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 ${\displaystyle S^{n}}$  as the union of the upper and lower hemispheres ${\displaystyle D_{+}^{n}}$  and ${\displaystyle D_{-}^{n}}$  along their intersection, the equator, an ${\displaystyle S^{n-1}}$ .

Given trivialized fiber bundles with fiber ${\displaystyle F}$  and structure group ${\displaystyle G}$  over the two hemispheres, then given a map ${\displaystyle f\colon S^{n-1}\to G}$  (called the clutching map), glue the two trivial bundles together via f.

Formally, it is the coequalizer of the inclusions ${\displaystyle S^{n-1}\times F\to D_{+}^{n}\times F\coprod D_{-}^{n}\times F}$  via ${\displaystyle (x,v)\mapsto (x,v)\in D_{+}^{n}\times F}$  and ${\displaystyle (x,v)\mapsto (x,f(x)(v))\in D_{-}^{n}\times F}$ : glue the two bundles together on the boundary, with a twist.

Thus we have a map ${\displaystyle \pi _{n-1}G\to {\text{Fib}}_{F}(S^{n})}$ : clutching information on the equator yields a fiber bundle on the total space.

In the case of vector bundles, this yields ${\displaystyle \pi _{n-1}O(k)\to {\text{Vect}}_{k}(S^{n})}$ , and indeed this map is an isomorphism (under connect sum of spheres on the right).

Generalization

The above can be generalized by replacing ${\displaystyle D_{\pm }^{n}}$  and ${\displaystyle S^{n}}$  with any closed triad ${\displaystyle (X;A,B)}$ , that is, a space X, together with two closed subsets A and B whose union is X. Then a clutching map on ${\displaystyle A\cap B}$  gives a vector bundle on X.

Classifying map construction

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

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

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

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

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

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

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

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

Contrast with twisted spheres

See also: Twisted sphere

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 ${\displaystyle S^{n-1}\to S^{n-1}}$ : 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 ${\displaystyle S^{n-1}\to G}$ : 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 ${\displaystyle \pi _{3}.}$ )

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

See also

• Alexander trick

References

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