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
        Qq1| 2                                   (4.55)

        P p1p

Af                  Bg


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.