Clutching construction explained

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

Sⁿ

as the union of the upper and lower hemispheres

	n
D
	+

and

	n
D
	-

along their intersection, the equator, an

S^n-1

Given trivialized fiber bundles with fiber

and structure group

over the two hemispheres, then given a map

f\colonS^n-1\toG

(called the clutching map), glue the two trivial bundles together via f.

Formally, it is the coequalizer of the inclusions

S^n-1 x F\to

	n
D
	+

x F\coprod

	n
D
	-

x F

via

(x,v)\mapsto(x,v)\in

	n
D
	+

x F

and

(x,v)\mapsto(x,f(x)(v))\in

	n
D
	-

x F

: glue the two bundles together on the boundary, with a twist.

Thus we have a map

\pi_n-1G\to

	n)
Fib
	F(S

: clutching information on the equator yields a fiber bundle on the total space.

In the case of vector bundles, this yields

\pi_n-1O(k)\to

	n)
Vect
	k(S

, and indeed this map is an isomorphism (under connect sum of spheres on the right).

Generalization

The above can be generalized by replacing

	n
D
	\pm

and

Sⁿ

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\capB

gives a vector bundle on X.

Classifying map construction

Let

p\colonM\toN

be a fibre bundle with fibre

. Let

be a collection of pairs

(U_i,q_i)

such that

q_i\colonp^-1(U_i)\toN x F

is a local trivialization of

over

U_i\subsetN

. Moreover, we demand that the union of all the sets

U_i

(i.e. the collection is an atlas of trivializations

\coprod_iU_i=N

Consider the space

\coprod_iU_{i x}F

modulo the equivalence relation

(u_i,f_i)\inU_i x F

is equivalent to

(u_j,f_j)\inU_j x F

if and only if

U_i\capU_j ≠ \phi

and

q_i\circ

	-1
q
	j

(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

Consider the space

\coprod_iU_{i x}\operatorname{Homeo}(F)

modulo the equivalence relation

(u_i,h_i)\inU_i x \operatorname{Homeo}(F)

is equivalent to

(u_j,h_j)\inU_j x \operatorname{Homeo}(F)

if and only if

U_i\capU_j ≠ \phi

and consider

q_i\circ

	-1
q
	j

to be a map

q_i\circ

	-1
q
	j

:U_i\capU_j\to\operatorname{Homeo}(F)

then we demand that

q_i\circ

	-1
q
	j

(u_j)(h_j)=h_i

. That is, in our re-construction of

we are replacing the fibre

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

with fibre

\operatorname{Homeo}(F)

and is a principal bundle. Denote it by

p\colonM_p\toN

. The relation to the previous bundle is induced from the principal bundle:

(M_p x F)/\operatorname{Homeo}(F)=M

So we have a principal bundle

\operatorname{Homeo}(F)\toM_p\toN

. The theory of classifying spaces gives us an induced push-forward fibration

M_p\toN\toB(\operatorname{Homeo}(F))

where

B(\operatorname{Homeo}(F))

is the classifying space of

\operatorname{Homeo}(F)

. Here is an outline:

Given a

-principal bundle

G\toM_p\toN

, consider the space

M_p x _GEG

. This space is a fibration in two different ways:

1) Project onto the first factor:

M_p x _GEG\toM_p/G=N

. The fibre in this case is

, which is a contractible space by the definition of a classifying space.

2) Project onto the second factor:

M_p x _GEG\toEG/G=BG

. The fibre in this case is

M_p

Thus we have a fibration

M_p\toN\simeqM_{p x}_GEG\toBG

. This map is called the classifying map of the fibre bundle

p\colonM\toN

since 1) the principal bundle

G\toM_p\toN

is the pull-back of the bundle

G\toEG\toBG

along the classifying map and 2) The bundle

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\toS^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\toG

: 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.

References

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

External links

clutching construction on nLab