Pullback bundle

Last updated

In mathematics, a pullback bundle or induced bundle [1] [2] [3] is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle π : E  B and a continuous map f : B  B one can define a "pullback" of E by f as a bundle f*E over B. The fiber of f*E over a point b in B is just the fiber of E over f(b). Thus f*E is the disjoint union of all these fibers equipped with a suitable topology.

Formal definition

Let π : EB be a fiber bundle with abstract fiber F and let f : BB be a continuous map. Define the pullback bundle by

${\displaystyle f^{*}E=\{(b',e)\in B'\times E\mid f(b')=\pi (e)\}\subseteq B'\times E}$

and equip it with the subspace topology and the projection map π : f*EB given by the projection onto the first factor, i.e.,

${\displaystyle \pi '(b',e)=b'.\,}$

The projection onto the second factor gives a map

${\displaystyle h\colon f^{*}E\to E}$

such that the following diagram commutes:

${\displaystyle {\begin{array}{ccc}f^{\ast }E&{\stackrel {h}{\longrightarrow }}&E\\{\pi }'\downarrow &&\downarrow \pi \\B'&{\stackrel {f}{\longrightarrow }}&B\end{array}}}$

If (U, φ) is a local trivialization of E then (f1U, ψ) is a local trivialization of f*E where

${\displaystyle \psi (b',e)=(b',{\mbox{proj}}_{2}(\varphi (e))).\,}$

It then follows that f*E is a fiber bundle over B with fiber F. The bundle f*E is called the pullback of E by f or the bundle induced by f. The map h is then a bundle morphism covering f.

Properties

Any section s of E over B induces a section of f*E, called the pullback sectionf*s, simply by defining

${\displaystyle f^{*}s=s\circ f}$.

If the bundle EB has structure group G with transition functions tij (with respect to a family of local trivializations {(Ui, φi)}) then the pullback bundle f*E also has structure group G. The transition functions in f*E are given by

${\displaystyle f^{*}t_{ij}=t_{ij}\circ f.}$

If EB is a vector bundle or principal bundle then so is the pullback f*E. In the case of a principal bundle the right action of G on f*E is given by

${\displaystyle (x,e)\cdot g=(x,e\cdot g)}$

It then follows that the map h covering f is equivariant and so defines a morphism of principal bundles.

In the language of category theory, the pullback bundle construction is an example of the more general categorical pullback. As such it satisfies the corresponding universal property.

The construction of the pullback bundle can be carried out in subcategories of the category of topological spaces, such as the category of smooth manifolds. The latter construction is useful in differential geometry and topology.

Examples: It is illuminating to consider the pullback of the degree 2 map from the circle to itself over the degree 3 or 4 map from the circle to itself. In such examples one sometimes gets a connected (e.g. choosing degree 3) and sometimes disconnected space (degree 4), but always several copies of the circle.

Bundles and sheaves

Bundles may also be described by their sheaves of sections. The pullback of bundles then corresponds to the inverse image of sheaves, which is a contravariant functor. A sheaf, however, is more naturally a covariant object, since it has a pushforward, called the direct image of a sheaf. The tension and interplay between bundles and sheaves, or inverse and direct image, can be advantageous in many areas of geometry. However, the direct image of a sheaf of sections of a bundle is not in general the sheaf of sections of some direct image bundle, so that although the notion of a 'pushforward of a bundle' is defined in some contexts (for example, the pushforward by a diffeomorphism), in general it is better understood in the category of sheaves, because the objects it creates cannot in general be bundles.

Related Research Articles

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In mathematics, a sheaf is a tool for systematically tracking data attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set.

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X, which is then called a vector bundle over X.

In mathematics, and particularly topology, a fiber bundle is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map, :, that in small regions of E behaves just like a projection from corresponding regions of to . The map , called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space is known as the total space of the fiber bundle, as the base space, and the fiber.

In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X × G of a space X with a group G. In the same way as with the Cartesian product. A principal bundle P is equipped with

1. An action of G on P, analogous to (x, g)h = for a product space.
2. A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) ↦ x.

In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space. A fibration is like a fiber bundle, except that the fibers need not be the same space, nor even homeomorphic; rather, they are just homotopy equivalent. Weak fibrations discard even this equivalence for a more technical property.

In the mathematical field of topology, a section of a fiber bundle is a continuous right inverse of the projection function . In other words, if is a fiber bundle over a base space, :

In mathematics, the theory of fiber bundles with a structure group allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of . For a fiber bundle F with structure group G, the transition functions of the fiber in an overlap of two coordinate systems Uα and Uβ are given as a G-valued function gαβ on UαUβ. One may then construct a fiber bundle F′ as a new fiber bundle having the same transition functions, but possibly a different fiber.

Suppose that φ : MN is a smooth map between smooth manifolds M and N. Then there is an associated linear map from the space of 1-forms on N to the space of 1-forms on M. This linear map is known as the pullback, and is frequently denoted by φ. More generally, any covariant tensor field – in particular any differential form – on N may be pulled back to M using φ.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.

In mathematics, a gerbe is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analogue of fibre bundles where the fibre is the classifying stack of a group. Gerbes provide a convenient, if highly abstract, language for dealing with many types of deformation questions especially in modern algebraic geometry. In addition, special cases of gerbes have been used more recently in differential topology and differential geometry to give alternative descriptions to certain cohomology classes and additional structures attached to them.

In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. The pullback is often written

In mathematics, local coefficients is an idea from algebraic topology, a kind of half-way stage between homology theory or cohomology theory with coefficients in the usual sense, in a fixed abelian group A, and general sheaf cohomology which, roughly speaking, allows coefficients to vary from point to point in a topological space X. Such a concept was introduced by Norman Steenrod in 1943.

In differential geometry, an Ehresmann connection is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it does not rely on the possible vector bundle structure of the underlying fiber bundle, but nevertheless, linear connections may be viewed as a special case. Another important special case of Ehresmann connections are principal connections on principal bundles, which are required to be equivariant in the principal Lie group action.

In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere bundle. The Gysin sequence is a useful tool for calculating the cohomology rings given the Euler class of the sphere bundle and vice versa. It was introduced by Gysin (1942), and is generalized by the Serre spectral sequence.

In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.

In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist.

This is a glossary of algebraic geometry.

This is a glossary of properties and concepts in algebraic topology in mathematics.

In mathematics, the base change theorems relate the direct image and the pull-back of sheaves. More precisely, they are about the base change map, given by the following natural transformation of sheaves:

References

1. Steenrod 1951 , p. 47
2. Husemoller 1994 , p. 18
3. Lawson & Michelsohn 1989 , p. 374

Sources

• Steenrod, Norman (1951). . Princeton: Princeton University Press. ISBN   0-691-00548-6.
• Husemoller, Dale (1994). Fibre Bundles. Graduate Texts in Mathematics. 20 (Third ed.). New York: Springer-Verlag. ISBN   978-0-387-94087-8.
• Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN   978-0-691-08542-5.
• Sharpe, R. W. (1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Graduate Texts in Mathematics. 166. New York: Springer-Verlag. ISBN   0-387-94732-9.