Whitehead’s asphericity problem
Determine whether every subcomplex of every aspherical -dimensional CW complex is aspherical; that is, whether for every whenever for every .
References
Primary source
Additional references
- Finite Chains of Two-Complexes and Acyclic Covers — arXiv — Laurent Bartholdi, Roman Mikhailov
Progress summary
New work constructs arbitrarily long finite approximations related to Whitehead’s problem, but no infinite example or solution has been found.
J. H. C. Whitehead posed the problem in 1941: whether every subcomplex of an aspherical two-dimensional complex is aspherical. It remains open.
Known results
- Howie’s 1979 reduction shows that any counterexample has one of two specified forms.
- Lüft’s 1996 strengthening shows that only the second form can occur.
- Bestvina and Brady’s 1997 construction gives a conditional connection with the Eilenberg–Ganea conjecture.
- Positive results are known for certain quasi-constructible complexes.
October 2026 finite-chain construction
Laurent Bartholdi and Roman Mikhailov characterize when a two-complex admits chains of every finite length via a connected acyclic regular cover, and exhibit a presentation complex of with this property. Their construction does not produce an infinite chain and therefore does not settle Whitehead’s problem.
Current status (as of October 2026): Whitehead’s asphericity problem remains open; the new finite-chain construction is claimed progress, not a counterexample or proof.
Solutions 0
No solutions have been posted yet.