---
tag: 3R064KFF68
kind: statement
role: theorem
name: Poincaré conjecture
msc: [57K30, 57R60, "53E20"]
uses: [R20TB2KEQQ, T62369RFH6, S1XK3HXHFX, 0QT00PB5G2]
about: [R20TB2KEQQ, T62369RFH6]
sources:
  - url: https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture
same_as:
  wikidata: Q203586
  wikipedia: Poincaré conjecture
---

## Statement

Let $M$ be a closed connected [[3-manifold|R20TB2KEQQ]] whose [[fundamental group|S1XK3HXHFX]] is trivial.
Then $M$ is [[homeomorphic|0QT00PB5G2]] to the [[3-sphere|T62369RFH6]] $S^3$.

## Remarks

Poincaré asked this question in 1904.
Perelman proved it in 2002 and 2003, following Hamilton's program of Ricci flow.
One runs the Ricci flow from any Riemannian metric on $M$ and cuts out regions where the curvature blows up (surgery).
For simply connected $M$ the flow with surgery becomes extinct in finite time, and this shows that $M$ is a sphere.
It is also a special case of the [[geometrization conjecture|B27HB43K86]].
Both hypotheses matter: $\R^3$ is simply connected but not closed, and the [[Poincaré homology sphere|NHERCK7C43]] is closed with the homology of $S^3$ but has nontrivial fundamental group.
The analogous statement in dimensions $n \ge 5$ is due to Smale and in dimension $4$ to Freedman, in their topological versions.

## Explanations

### L1 human

A closed $3$-dimensional space in which every loop can be shrunk to a point must be the $3$-sphere.
It was the most famous open problem of topology for a century.
It was one of the Clay Millennium Prize problems.

### L2 human

The statement is a recognition theorem: a single algebraic condition identifies $S^3$ among all closed $3$-manifolds.
In practice it is now used as part of geometrization, which gives every closed $3$-manifold with finite fundamental group the form $S^3 / \Gamma$ for a finite group $\Gamma$ of isometries.
In the smooth category, it also follows that every smooth homotopy $3$-sphere is diffeomorphic to $S^3$, because topological $3$-manifolds have unique smooth structures.