
Clutching construction
Encyclopedia
In topology
, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres.
as the union of the upper and lower hemispheres
and
along their intersection, the equator, an
.
Given trivialized fiber bundle
s with fiber F and structure group G over the two disks, then given a map
(called the clutching map), glue the two trivial bundles together via f.
Formally, it is the coequalizer
of the inclusions
via
and
: glue the two bundles together on the boundary, with a twist.
Thus we have a map
: clutching information on the equator yields a fiber bundle on the total space.
In the case of vector bundles, this yields
, and indeed this map is an isomorphism (under connect sum of spheres on the right).
, that is, a space X, together with two closed subsets A and B whose union is X. Then a clutching map on
gives a vector bundle on X.
be a fibre bundle with fibre
. Let
be a collection of pairs
such that
is a local trivialization of
over
. Moreover, we demand that the union of all the sets
is
(ie: the collection is an atlas of trivializations
).
Consider the space
modulo the equivalence relation
is equivalent to
if and only if
and
. By design, the local trivializations
give a fibrewise equivalence between this quotient space and the fibre bundle
.
Consider the space
modulo the equivalence relation
is equivalent to
if and only if
and consider
to be a map
then we demand that
.
Ie: in our re-construction of
we are replacing the fibre
by the topological group of homeomorphisms of the fibre,
. If the structure group of the bundle is known to reduce, you could replace
with the reduced structure group. This is a bundle over
with fibre
and is a principal bundle. Denote it by
. The relation to the previous bundle is induced from the principal bundle:
.
So we have a principal bundle
. The theory of classifying spaces gives us an induced push-forward fibration
where
is the classifying space of
. Here is an outline:
Given a
-principal bundle
, consider the space
. This space is a fibration in two different ways:
1) Project onto the first factor:
. The fibre in this case is
, which is a contractible space by the definition of a classifying space.
2) Project onto the second factor:
. The fibre in this case is
.
Thus we have a fibration
. This map is called the classifying map of the fibre bundle
since 1) the principal bundle
is the pull-back of the bundle
along the classifying map and 2) The bundle
is induced from the principal bundle as above.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres.
Definition
Consider the sphere



Given trivialized fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
s with fiber F and structure group G over the two disks, then given a map

Formally, it is the coequalizer
Coequalizer
In category theory, a coequalizer is a generalization of a quotient by an equivalence relation to objects in an arbitrary category...
of the inclusions



Thus we have a map

In the case of vector bundles, this yields

Generalization
The above can be generalized by replacing the disks and sphere with any closed triad

Classifying map construction
Let









Consider the space







Consider the space







Ie: in our re-construction of








So we have a principal bundle




Given a



1) Project onto the first factor:


2) Project onto the second factor:


Thus we have a fibration





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 disks along their boundary. The disks are a priori identified (with the standard disk), and points on the boundary sphere do not in general go to their corresponding points on the other boundary sphere. This is a map
: the gluing is non-trivial in the base.
- In the clutching construction, you glue two bundles together over the boundary of their base disks. 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
: the gluing is trivial in the base, but not in the fibers.