5.2 (Agol) — Large injectivity radius in hyperbolic homology manifolds

Agol’s problem asks whether there are fibered hyperbolic $3$-manifolds with homology $S^2\times S^1$ and arbitrarily large injectivity radius, and whether hyperbolic homology $3$-spheres can have the same property.

Known results

• A 2013 paper constructs integer homology $3$-spheres whose injectivity radius exceeds any fixed $R$ at all but an arbitrarily small proportion of points, but not everywhere.
• The same paper constructs closed hyperbolic mapping tori with $\operatorname{inj}(M)>R$ for every $R$, without establishing the homology $S^2\times S^1$ condition.
• A 2005 result gives towers of hyperbolic rational homology $3$-spheres with injectivity radius tending to infinity, conditional on the Generalized Riemann Hypothesis and an additional $\mathrm{GL}_2$ conjecture.

2022 confirmation of openness

A 2022 source explicitly states that homology spheres with arbitrarily large injectivity radius remain unknown. The supplied scan found no subsequent proof, counterexample, verification, or claimed solution for either part.

Current status (as of September 2026): Both questions remain open; known constructions give only weakened density results, conditional rational-homology results, or mapping tori without the required homology condition.