Fibers

4.5 Fibers

Having deﬁned topological spaces and manifolds, which are the setting of further considerations, it is also required to deﬁne how such spaces may be combined and attached to each other. The next deﬁnitions provide the fundamentals of these endeavours and also provide the major abstraction for the storage of values in digital computers.

Deﬁnition 39 (Fibration) A continuous mapping (Deﬁnition 30)

p : ℰ → ℬ (4.40)

is a ﬁbration, if the following assertions hold. For a space $𝒳$ every homotopy (Deﬁnition 32)

f : 𝒳 × [0;1] → ℬ (4.41)

and a mapping

¯f : 𝒳 × {0} → ℰ (4.42)

the diagram

¯ 𝒳 × {0}fidX×{0}-----ℰ p (4.43) | | | fF ------- 𝒳 × [0;1] ℬ

commutes, especially noting $p ∘ F = f$ .

It is also commonly written as ﬁbration sequence

ℱ → ℰ → ℬ (4.44)

The ﬁrst arrow indicates the inclusion of the so called ﬁber $ℱ$ by means of a mapping into the total space $ℰ$ , while the second map is the ﬁbration to the base space $ℬ$ .

An important specialization, with an additional local requirement, can be deﬁned which has important realizations as demonstrated in Deﬁnition 52.

Deﬁnition 40 (Fiber bundle) The topological spaces $ℬ$ , called the base space, and $ℰ$ , called the total space, along with a surjective (Deﬁnition 23) projection (Deﬁnition 25) $π:ℰ→ℬ$ is known as a ﬁber bundle $(ℰ ,ℬ,π, ℱ )$ , if it locally satisﬁes the following condition:

For every element $x ∈ ℬ$ there exists an open neighbourhood $U$ such that the preimages of the projection $π$ , $π− 1(U )$ are homeomorphic to a product space $U × F$ , such that the following diagram commutes:

π −1(U )φ π-----U × F proj1 (4.45) | U ∈ ℬ

$F$ is called a ﬁber over $x ∈ ℬ$ .

A ﬁber bundle is also commonly given as

ℱ → ℰ → π ℬ, (4 .46)

which also marks ﬁber bundles as a special case of ﬁbrations. The speciality compared to Deﬁnition 39 is the demand for an at least local trivialization of the form $U × F$ . It should not be omitted to point out a certain kinship with Deﬁnition 35, which also features a local requirement at its core. It is thus not surprising that manifolds can be represented as a ﬁber bundle, which, while maybe not theoretically exciting, provides a guide to the storage and representation of even non-trivial manifolds in digital form by using the ﬁber bundle as the main abstraction [75]. The ﬁber bundle as an abstraction mechanism should not be underestimated, especially with respect to guiding the implementations on digital computers, as the utilized memory structures represent ﬁber bundles. The set of memory cells, along with a topology deﬁned on it, acts as a base space, on which the ﬁber space of values is attatched to. Fiber bundles, however, also provide elegant and powerful solutions in the theoretical setting. Several topics covered in Section 4.6.4 have a simple representation using ﬁber bundles, when also considering the following deﬁnition.

Deﬁnition 41 (Section of a ﬁber bundle) A mapping

f : U ⊆ ℬ → ℰ (4.47)

is called a section of a ﬁber bundle, if

∀x ∈ U : π ∘ f = π (f (x)) = x. (4.48)

Beside the already presented mechanisms it is also desirable to ﬁrmly establish a formal manner in which to transport properties of mappings to various topological spaces, where they have previously not been deﬁned. To provide Deﬁnition 44 with in depth backing, ﬁrst very general notions are introduced.

Deﬁnition 42 (Fiber product) Given two mappings with identical codomain

f : 𝒜 → 𝒞 (4.49a) g : ℬ → 𝒞 (4.49b)

the ﬁber product over $ℬ$ consists of two mappings

p1 : 𝒫 → A (4.50a) p2 : 𝒫 → B (4.50b)

such that $p1∘f= p2 ∘ g$ , which may also be expressed by saying that the following diagram commutes:

f -------C (4.51) A | | | | ----- |g P p1p2 B

Additionally, it is required that given additional mappings

q1 : Q → A (4.52) q2 : Q → B (4.53)

which also satisfy the relation $q1 ∘ f = q2 ∘ g$ , deﬁnes a mapping

q : Q → P (4.54)

as is illustrated in the following diagram

qq Qq1| 2 (4.55) | P p1p 2 Af Bg C

This demand ensures that a tuple $(P, p ,p ) 1 2$ is deﬁned uniquely up to an isomorphism. It is also common to ﬁnd the notation

P = A ×C B. (4.56)

Figure 4.2: Illustration of a pullback bundle.

The general notion just deﬁned can be applied to ﬁber bundles to attach ﬁbers originally situated in one topological space to another one using a simple mapping. The formalization of this is presented next.

Deﬁnition 43 (Pullback bundle) Given a ﬁber bundle $π : ℰ → ℬ$ and a mapping $f : 𝒜 → ℬ$ it is possible to deﬁne a ﬁberbundle, the so called pullback bundle $∗ f ℰ$ , which uses $𝒜$ as a base space by attaching at every element $x ∈ 𝒜$ the ﬁber corresponding to the element $f(x) ∈ ℬ$ (the position of attachment is given by the index):

∗ (f ℰ )x = ℰf(x)∀x ∈ 𝒜. (4.57)

f ∗ℰ ∗ π∗f ----- π (4.58) f|π ℰ| | | | 𝒜f ---------ℬ

In short, the pullback bundle, as sketched in Figure 4.2, is simply the ﬁber product (Deﬁnition 42) $𝒜×ℬℰ$ . It should not go unnoticed that this construct is compatible with sections of ﬁber bundles (Deﬁnition 41); therefore entities which appear as the section of a ﬁber bundle, such as presented in Deﬁnition 61 and Deﬁnition 62, will be pulled back and appear again as sections of the pullback bundle (Deﬁnition 43).

Considering two topological spaces $𝔐$ and $𝔑$ and the mappings

f : 𝔐 → 𝔑 (4.59) φ : 𝔑 → ℝ (4.60)

it is possible by using the composition of the maps to deﬁne a mapping

∗ φ ∘ f = φ (f (⋅)) = f φ : 𝔐 → ℝ, (4 .61)

which eﬀectively transports the function $φ$ from $𝔑$ to $𝔐$ against the direction of $f$ . It is said that $φ$ is pulled back from $𝔑$ via $f$ .

Deﬁnition 44 (Pullback (of functions)) The mapping $∗ f$ resulting from a mapping between two topological spaces

f : 𝔐 → 𝔑, (4 .62)

which transports functions from the codomain of $f$ to its domain is called a pullback (of functions).

The pullback of functions is a particular case of the of the pullback bundle (Deﬁnition 43), which illustrates the concept in a relatively simple fashion.