---
tag: 8KDV43WZ3Z
kind: statement
role: theorem
name: Generalized Poincaré conjecture
msc: [57R60, 57N99]
uses: [QQQ9B1DK4P, T62369RFH6, XZG6MTSWCN, 0TVJZYB6SH, 0QT00PB5G2]
about: [QQQ9B1DK4P, T62369RFH6]
sources:
  - url: https://en.wikipedia.org/wiki/Generalized_Poincar%C3%A9_conjecture
same_as:
  wikidata: Q5532452
  wikipedia: Generalized Poincaré conjecture
---

## Statement

Let $n \ge 1$ and let $M$ be a [[compact|0TVJZYB6SH]] [[topological manifold|QQQ9B1DK4P]] of dimension $n$ with empty boundary.
If $M$ is [[homotopy equivalent|XZG6MTSWCN]] to the [[$n$-sphere|T62369RFH6]] $S^n$, then $M$ is [[homeomorphic|0QT00PB5G2]] to $S^n$.

## Remarks

For $n \le 2$ the statement follows from the classification of curves and surfaces.
For $n \ge 5$ it was proved by Stephen Smale for smooth manifolds, through handle decompositions and the h-cobordism theorem.
Max Newman extended it to topological manifolds, and John Stallings and Christopher Zeeman gave other proofs.
For $n = 4$ it was proved by Michael Freedman in 1982, and for $n = 3$ it is the Poincaré conjecture, proved by Grigori Perelman with Ricci flow.
The analogous statement with "diffeomorphic" in place of "homeomorphic" is false: John Milnor found smooth manifolds homeomorphic but not diffeomorphic to $S^7$, the exotic spheres.
The smooth version for $n = 4$ is an open problem.
For $n \ge 2$, a closed $n$-manifold is homotopy equivalent to $S^n$ as soon as it is simply connected and has the integer homology of $S^n$, by the Hurewicz and Whitehead theorems.

## Explanations

### L1 human

If a closed manifold cannot be told apart from a sphere by deforming loops, surfaces and higher spheres inside it, then it really is a sphere.
The statement is true in every dimension, but each dimension needed its own ideas.
Strangely, dimensions five and higher were settled before dimensions three and four, and dimension three was done last.

### L2 human

The theorem is the starting point of the classification of manifolds up to homeomorphism.
In dimensions at least five, it comes out of the h-cobordism theorem, which turns homotopy information into homeomorphisms using handle moves.
In the smooth category the question becomes the study of exotic spheres, which Kervaire and Milnor organized into groups of homotopy spheres under connected sum.