On the Diophantine equation 1 = ∑ 1∕n_i + 1∕∏ n_i and a class of homologically trivial complex surface singularities

Lawrence James Brenton and Richard Hill

Vol. 133 (1988), No. 1, 41–67

DOI: 10.2140/pjm.1988.133.41

Abstract

Let n₁,…,n_N be integers ≥ 2, and let x ∈ X be an isolated two-dimensional complex singularity whose dual intersection graph is a star with central weight 1 and with weights n_i on the arms. Then X is locally the cone on a homology 3-sphere if and only if ∑ n_i⁻¹ + ∏ n_i⁻¹ = 1. All such unit fraction expressions for 1 are given for N ≤ 7, and properties of such sequences {n_i} are discussed in general.

Mathematical Subject Classification 2000

Primary: 32C40, 32C40

Secondary: 14B05, 32B30

Milestones

Received: 29 April 1985

Published: 1 May 1988