  Data on the beam asymmetry $\Sigma$ in the photoproduction of $\eta$ mesons
off protons are reported for tagged photon energies from 1130 to 1790 MeV (mass
range from $W=$ 1748 MeV to $W=$ 2045 MeV). The data cover the full solid angle
that allows for a precise moment analysis. For the first time, a strong cusp
effect in a polarization observable has been observed that is an effect of a
branch-point singularity at the $p\eta'$ threshold [$E_\gamma=$ 1447 MeV ($W=$
1896 MeV)]. The latest BnGa partial wave analysis includes the new beam
asymmetry data and yields a strong indication for the $N(1895)\frac{1}{2}^-$
nucleon resonance, demonstrating the importance of including all singularities
for a correct determination of partial waves and resonance parameters.
