Topology MOC

Fibre bundle

Let 𝐵,𝐹 be spaces in Category of topological spaces or an appropriate subcategory 𝖢,¹ which we call the base space and fibre space respectively. A fibre bundle²

𝐹→𝐸𝜋↠𝐵

is a “total space” 𝐸 equipped with a surjective morphism 𝜋 :𝐸 ↠𝐵 such that (𝐸,𝜋) is locally the product space (𝐵 ×𝐹,proj1). topology This is formalized as follows: For every 𝑝 ∈𝐵, there is an open neighbourhood 𝑈 ⊆𝐵 of 𝑝 with an isomorphism

𝜑:𝑈×𝐹∼⟶𝜋−1(𝑈)

called a local trivialization at 𝑝 such that

commutes. An open cover 𝒯 of 𝐵 with local trivializations is called a local trivialization of 𝐸.

Further terminology

• We denote thr fibre above a base point 𝑝 ∈𝐵 as 𝐸𝑝 :=𝜋−1{𝑝}.
• A bundle for which 𝐸 =𝐹 ×𝐵 and 𝜋 =proj1 is called trivial.
• A Bundle map over a fixed based space is a morphism of bundles, and we can thus form the category Category of fibre bundles.
• A Bundle section is a section (right-inverse) to 𝜋, or equivalently a bundle map from 𝐵 to 𝐸.

Further structure

• Vector bundle

Examples

• The Möbius strip 𝑀 is a bundle (0,1) ↪𝑀 ↠𝕊1 .
• The Klein bottle 𝐾 is a bundle 𝕊1 ↪𝐾 ↠𝕊1.

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Footnotes

1. Often we take Category of manifolds or Holomorphic category. ↩

2. Despite the notation, which is chosen to resemble a short exact sequence, the morphism 𝐹 ↪𝐸 should not be taken too literally, since there exists an isomorphism 𝐹 ≅𝜋−1{𝑥} for every fibre. ↩