BUNDLES

1.1. Definition. A bundle (E, B, p) consists of a total space E, a base space B, and a projection map p : E → B such that: for every b ∈ B, there exists an open neighborhood U of b, a space F, and a homeomorphism ϕ : U × F → p^{−1}(U) such that the following diagram commutes:

U × F ⟶^{ϕ} p^{−1}(U)
pr^{1}↓  ↓^{p}
U ⟶^{id} U

The condition above is referred to as local triviality of the bundle, and the pair (U, ϕ) as local coordinates about b.

Set E_{b} := p^{−1}(b); this is called the fiber over b, and is identified to F via ϕ : {b} × F ⟶^{∼} E_{b}. If b′ is another point in U, we also have ϕ : {b′} × F →^{∼} E_{b′}, and so E_{b} ≈ E_{b′}. Hence, if B is path-connected (which we henceforth will allways assume), all the fibers are homeomorphic to F, the typical fiber of the bundle. We will often write the bundle as F → E →^{p} B and say that E fibers over B with fiber F.

1.2. Remark. The map p is onto. That’s because p^{−1}(b) ≈ F ≠ ∅.

1.3. Remark. The map p is open. To see this, it is enough to show that the restriction p : p^{−1}(U) → U is open, or, equivalently, pr_{1} : U × F → U is open. But pr_{1} is the composite U × F ⟶^{id ×c} U × ∗ ⟶^{∼} U, where c is the map that collapses F to a point. As c is obviously open, and the product of two open maps is open, we are done.

1.4. Definition. A bundle morphism (u, f) : (E, B, p) → (E′, B′, p) consists of maps u : E → E′, f : B → B′ such that the following diagram commutes:

E ⟶^{u} E′
p↓  ↓^{p′}
B ⟶^{f} B′

Note that u maps the fiber over b ∈ B to the fiber over f(b) ∈ B′. Hence the restriction of u to E_{b} defines a map

u_{b} : E_{b} → E_{f(b)}.

Note also that u determines f; indeed, f(b) = p′(u(x)), for any x ∈ E_{b}.

A morphism for which B′ = B and f = id is called a B-morphism. As the requirement is this case is p′ ∘ u = p, or u(E_{b}) ⊂ E_{b}, we also say that u : E → E′ is a fiber-preserving map. If u is injective, we say that ξ = (E, B, p) is a sub-bundle of ξ′ = (E′, B, p′), and write ξ ⊂ ξ′.

A B-morphism u : E → E′ which is a homeomorphism is called a bundle isomorphism; if such a morphism exists, we say the bundles ξ = (E, B, p) and ξ′ = (E′, B, p′) are equivalent, and write ξ ≅ ξ′. The automorphisms of a bundle ξ are also called gauge equivalences; they form a group, G(ξ), called the gauge group of ξ.

The following lemma is a useful criterion for bundle equivalence.

1.5. Lemma. Let u : E → E′ be a B-morphism such that for each b ∈ B, u_{b} : E_{b} → E′_{b} is a homeomorphism. If the fiber F is a locally connected, locally compact, Hausdorff space, then u is a bundle isomorphism.

Proof. Since the restriction of u to any fiber is a bijection, u itself is a bijection. All we have to prove is that the inverse of u is continuous. It is enough to do that in a local coordinate chart.

So let u : U × F → U × F be a map given by (b, x) ↦ (b, u_{b}(x)), where u_{b} ∈ Homeo(F). Endow Homeo(F) with the compact-open topology. Since F is locally compact and Hausdorff, and since the map (b, x) ↦ u_{b}(x) is continuous, the map α : U → Homeo(F), b ↦ u_{b} is also continuous (see [Bourbaki]). On the other hand, since F is locally connected, locally compact and Hausdorff, the group structure on Homeo(F) (given by composition of maps) is compatible with the chosen topology (see [Bourbaki]). In particular, the map β : Homeo(F) → Homeo(F), g ↦ g^{−1} is continuous. Thus β ∘ α : U → Homeo(F), b ↦ u_{b}^{−1} is continuous. This implies that u^{−1} : U ×F → U ×F, u^{−1}(b, x) = (b, u_{b}^{−1}(x)) is also continuous, and we are done. □

1.6. Definition. A map s : B → E is called a section of the bundle (E, B, p) if p ∘ s = id_{B}.

1.7. Definition. A bundle (B × F, B, pr_{1}) is called trivial. A bundle (E, B, p) is trivializable if it is equivalent to a trivial one; a B-isomorphism (E, B, p) →^{∼} (B × F, B, pr_{1}) is called a trivialization.

A section of the trivial bundle (B × F, B, pr_{1}) has the form s : B → B × F, s(b) = (b, f(b)). We thus have a bijection {sections of trivial bundle} ⟷ Map(B, F), given by s ↔ f.

1.8. Definition. Let ξ = (E, B, p) be a bundle, and f : B′ → B a map. The pull-back of ξ by f is the bundle f^{∗}(ξ) = (Ê, B′, p̂), where Ê = {(b′, x) ∈ B′ × E | f(b′) = p(x)} and p̂ is the restriction of pr_{1} : B′ × E → B′ to Ê.

If ξ has local coordinates (U, ϕ), we may choose the local coordinates of f^{∗}(ξ) to be (U′, ϕ′), where U′ = f^{−1}(U) and ϕ′ : U′ × F → p^{′−1}(U′) is given by ϕ′(b′, y) = (b′, ϕ(f(b′), y)).

