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 [math]\displaystyle{ S^n }[/math] as the union of the upper and lower hemispheres [math]\displaystyle{ D^n_+ }[/math] and [math]\displaystyle{ D^n_- }[/math] along their intersection, the equator, an [math]\displaystyle{ S^{n-1} }[/math].
Given trivialized fiber bundles with fiber [math]\displaystyle{ F }[/math] and structure group [math]\displaystyle{ G }[/math] over the two hemispheres, then given a map [math]\displaystyle{ f\colon S^{n-1} \to G }[/math] (called the clutching map), glue the two trivial bundles together via f.
Formally, it is the coequalizer of the inclusions [math]\displaystyle{ S^{n-1} \times F \to D^n_+ \times F \coprod D^n_- \times F }[/math] via [math]\displaystyle{ (x,v) \mapsto (x,v) \in D^n_+ \times F }[/math] and [math]\displaystyle{ (x,v) \mapsto (x,f(x)(v)) \in D^n_- \times F }[/math]: glue the two bundles together on the boundary, with a twist.
Thus we have a map [math]\displaystyle{ \pi_{n-1} G \to \text{Fib}_F(S^n) }[/math]: clutching information on the equator yields a fiber bundle on the total space.
In the case of vector bundles, this yields [math]\displaystyle{ \pi_{n-1} O(k) \to \text{Vect}_k(S^n) }[/math], and indeed this map is an isomorphism (under connect sum of spheres on the right).
Generalization
The above can be generalized by replacing [math]\displaystyle{ D^n_\pm }[/math] and [math]\displaystyle{ S^n }[/math] with any closed triad [math]\displaystyle{ (X;A,B) }[/math], that is, a space X, together with two closed subsets A and B whose union is X. Then a clutching map on [math]\displaystyle{ A \cap B }[/math] gives a vector bundle on X.
Classifying map construction
Let [math]\displaystyle{ p \colon M \to N }[/math] be a fibre bundle with fibre [math]\displaystyle{ F }[/math]. Let [math]\displaystyle{ \mathcal U }[/math] be a collection of pairs [math]\displaystyle{ (U_i,q_i) }[/math] such that [math]\displaystyle{ q_i \colon p^{-1}(U_i) \to N \times F }[/math] is a local trivialization of [math]\displaystyle{ p }[/math] over [math]\displaystyle{ U_i \subset N }[/math]. Moreover, we demand that the union of all the sets [math]\displaystyle{ U_i }[/math] is [math]\displaystyle{ N }[/math] (i.e. the collection is an atlas of trivializations [math]\displaystyle{ \coprod_i U_i = N }[/math]).
Consider the space [math]\displaystyle{ \coprod_i U_i\times F }[/math] modulo the equivalence relation [math]\displaystyle{ (u_i,f_i)\in U_i \times F }[/math] is equivalent to [math]\displaystyle{ (u_j,f_j)\in U_j \times F }[/math] if and only if [math]\displaystyle{ U_i \cap U_j \neq \phi }[/math] and [math]\displaystyle{ q_i \circ q_j^{-1}(u_j,f_j) = (u_i,f_i) }[/math]. By design, the local trivializations [math]\displaystyle{ q_i }[/math] give a fibrewise equivalence between this quotient space and the fibre bundle [math]\displaystyle{ p }[/math].
Consider the space [math]\displaystyle{ \coprod_i U_i\times \operatorname{Homeo}(F) }[/math] modulo the equivalence relation [math]\displaystyle{ (u_i,h_i)\in U_i \times \operatorname{Homeo}(F) }[/math] is equivalent to [math]\displaystyle{ (u_j,h_j)\in U_j \times \operatorname{Homeo}(F) }[/math] if and only if [math]\displaystyle{ U_i \cap U_j \neq \phi }[/math] and consider [math]\displaystyle{ q_i \circ q_j^{-1} }[/math] to be a map [math]\displaystyle{ q_i \circ q_j^{-1} : U_i \cap U_j \to \operatorname{Homeo}(F) }[/math] then we demand that [math]\displaystyle{ q_i \circ q_j^{-1}(u_j)(h_j)=h_i }[/math]. That is, in our re-construction of [math]\displaystyle{ p }[/math] we are replacing the fibre [math]\displaystyle{ F }[/math] by the topological group of homeomorphisms of the fibre, [math]\displaystyle{ \operatorname{Homeo}(F) }[/math]. If the structure group of the bundle is known to reduce, you could replace [math]\displaystyle{ \operatorname{Homeo}(F) }[/math] with the reduced structure group. This is a bundle over [math]\displaystyle{ N }[/math] with fibre [math]\displaystyle{ \operatorname{Homeo}(F) }[/math] and is a principal bundle. Denote it by [math]\displaystyle{ p \colon M_p \to N }[/math]. The relation to the previous bundle is induced from the principal bundle: [math]\displaystyle{ (M_p \times F)/\operatorname{Homeo}(F) = M }[/math].
So we have a principal bundle [math]\displaystyle{ \operatorname{Homeo}(F) \to M_p \to N }[/math]. The theory of classifying spaces gives us an induced push-forward fibration [math]\displaystyle{ M_p \to N \to B(\operatorname{Homeo}(F)) }[/math] where [math]\displaystyle{ B(\operatorname{Homeo}(F)) }[/math] is the classifying space of [math]\displaystyle{ \operatorname{Homeo}(F) }[/math]. Here is an outline:
Given a [math]\displaystyle{ G }[/math]-principal bundle [math]\displaystyle{ G \to M_p \to N }[/math], consider the space [math]\displaystyle{ M_p \times_{G} EG }[/math]. This space is a fibration in two different ways:
1) Project onto the first factor: [math]\displaystyle{ M_p \times_G EG \to M_p/G = N }[/math]. The fibre in this case is [math]\displaystyle{ EG }[/math], which is a contractible space by the definition of a classifying space.
2) Project onto the second factor: [math]\displaystyle{ M_p \times_G EG \to EG/G = BG }[/math]. The fibre in this case is [math]\displaystyle{ M_p }[/math].
Thus we have a fibration [math]\displaystyle{ M_p \to N \simeq M_p\times_G EG \to BG }[/math]. This map is called the classifying map of the fibre bundle [math]\displaystyle{ p \colon M \to N }[/math] since 1) the principal bundle [math]\displaystyle{ G \to M_p \to N }[/math] is the pull-back of the bundle [math]\displaystyle{ G \to EG \to BG }[/math] along the classifying map and 2) The bundle [math]\displaystyle{ p }[/math] 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 [math]\displaystyle{ S^{n-1} \to S^{n-1} }[/math]: 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 [math]\displaystyle{ S^{n-1} \to G }[/math]: 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 [math]\displaystyle{ \pi_3. }[/math])
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.
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