---
tag: R20TB2KEQQ
kind: concept
name: 3-manifold
msc: [57K30]
specializes: [QQQ9B1DK4P]
sources:
  - url: https://en.wikipedia.org/wiki/3-manifold
same_as:
  wikidata: Q526901
  wikipedia: 3-manifold
---

## Summary

A $3$-manifold is a [[topological manifold|QQQ9B1DK4P]] of dimension $3$: a space that looks like ordinary space $\R^3$ near each of its points.
By Moise's theorem every $3$-manifold has an essentially unique triangulation.
It also has an essentially unique smooth structure, so the topological, piecewise linear and smooth theories agree.
Their study combines topology with geometry, through the [[geometrization conjecture|B27HB43K86]], proved by Perelman, which includes the [[Poincaré conjecture|3R064KFF68]].

## Explanations

### L1 human

A $3$-manifold is a possible shape for a universe: around every point, space looks like the space we live in.
The space $\R^3$, the $3$-sphere and the $3$-torus are examples.
The $3$-torus is a cube whose opposite faces are glued: leaving through the right face, one comes back through the left face.
Globally such spaces can be very different, even though they all look the same locally.