arXiv math/05064322005-06math.GTmath.AG

The geometry of continued fractions and the topology of surface singularities

Patrick Popescu-Pampu

备注：: 54 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
发表信息：: In ``Singularities in Geometry and Topology 2004'', J.-P. Brasselet, T. Suwa eds. Advanced Studies in Pure Mathematics 46, 2007, 119-195.

英文摘要：

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).

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