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Milnor Fiber Boundary of a Non-isolated Surface Singularity

  • Book
  • © 2012

Overview

Authors:
  1. András Némethi

    1. Algebraic Geometry and Differential Topo, Rényi Institute of Mathematics, Budapest, Hungary

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  2. Ágnes Szilárd

    1. Algebraic Geometry and Differential Topo, Rényi Institute of Mathematics, Budapest, Hungary

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    You can also search for this author in PubMed   Google Scholar

  • Presents a new approach in the study of non-isolated hypersurface singularities
  • The first book about non-isolated hypersurface singularities
  • Conceptual and comprehensive description of invariants of non-isolated singularities
  • Key connections between singularity theory and low-dimensional topology
  • Numerous explicit examples for plumbing representation of the boundary of the Milnor fiber Numerous explicit examples for the Jordan block structure of different monodromy operators

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2037)

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Table of contents (24 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    • András Némethi, Ágnes Szilárd
    Pages 1-7

  3. Preliminaries

    1. Front Matter

      Pages 9-9
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    2. The Topology of a Hypersurface Germ f in Three Variables

      • András Némethi, Ágnes Szilárd
      Pages 11-15

    3. The Topology of a Pair (f,g)

      • András Némethi, Ágnes Szilárd
      Pages 17-23

    4. Plumbing Graphs and Oriented Plumbed 3-Manifolds

      • András Némethi, Ágnes Szilárd
      Pages 25-43

    5. Cyclic Coverings of Graphs

      • András Némethi, Ágnes Szilárd
      Pages 45-54

    6. The Graph \(\mathit\Gamma_{\mathcal{C}}\) of a pair (f,g): The Definition

      • András Némethi, Ágnes Szilárd
      Pages 55-61

    7. The Graph \(\mathit\Gamma_{\mathcal{C}}\) : Properties

      • András Némethi, Ágnes Szilárd
      Pages 63-77

    8. Examples: Homogeneous Singularities

      • András Némethi, Ágnes Szilárd
      Pages 79-82

    9. Examples: Families Associated with Plane Curve Singularities

      • András Némethi, Ágnes Szilárd
      Pages 83-97

  4. Plumbing Graphs Derived from $$\mathit\Gamma_{\mathcal{C}}$$

    1. Front Matter

      Pages 99-99
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  5. Plumbing graphs derived from GC

    1. The Main Algorithm

      • András Némethi, Ágnes Szilárd
      Pages 101-115

    2. Proof of the Main Algorithm

      • András Némethi, Ágnes Szilárd
      Pages 117-130

    3. The Collapsing Main Algorithm

      • András Némethi, Ágnes Szilárd
      Pages 131-138

    4. Vertical/Horizontal Monodromies

      • András Némethi, Ágnes Szilárd
      Pages 139-151

    5. The Algebraic Monodromy of H 1(∂ F): Starting Point

      • András Némethi, Ágnes Szilárd
      Pages 153-156

    6. The Ranks of H1(∂ F) and H 1(∂ F \ Vg) via plumbing

      • András Némethi, Ágnes Szilárd
      Pages 157-160

    7. The Characteristic Polynomial of ∂F via P # and \({P}^\sharp_{j}\)

      • András Némethi, Ágnes Szilárd
      Pages 161-166

    8. The Proof of the Characteristic Polynomial Formulae

      • András Némethi, Ágnes Szilárd
      Pages 167-172
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Keywords

About this book

In the study of algebraic/analytic varieties a key aspect is the description of the invariants of their singularities. This book targets the challenging non-isolated case. Let f be a complex analytic hypersurface germ in three variables whose zero set has a 1-dimensional singular locus. We develop an explicit procedure and algorithm that describe the boundary M of the Milnor fiber of f as an oriented plumbed 3-manifold. This method also provides the characteristic polynomial of the algebraic monodromy. We then determine the multiplicity system of the open book decomposition of M cut out by the argument of g for any complex analytic germ g such that the pair (f,g) is an ICIS. Moreover, the horizontal and vertical monodromies of the transversal type singularities associated with the singular locus of f and of the ICIS (f,g) are also described. The theory is supported by a substantial amount of examples, including homogeneous and composed singularities and suspensions. The properties peculiar to M are also emphasized.

Reviews

From the reviews:

“The aim of this book is to study the topological types of the oriented smooth 3-manifolds appearing as boundaries ∂F of the Milnor fibers of complex surface singularities of embedding dimension 3, as well as the monodromy actions on their homology. … It is clearly invaluable for anybody interested in the topology of non-isolated complex surface singularities and even of singularities of real analytic spaces of dimension 4.” (Patrick Popescu-Pampu, Mathematical Reviews, January, 2014)

“The book describes three manifolds which occur in relation with complex hypersurfaces in C3 near singular points. … I recommend it to all students and researchers who are interested in the local topology of algebraic varieties. It contains a good description of techniques, such as plumbing, cyclic coverings, monodromy, et cetera. The book is well written and ends with several topics for future research.” (Dirk Siersma, Nieuw Archief voor Wiskunde, Vol. 14 (2), June, 2013)

Authors and Affiliations

  • Algebraic Geometry and Differential Topo, Rényi Institute of Mathematics, Budapest, Hungary

    András Némethi, Ágnes Szilárd

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