---
tag: 702SNFQG0W
kind: concept
name: Orientability
msc: [57N65, 57R99]
sources:
  - url: https://en.wikipedia.org/wiki/Orientability
same_as:
  wikidata: Q2748415
  wikipedia: Orientability
---

## Summary

A manifold is orientable when one can choose an orientation at every point, varying continuously over the whole manifold.
An orientation is a sense of "clockwise" in dimension $2$, or of "right-handed" in dimension $3$.
The sphere and the torus are orientable; the Möbius strip, the Klein bottle and the real projective plane are not.
Orientability is needed to integrate differential forms of top degree, to define the fundamental class and to state Poincaré duality with integer coefficients.

## Explanations

### L1 human

Draw a small clockwise circle on a surface and slide it around along any closed path.
On a sphere, the circle always comes back clockwise.
On a [[Möbius strip|04AJE0M271]], one trip around the strip brings it back turning the other way.
A surface is orientable when such a reversal never happens, so that "clockwise" has one meaning everywhere.
In the same way, a $3$-dimensional manifold is orientable when a right-handed glove can never be carried around a loop and come back as a left-handed one.