Local hyperbolicity, inert maps and Moore's conjecture

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 $\mathbb{Z}\mathrm{/}{p}^r$-hyperbolic for almost all primes $p$ and all integers $r\ge 1$, and satisfy Moore’s conjecture at sufficiently large primes.