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

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.

References

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