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\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 D^n_\pm 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 \colon 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 \colon 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 \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 \operatorname{Homeo}(F) modulo the equivalence relation (u_i,h_i)\in U_i \times \operatorname{Homeo}(F) is equivalent to (u_j,h_j)\in U_j \times \operatorname{Homeo}(F) if and only if U_i \cap U_j \neq \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 \operatorname{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, \operatorname{Homeo}(F). If the structure group of the bundle is known to reduce, you could replace \operatorname{Homeo}(F) with the reduced structure group. This is a bundle over N with fibre \operatorname{Homeo}(F) and is a principal bundle. Denote it by p \colon M_p \to N. The relation to the previous bundle is induced from the principal bundle: (M_p \times F)/\operatorname{Homeo}(F) = M.

So we have a principal bundle \operatorname{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(\operatorname{Homeo}(F)) where B(\operatorname{Homeo}(F)) is the classifying space of \operatorname{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 \colon 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

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 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.[1]

See also

References

  1. ^ Yao, Yuan; Hsieh, Chang-Tse; Oshikawa, Masaki (2019). "Anomaly Matching and Symmetry-Protected Critical Phases in SU(N) Spin Systems in 1+1 Dimensions". Physical Review Letters. 123 (18). arXiv:1805.06885. doi:10.1103/PhysRevLett.123.180201. PMID 31763914