We investigate Landau-Ginzburg string theory with the singular superpotential X^{-1} on arbitrary Riemann surfaces. This theory, which is a topological version of the c=1 string at the self-dual radius, is solved using results from intersection theory and from the analysis of matter Landau-Ginzburg systems, and consistency requirements. Higher-genus amplitudes decompose as a sum of contributions from the bulk and the boundary of moduli space. These amplitudes generate the W-infinity algebra.