Fibre space

An object $ (X,\pi,B) $, where $ \pi: X \to B $ is a continuous surjective mapping of a topological space $ X $ onto a topological space $ B $ (i.e., a fibration). Note that $ X $, $ B $ and $ \pi $ are also called the total space, the base space and the projection of the fibre space, respectively, and $ {\pi^{\leftarrow}}[\{ b \}] $ is called the fibre above $ b $. A fibre space can be regarded as the union of the fibres $ {\pi^{\leftarrow}}[\{ b \}] $, parametrized by the base space $ B $ and glued by the topology of $ X $. For example, there is the product $ \pi: B \times F \to B $, where $ \pi $ is the projection onto the first factor; the fibration-base $ \pi: B \to B $, where $ \pi = \operatorname{id} $ and $ X $ is identified with $ B $; and the fibre space over a point, where $ X $ is identified with a (unique) space $ F $.

A section of a fibration (fibre space) is a continuous mapping $ s: B \to X $ such that $ \pi \circ s = \operatorname{id}_{B} $.

The restriction of a fibration (fibre space) $ \pi: X \to B $ to a subset $ A \subseteq B $ is the fibration $ \pi': X' \to A $, where $ X' \stackrel{\text{df}}{=} {\pi^{\leftarrow}}[A] $ and $ \pi' \stackrel{\text{df}}{=} \pi|_{X'} $. A generalization of the operation of restriction is the construction of an induced fibre bundle.

A mapping $ F: X \to X_{1} $ is called a morphism of a fibre space $ \pi: X \to B $ into a fibre space $ \pi_{1}: X_{1} \to B_{1} $ if and only if it maps fibres into fibres, i.e., if for each point $ b \in B $, there exists a point $ b_{1} \in B_{1} $ such that $ F[{\pi^{\leftarrow}}[\{ b \}]] \subseteq {\pi^{\leftarrow}}[\{ b_{1} \}] $. Such an $ F $ determines a mapping $ f: B \to B_{1} $, given by $ f(b) \stackrel{\text{df}}{=} (\pi \circ F)[{\pi^{\leftarrow}}[\{ b \}]] $. Note that $ F $ is a covering of $ f $ and that $ \pi_{1} \circ F = f \circ \pi $; the restrictions $ F_{b}: {\pi^{\leftarrow}}[\{ b \}] \to {\pi_{1}^{\leftarrow}}[\{ b_{1} \}] $ are mappings of fibres. If $ B = B_{1} $ and $ f = \operatorname{id} $, then $ F $ is called a $ B $-morphism. Fibre spaces with their morphisms form a category — one that contains fibre spaces over $ B $ with their $ B $-morphisms as a subcategory.

Any section of a fibration $ \pi: X \to B $ is a fibre-space $ B $-morphism $ s: B \to X $ from $ (B,\operatorname{id},B) $ into $ (X,\pi,B) $. If $ A \subseteq B $, then the canonical imbedding $ i: {\pi^{\leftarrow}}[A] \to B $ is a fibre-space morphism from $ \pi|_{A} $ to $ \pi $.

When $ F $ is a homeomorphism, it is called a fibre-space isomorphism. A fibre space isomorphic to a product is called a trivial fibre space. An isomorphism $ \theta: X \to B \times F $ is called a trivialization of $ \pi $.

If each fibre $ {\pi^{\leftarrow}}[\{ b \}] $ is homeomorphic to a space $ F $, then $ \pi $ is called a fibration with fibre $ F $. For example, in any locally trivial fibre space over a connected base space $ B $, all the fibres $ {\pi^{\leftarrow}}[\{ b \}] $ are homeomorphic to one another, and one can take $ F $ to be any $ {\pi^{\leftarrow}}[\{ b_{0} \}] $; this determines homeomorphisms $ \phi_{b}: F \to {\pi^{\leftarrow}}[\{ b \}] $.

Comments

Both the notations $ \pi: X \to B $ and $ (X,\pi,B) $ are used to denote a fibration, a fibre space or a fibre bundle.

In the West, a mapping $ \pi: X \to B $ would only be called a fibration if it satisfied some suitable condition, for example, the homotopy lifting property for cubes (such a fibration is known as a Serre fibration; see Covering homotopy for the homotopy lifting property ([a3])). A mapping $ F: X \to X_{1} $ would be called a morphism (respectively, an isomorphism) only if the induced function $ f: B \to B_{1} $ were continuous (respectively, a homeomorphism).

References

[a1]	A. Dold, “Partitions of unity in the theory of fibrations”, Ann. of Math., 78 (1963), pp. 223–255.
[a2]	D. Husemoller, “Fibre bundles”, McGraw-Hill (1966).
[a3]	J.-P. Serre, “Homologie singulière des èspaces fibrés”, Ann. of Math., 54 (1951), pp. 425–505.
[a4]	E.H. Spanier, “Algebraic topology”, McGraw-Hill (1966), Chapt. 2.
[a5]	N.E. Steenrod, “The topology of fibre bundles”, Princeton Univ. Press (1951).