For every simply connected finite CW complex X, if X is rationally hyperbolic, then there exists a finite set of primes P_X such that, for every prime p notin P_X and every integer r≥ 1, X is hyperbolic over Z/p^rZ.

Moore’s conjecture

Moore’s conjecture predicts that rationally hyperbolic spaces should also be hyperbolic modulo $p^r$ for all but finitely many primes. The general problem remains open, including the mod- $2$ Moore-space case.

Known results

Hao, Sun, and Theriault (2017) proved the conjecture for generalized moment-angle complexes.
A 2023 result extended it to Poincaré duality complexes whose loop spaces have specified connected-sum types.
Huang’s localized theorem (2025) proves the conclusion for suitable cofibrations and Poincaré duality complexes for all but finitely many primes.
A connected-sum theorem verifies a weak version: under stated hypotheses, $M\#N$ is hyperbolic outside finitely many primes.

August 2026 localized inertness theorem

A new theorem proves inertness for top-cell attaching maps of certain Poincaré duality complexes after localization away from a finite set of primes. It yields rational and $\mathbb{Z}/p^r\mathbb{Z}$ -hyperbolicity outside that set and produces new non-inert sphere maps; it does not settle Moore’s conjecture generally.

Current status (as of August 2026): localized and family-specific cases are established, but the full conjecture and the mod- $2$ Moore-space case remain open.

Moore’s conjecture

Primary source

Additional references

Progress summary

Solutions 0