Editing Lecture 6 - Thursday Week 3 (section)

== Fibre Bundles ==

def: Let $F, M$ be manifolds. Then a fibre bundle of type $F$ (or an $F$ fibre bundle) is a manifold $E$ with a map $E \rightarrow M$ such that there exists an open cover of $M$ and morphisms from each preimage $\chi_\alpha : \pi^{-1}(u_\alpha) \rightarrow U_\alpha \times F$, the triangle commutes ($\pi = \text{pr}_1 \circ \chi_\alpha$).

terminoligy:
* total space
* base
* proj
* sections

remark: sections may not exist, but local sections always exist

From a local trivialisation, we can construct a fibre bundle by glueing local trivialisations using transition functions. $\rho_{\beta\alpha}: U_\alpha \cap U_\beta \rightarrow \text{Diffeo}(F)$ with

* $\rho_{\beta\alpha} = \rho_{\alpha\beta}^{-1}$
* cocycle condition.

A vector bundle is a fibre bundle where each fibre is a vector space. These are the ones we will be most interested in, where $F$ has extra structure and the trivialisations preserve this, and so the transition functions preserve this extra structure. e.g. for vector spaces, where the transition functions $\rho_{\beta\alpha} \subset \text{GL}(V) \subset \text{Diffeo}(V)$.