Clutching construction

Short description: Topological construct

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 $f:S^{n-1}\to G$ (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 $D_{\pm }^n$ 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 $p:M\to N$ be a fibre bundle with fibre $F$. Let ${𝒰}$ 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 $U_i\cap U_j\ne \varnothing$ 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 $\coprod _i{U}_i\times \mathrm{Homeo}(F)$ modulo the equivalence relation $(u_i,h_i)\in U_i\times \mathrm{Homeo}(F)$ is equivalent to $(u_j,h_j)\in U_j\times \mathrm{Homeo}(F)$ if and only if $U_i\cap U_j\ne \varnothing$ and consider $q_i\circ q_j^{-1}$ to be a map $q_i\circ q_j^{-1}:U_i\cap U_j\to \mathrm{Homeo}(F)$ 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, $\mathrm{Homeo}(F)$. If the structure group of the bundle is known to reduce, you could replace $\mathrm{Homeo}(F)$ with the reduced structure group. This is a bundle over $N$ with fibre $\mathrm{Homeo}(F)$ 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: $(M_p\times F)/\mathrm{Homeo}(F)=M$.

So we have a principal bundle $\mathrm{Homeo}(F)\to M_p\to N$. The theory of classifying spaces gives us an induced push-forward fibration $M_p\to N\to B(\mathrm{Homeo}(F))$ where $B(\mathrm{Homeo}(F))$ is the classifying space of $\mathrm{Homeo}(F)$. 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 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.

See also

• Alexander trick

References

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

External links

• clutching construction on nLab