Algebraic Montgomery-Yang Problem: the noncyclic case

TL;DR

This paper proves a case of the algebraic Montgomery-Yang problem for projective surfaces with at least one noncyclic singularity, classifying certain surfaces and linking their properties to the icosahedral group.

Contribution

It confirms the conjecture for surfaces with noncyclic singularities and classifies surfaces with specific topological properties and multiple singular points.

Findings

Surfaces with at least one noncyclic singularity satisfy the conjecture.

Classified surfaces with specific invariants and multiple singular points, showing their fundamental group is the icosahedral group.

Proved that certain surfaces with four or more singular points have fundamental group isomorphic to .

Abstract

Montgomery-Yang problem predicts that every pseudofree differentiable circle action on the 5-dimensional sphere ${\mathbb S}^5$ 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 $b_2(S) = 1$ has at most 3 singular points if its smooth locus $S^0$ is simply-connected. In this paper, we prove the conjecture under the assumption that $S$ has at least one noncyclic singularity. In the course of the proof, we classify projective surfaces $S$ with quotient singularities such that (i) $b_2(S) = 1$, (ii) $H_1(S^0, \mathbb{Z}) = 0$, and (iii) $S$ has 4 or more singular points, not all cyclic, and prove that all such surfaces have $\pi_1(S^0)\cong \mathfrak{A}_5$, the icosahedral group.