On the Čech cohomology of Morse boundaries

We consider cusped hyperbolic $n-$manifolds, and compute Čech cohomology groups of the Morse boundaries of their fundamental groups. In particular, we show that the reduced Čech cohomology with real coefficients vanishes in dimension at most $n-3$ and does not vanish in dimension $n-2$. A similar result holds for relatively hyperbolic groups with virtually nilpotent peripherals and Bowditch boundary homeomorphic to a sphere; these include all non-uniform lattices in rank$-1$ simple Lie groups.