---
tag: 913WEQERBE
kind: concept
name: Orientation of a manifold
msc: [57N65, 55M05]
sources:
  - url: https://en.wikipedia.org/wiki/Orientability
---

## Summary

An orientation of an $n$-dimensional [[manifold|QQQ9B1DK4P]] is a consistent choice, at every point, of one of the two local directions of rotation, or "handedness".
In algebraic topology it is a continuous choice of generator of the local homology group $H_n(M, M \setminus \set{x}) \cong \Z$ at each point $x$.
For smooth manifolds it is the same as a continuous choice of [[orientation|0QJJAQ6ZYZ]] of every tangent space.
Orientations are needed for fundamental classes, Poincaré duality with integer coefficients, and integration of differential forms.

## Explanations

### L1 human

On a small piece of surface you can always decide which way is clockwise.
A surface is orientable if you can make this decision everywhere so that nearby choices agree.
A sphere is orientable: clockwise as seen from outside works everywhere.
A Möbius band is not: carry a small clockwise circle once around the band and it comes back counterclockwise.
An [[orientable surface|0TD904MG06]] is a surface where such a global choice exists.