In
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres.
Definition
Consider the sphere
as the union of the upper and lower hemispheres
and
along their intersection, the equator, an
.
Given trivialized
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
s with fiber
and structure group
over the two hemispheres, then given a map
(called the ''clutching map''), glue the two trivial bundles together via ''f''.
Formally, it is the
coequalizer
In category theory, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer.
Definition
A coequalizer is a co ...
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).
Generalization
The above can be generalized by replacing
and
with any closed triad
, 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''.
Classifying map construction
Let
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
(i.e. 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
. That is, 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.
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
: 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
: the gluing is trivial in the base, but not in the fibers.
Examples
The clutching construction is used to form the
chiral anomaly
In theoretical physics, a chiral anomaly is the anomalous nonconservation of a chiral current. In everyday terms, it is equivalent to a sealed box that contained equal numbers of left and right-handed bolts, but when opened was found to have mor ...
, 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
)
Similar constructions can be found for various
instanton
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...
s, including the
Wess–Zumino–Witten model
In theoretical physics and mathematics, a Wess–Zumino–Witten (WZW) model, also called a Wess–Zumino–Novikov–Witten model, is a type of two-dimensional conformal field theory named after Julius Wess, Bruno Zumino, Sergei Novikov and Edwa ...
.
See also
*
Alexander trick
Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander.
Statement
Two homeomorphisms of the ''n''-dimensional ball D^n which agree on the boundary sphere S^ are isotopic.
Mo ...
References
*
Allen Hatcher Allen, Allen's or Allens may refer to:
Buildings
* Allen Arena, an indoor arena at Lipscomb University in Nashville, Tennessee
* Allen Center, a skyscraper complex in downtown Houston, Texas
* Allen Fieldhouse, an indoor sports arena on the Univer ...
's book-in-progres
Vector Bundles & K-Theoryversion 2.0, p. 22.
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