The geometry of continued fractions and the topology of surface singularities

Abstract

We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing structure on the abstract boundaries (also called links) of normal surface singularities. The duality between supplementary cones gives in particular a geometric interpretation of a duality discovered by Hirzebruch between the continued fraction expansions of two numbers l >1 and l/(l - 1).

arXiv comment54 pages, 25 figures. To appear in Singularities-Sapporo 2004, Advanced Studies in Pure Mathematics, Kinokuniya, Tokyo, 2006. With respect to the first version, I added references to papers by Dais, Haus and Henk; Giroux; Gonzalez-Sprinberg. They are alluded to at the end of section 5. I also added explanations at the beginning of section 5.1