Abstract. Montgomery-Yang problem predicts that every pseudofree circle action on the 5-
dimensional sphere has at most 3 non-free orbits. Using a certain one-to-one correspondence,
Koll´ar formulated the algebraic version of the Montgomery-Yang problem: every projective surface
S with quotient singularities such that the second Betti number b2(S) = 1 has at most 3
singular points if its smooth locus S0 is simply connected.
We prove the conjecture under the assumption that S has at least one non-cyclic singularity.
In the course of the proof, we classify projective surfaces S with quotient singularities such that
(i) b2(S) = 1, (ii) H1(S0; Z) = 0, and (iii) S has 4 or more singular points, not all cyclic, and
prove that all such surfaces have 1(S0) = A5, the icosahedral group.