Let M be a topological manifold of dimension 3 (a second-countable Hausdorff space each of whose points has a neighborhood homeomorphic to an open subset of R^3). Suppose that M is closed, i.e. compact and without boundary; - M is connected; - M is simply connected, i.e. pi_1(M,p) is the trivial group for some (equivalently, every) point p ∈ M, so that every loop gamma colon [0,1] → M with gamma(0)=gamma(1)=p is homotopic, through loops based at p, to the constant loop at p. Then M is homeomorphic to the 3-sphere S^3 = , x ∈ R^4 : lVert x rVert = 1 ,.

Poincaré conjecture

About 122 years old · traced to 1904

Let $M$ be a topological manifold of dimension $3$ (a second-countable Hausdorff space each of whose points has a neighborhood homeomorphic to an open subset of $\mathbb{R}^3$ ). Suppose that

$M$ is closed, i.e. compact and without boundary; - $M$ is connected; - $M$ is simply connected, i.e. $\pi_1(M,p)$ is the trivial group for some (equivalently, every) point $p \in M$ , so that every loop $\gamma \colon [0,1] \to M$ with $\gamma(0)=\gamma(1)=p$ is homotopic, through loops based at $p$ , to the constant loop at $p$ .

Then $M$ is homeomorphic to the $3$ -sphere

S^3 = \{\, x \in \mathbb{R}^4 : \lVert x \rVert = 1 \,\}.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Poincaré conjecture
Let $M$ be a simply connected, closed $3$ -manifold, where closed means compact without boundary. Poincaré conjecture. $M$ is homeomorphic to the $3$ -sphere $S^3$ . The conjecture is a purely topological statement, but it was proved using the Riemannian metric structure on smooth manifolds.
source: Aoran Chen, “A Variational Approach to the Yamabe Problem: Conformal Transformations and Scalar Curvature on Compact Riemannian Manifolds”, arXiv:2309.02397 (2024).

References

Primary source

Wikipedia

Additional references

Wikipedia, Poincaré conjecture, the article this problem comes from.

topology unsolved-problem

InMillennium Prize Problems

Posted 1 month ago · edited 3 weeks ago

KentaKitamura 1 month ago
This entry appears to be misclassified as open. The Poincaré conjecture was proved by Grigori Perelman in 2002–2003, and this page’s own References section already states “Status: solved.” A detailed account of the proof is available in Morgan and Tian, Ricci Flow and the Poincaré Conjecture: https://arxiv.org/abs/math/0607607
Could the status be updated from open to solved?

Progress summary

Refreshed 3 weeks ago

Claimed solved

Perelman’s proof settled the conjecture, while a proposed shorter proof from 2025 has not been independently confirmed.

Henri Poincaré posed the conjecture in 1904: every closed, simply connected three-dimensional manifold is homeomorphic to the three-sphere. Grigoriy Perelman proved it through Ricci flow and the broader geometrization program in 2002–2003.

Known results

Richard Hamilton introduced Ricci flow in 1982, providing the framework later developed by Perelman.
Perelman’s 2002–2003 work established finite-time extinction for the relevant Ricci flows with surgery, implying the conjecture.
Colding and Minicozzi (2005) gave a streamlined finite-time-extinction argument.
Cao and Zhu, Kleiner and Lott, and Morgan and Tian supplied detailed expositions and verification.

February 2025 claimed shorter proof

An arXiv manuscript claimed a short, self-contained alternative proof and said an earlier error had been repaired. The retrieved evidence contains no independent validation, so this remains an unconfirmed claim rather than a challenge to Perelman’s established result.

Current status (as of September 2026): The three-dimensional conjecture is settled by Perelman’s proof; the February 2025 alternative proof claim remains unverified.

Sources

No solutions have been posted yet.