Local Hyperbolicity, Inert Maps, and Applications in Moore’s Conjecture

Abstract We show that the base space of a homotopy cofibration is locally hyperbolic under various conditions. In particular, if these manifolds admit a rationally elliptic closure, then almost all punctured manifolds and almost all manifolds with rationally spherical boundary are $\mathbb{Z}/p^{r}$-hyperbolic for almost all primes $p$ and all integers $r \geq 1$, and satisfy Moore’s conjecture at sufficiently large primes.