Pullback bundle explained

In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle

\pi:E → B

and a continuous map

f:B' → B

one can define a "pullback" of

as a bundle

f^*E

over

. The fiber of

f^*E

over a point

is just the fiber of

over

f(b')

. Thus

f^*E

is the disjoint union of all these fibers equipped with a suitable topology.

Formal definition

Let

\pi:E → B

be a fiber bundle with abstract fiber

and let

f:B' → B

be a continuous map. Define the pullback bundle by

f^*E=\{(b',e)\inB' x E\midf(b')=\pi(e)\}\subseteqB' x E

\pi':f^{*E →}B'

given by the projection onto the first factor, i.e.,

\pi'(b',e)=b'.

The projection onto the second factor gives a map

h\colonf^*E\toE

such that the following diagram commutes:

\begin{array}{ccc} f^\astE&\stackrel{h}{\longrightarrow}&E\\ {\pi}'\downarrow&&\downarrow\pi\\ B'&\stackrelf{\longrightarrow}&B \end{array}

(U,\varphi)

is a local trivialization of

then

(f^-1U,\psi)

is a local trivialization of

f^*E

where

\psi(b',e)=(b',proj_{2(\varphi(e))).}

It then follows that

f^*E

is a fiber bundle over

with fiber

. The bundle

f^*E

is called the pullback of E by or the bundle induced by . The map

is then a bundle morphism covering

Properties

over

induces a section of

f^*E

over

, called the pullback section

f^*s

, simply by defining

f^{*s(b'):=(b',s(f(b')))}

for all

b'\inB'

If the bundle

E → B

has structure group

with transition functions

t_ij

(with respect to a family of local trivializations

\{(U_i,\varphi_i)\}

) then the pullback bundle

f^*E

also has structure group

. The transition functions in

f^*E

are given by

$f^t_ = t_ \circ f.$

E → B

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

f^*E

is given by

$(x,e)\cdot g = (x,e\cdot g)$

It then follows that the map

covering

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.

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.

Sources

Book: Steenrod, Norman . Norman Steenrod

. Norman Steenrod . The Topology of Fibre Bundles . registration . . Princeton . 1999 . 1951 . 9780691005485.

Book: Husemoller, Dale. Dale Husemoller

. Dale Husemoller . Fibre Bundles . Graduate Texts in Mathematics . 20 . . Third . New York . 1994 . 978-0-387-94087-8.

Book: Lawson . H. Blaine . H. Blaine Lawson . Michelsohn . Marie-Louise Michelsohn. Marie-Louise . Spin Geometry . . 978-0-691-08542-5 . 1989.

Pullback bundle explained

Formal definition

Properties

Bundles and sheaves

Sources

Further reading