Let $p : M \to N$ be a fibre bundle with fibre $F$. Let $\mathcal U$ be a collection of pairs $(U_i,q_i)$ such that $q_i : p^{-1}(U_i) \to N \times F$ 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 $\coprod_i U_i\times Homeo(F)$ 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, $Homeo(F)$. If the structure group of the bundle is known to reduce, you could replace $Homeo(F)$ with the reduced structure group. This is a bundle over $B$ with fibre $Homeo(F)$ 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 $B(Homeo(F))$ is the classifying space of $Homeo(F)$. 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 $p : M \to N$ 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 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.

References

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