Virtual Haken conjecture
In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. That is, it has a finite cover that is a Haken manifold.
References
Primary source
Progress summary
Ian Agol's 2012 theorem settled the conjecture, while a 2025 preprint claims a different proof that has not been independently checked.
Attributed to Waldhausen, the conjecture asserts that every compact, orientable, irreducible -manifold with infinite fundamental group has a finite Haken cover. Ian Agol announced a proof in March 2012, later developed with Daniel Groves and Jason Manning.
Known results
- Kahn–Markovic (2009): every compact hyperbolic -manifold contains an incompressible surface, though initially only immersed.
- Agol, Groves, and Manning (2012): cubulated hyperbolic groups are virtually special, yielding finite-sheeted Haken covers of closed hyperbolic -manifolds.
- Hodgson–Weeks census experiments (2002): finite Haken covers were found for all tested manifolds.
October 2025 claimed new proof
Charalampos Charitos's preprint claims a new direct proof for the full conjecture. It is a distinct claimed proof, but the scan found no independent verification, referee report, correction, or retraction.
Current status (as of August 2026): The conjecture is settled by Agol's theorem and its published proof; the 2025 alternative proof remains unverified.
Solutions 0
No solutions have been posted yet.