There is a canonical morphism from f^{∗}(ξ) to ξ, given by the commuting square

Ê \underrightarrow{f̂} E
p̂↓  ↓^{p}
B′ ⟶^{f} B

where f̂ is the restriction of pr_{2} : B′ × E → E to Ê.

The pull-back has the following universality property: for every bundle morphism (f′, f) : (E′, B′, p′) → (E, B, p) there exists a unique B′-morphism u : E′ → Ê such that (f′, f) = (f̂, f) ∘ (u, id_{B′}). The map u is given by u(x) = (p′(x), f′(x)).

For each b′ ∈ B′, the map f̂_{b′} : Ê_{b′} → E_{f(b′)} is a homeomorphism. Thus, if ξ has fiber F, so does f^{∗}(ξ). The following theorem says that, under mild restrictions on F, these properties characterize pull-backs.

1.9. Theorem. Let (f′, f) be a morphism from the bundle ξ′ = (E′, B′, p′) to the bundle ξ = (E, B, p) such that for each b′ ∈ B′, f′_{b′} : E′_{b′} → E_{f(b′)} is a homeomorphism. If the fiber F of ξ is a locally connected, locally compact, Hausdorff space, then ξ′ is equivalent to f^{∗}(ξ).

Proof. By the universality property of pull-backs, there is a morphism (u, id_{B′}) from ξ′ to f^{∗}(ξ) such that f′ = f̂ ∘ u. In particular, f′_{b′} = f̂_{b′} ∘ u_{b′}, and so u_{b′} is a homeomorphism. Thus, by Lemma 1.5., u : E′ → Ê is a bundle isomorphism. □

1.10. Definition. Let ξ = (E, B, p) be a bundle, A a subspace of B, and i : A → B the inclusion map. The restriction of ξ to A is the bundle ξ|_{A} = i^{∗}(ξ).

1.11. Definition. The product of the bundles ξ_{1} = (E_{1}, B_{1}, p_{1}) and ξ_{2} = (E_{2}, B_{2}, p_{2}) is the bundle ξ_{1} × ξ_{2} = (E_{1} × E_{2}, B_{1} × B_{2}, p_{1} × p_{2}).

If ξ_{i} has local coordinates (U_{i}, ϕ_{i}), we may choose the local coordinates of ξ_{1} × ξ_{2}

to be (U_{1} ×U_{2}, ϕ), where ϕ is the composite U_{1} ×U_{2} ⟶^{ϕ_{1}×ϕ_{2}} (U_{1} ×F_{1})×(U_{2} ×F_{2}) ⟶^{∼}

U_{1} × U_{2} × F_{1} × F_{2}. Note that the fiber of the product is the product of the fibers.

1.12. Definition. Let ξ_{1} = (E_{1}, B_{1}, p_{1}) and ξ_{2} = (E_{2}, B, p_{2}) be two bundles with the same base space. Their Whitney sum is the bundle ξ_{1} ⊕ ξ_{2} = Δ^{∗}(ξ_{1} × ξ_{2}), where Δ : B → B × B, Δ(b) = (b, b) is the diagonal map.

Write ξ_{1} ⊕ ξ_{2} = (E_{1} ⊕ E_{2}, B, p_{1} ⊕ p_{2}). We then have the following commuting square:

E_{1} ⊕ E_{2} \underrightarrow{Δ̂} E_{1} × E_{2}
p_{1}⊕p_{2}↓  ↓^{p_{1}×p_{2}}
B ⟶^{Δ} B × B

Note that ξ_{1} and ξ_{2} are sub-bundles of ξ_{1}⊕ξ_{2}, and that F (ξ_{1})×F (ξ_{2}) = F (ξ_{1}⊕ξ_{2}). Under suitable restrictions on the fibers, these properties characterize Whitney sums:

1.13. Theorem. Let ξ_{1} and ξ_{2} be sub-bundles of ξ such that F (ξ_{1})×F (ξ_{2}) = F (ξ). If F (ξ) is a locally connected, locally compact, Hausdorff space, then ξ′ is equivalent to ξ_{1} ⊕ ξ_{2}.

Proof. From the assumption and local triviality, we get a homeomorphism u_{b} : (E_{1})_{b} × (E_{2})_{b} → E_{b}, for every b ∈ B. This defines a morphism u : E_{1} ⊕ E_{2} → E by u(b, (x_{1}, x_{2})) = (b, u_{b}(x_{1}, x_{2})). By Lemma 1.5, u is a bundle isomorphism. □

Exercises

1. Show that (f ∘ g)^{∗}(ξ) ≅ g^{∗}(f^{∗}(ξ)) and id^{∗}(ξ) ≅ ξ.

2. If ξ ≅ η, then f^{∗}(ξ) ≅ f^{∗}(η).

3. If ξ is trivial, then f^{∗}(ξ) is trivial.

4. Let ξ = (B × F, B, pr_{1}) be a trivial bundle. Show that G(ξ) ≅ Map(B, Homeo(F)).

5. Show that the restriction of the compact-open topology on Homeo(ℝ^{n}) to GL(n, ℝ) coincides with the restriction of the Euclidean topology on ℝ^{n^{2}} to GL(n, ℝ).

6. Let X = {0} ∪ {2^{n}}_{n∈ℤ} and consider the space Homeo(X), endowed with the compact-open topology. Show that β : Homeo(X) → Homeo(X), β(g) = g^{−1} is not continuous.

7. Consider the space Homeo(ℝ^{2}), endowed with the topology of pointwise convergence. Show that the map Homeo(ℝ^{2}) × Homeo(ℝ^{2}) → Homeo(ℝ^{2}), (u, v) ↦ u ∘ v is not continuous.