On the diophantine equation \(1=\sum \frac{1}{n_ i}+\frac{1}{\prod n_ i}\quad and\) a class of homologically trivial complex surface singularities
In this paper we investigate the relation between an elementary problem in the theory of unit (or Egyptian) fractions, and the local topological structure of isolated singularities of complex surfaces, namely: Let \(n_ 1,...,n_ N\) be integers \(\geq 2\), and let \(x\in X\) be an isolated two-dimensional complex singularity whose dual intersection graph is a star with central \(weight\quad 1\) and with \(weights\quad n\) on the arms. Then X is locally the cone on a homology 3-sphere if and only if \(\sum n_ i^{-1}+(\prod n_ i)^{-1}=1.\quad All\) such unit fraction expressions for 1 are given for \(N\leq 7\), and properties of such sequences \(\{n_ i\}\) are discussed in general.
topological structure of isolated singularities of complex surfaces
scientific article; zbMATH DE number 3997987