  We describe some methods to compute fundamental groups, (co)homology, and
irregularity of semi-log-canonical surfaces.
  As an application, we show that there are exactly two irregular Gorenstein
stable surfaces with $K^2=1$, both of which have $\chi(X) = 0$ and
$\mathrm{Pic}^0(X)=\mathbb{C}^*$ but different homotopy type.
