Rational homology projective planes are normal projective complex surfaces whose Betti numbers agree with those of the complex projective plane. As these surfaces possess the smallest possible Betti numbers among algebraic surfaces, they are of natural interest in algebraic geometry. In this talk, we focus on surfaces admitting quotient singularities and demonstrate how tools from low-dimensional topology can be applied to their study.

First, we address the algebraic Montgomery-Yang problem, which conjectures that a rational homology projective plane with quotient singularities admits at most three singular points, provided its smooth locus is simply-connected. Second, we provide a classification of quotient singularity types for such surfaces of index up to three, under the assumption that their smooth loci have trivial first integral homology. These results are based on joint works with Jongil Park (Seoul National University) and Kyungbae Park (Kangwon National University).