---
tag: M58WRZ7PFM
kind: concept
name: Local coefficient system
msc: [55N25]
sources:
  - url: https://en.wikipedia.org/wiki/Local_system
same_as:
  wikidata: Q6664565
  wikipedia: Local system
---

## Summary

A local coefficient system on a space $X$ gives an abelian group for each point of $X$.
These groups may be transported along paths.
The transport depends only on the homotopy class of the path.
Homology and cohomology with local coefficients generalise ordinary homology and cohomology with a fixed coefficient group.
They are needed for Poincaré duality on non-orientable manifolds, for obstruction theory and for the Serre spectral sequence of a fibration with non-simply-connected base.

## Explanations

### L1 human

Ordinary cohomology attaches the same group, say $\Z$, to every point of a space.
A local system attaches a copy of a group to each point, but going around a loop may come back with a twist.
For example, on a circle, attach $\Z$ to each point and let one trip around the circle act by $n \mapsto -n$.
Locally nothing has changed, but globally the coefficients are twisted, as the Möbius band is a twisted version of a cylinder.