The geometry of continued fractions and the topology of surface singularities

Details The geometry of continued fractions and the topology of surface singularities The geometry of continued fractions and the topology of surface singularities

2005-06-21

arxiv.org/abs/math/0506432v2

In ``Singularities in Geometry and Topology 2004'', J.-P. Brasselet, T. Suwa eds. Advanced Studies in Pure Mathematics 46, 200

Mathematics

Geometric Topology

54 pages, 25 figures. To appear in Singularities-Sapporo 2004, Advanced Studies in Pure Mathematics, Kinokuniya, Tokyo, 2006.

Scientific paper

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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