Fibre bundle

Let $𝐵, 𝐹$ be spaces in $𝖳 𝗈 𝗉$ or an appropriate subcategory $𝖢$ ,¹ which we call the base space and fibre space respectively. A fibre bundle²

𝐹 \to 𝐸 𝜋 ↠ 𝐵

is a “total space” $𝐸$ equipped with a surjective morphism $𝜋 : 𝐸 ↠ 𝐵$ such that $(𝐸, 𝜋)$ is locally the product space $(𝐵 \times 𝐹, p r o j 1)$ . #m/def/topology This is formalized as follows: For every $𝑝 \in 𝐵$ , there is an open neighbourhood $𝑈 \subseteq 𝐵$ of $𝑝$ with an isomorphism

𝜑 : 𝑈 \times 𝐹 \sim ⟶ 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 $𝑝 \in 𝐵$ as $𝐸 𝑝 : = 𝜋 - 1 {𝑝}$ .
A bundle for which $𝐸 = 𝐹 \times 𝐵$ and $𝜋 = p r o j 1$ is called trivial.
A Bundle map over a fixed based space is a morphism of bundles, and we can thus form the category $𝖡 𝗎 𝗇 𝖽 𝖢 𝑋$ .
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 $(0, 1) ↪ 𝑀 ↠ 𝕊 1$ .
The Klein bottle $𝐾$ is a bundle $𝕊 1 ↪ 𝐾 ↠ 𝕊 1$ .

#state/tidy | #lang/en | #SemBr

Often we take $𝖬 𝖺 𝗇 𝛼$ or Holomorphic category. ↩
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. ↩

Fibre bundle

Further terminology

Further structure

Examples

Footnotes