---
tag: JTCQT4NQ79
kind: concept
name: Local trivialization
msc: [55R10, 57R22]
sources:
  - url: https://en.wikipedia.org/wiki/Fiber_bundle
---

## Summary

A local trivialization of a map $\pi \colon E \to B$ over an open set $U$ is a homeomorphism $\pi^{-1}(U) \to U \times F$ over $U$.
It says that over $U$ the map $\pi$ looks exactly like the projection of a product.
A [[fiber bundle|CX4M8K61AB]] is a map that has a local trivialization around every point of the base.
Comparing two local trivializations on an overlap gives the transition functions, which encode how the bundle twists.

## Explanations

### L1 human

A local trivialization is a way to straighten out a piece of a bundle so that it becomes a plain product.
On the Möbius band over the circle, the strip above any proper arc of the circle looks like an ordinary rectangle, arc times segment.
Choosing such an identification for one arc is a local trivialization.
No single identification works over the whole circle, because of the twist.