Connected components of the space of type-preserving representations

We complete the characterization of the connected components of the space of type-preserving representations of a punctured surface group into $\mathrm{PSL}(2,\mathbb{R})$. We show that the connected components are indexed by the relative Euler classes and the signs of the images of the peripheral elements satisfying a generalized Milnor-Wood inequality; and when the surface is a punctured sphere, there are additional connected components consisting of ``totally non-hyperbolic" representations. As a consequence, we count the total number of the connected components of the space of type-preserving representations.