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© 2012

Milnor Fiber Boundary of a Non-isolated Surface Singularity

Benefits

  • 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

Book

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

Table of contents

  1. Front Matter
    Pages i-xii
  2. András Némethi, Ágnes Szilárd
    Pages 1-7
  3. Preliminaries

    1. Front Matter
      Pages 9-9
    2. András Némethi, Ágnes Szilárd
      Pages 11-15
    3. András Némethi, Ágnes Szilárd
      Pages 17-23
    4. András Némethi, Ágnes Szilárd
      Pages 25-43
    5. András Némethi, Ágnes Szilárd
      Pages 45-54
    6. András Némethi, Ágnes Szilárd
      Pages 63-77
    7. András Némethi, Ágnes Szilárd
      Pages 79-82
    8. András Némethi, Ágnes Szilárd
      Pages 83-97
  4. Plumbing Graphs Derived from $$\mathit\Gamma_{\mathcal{C}}$$

    1. Front Matter
      Pages 99-99
    2. András Némethi, Ágnes Szilárd
      Pages 101-115
    3. András Némethi, Ágnes Szilárd
      Pages 117-130
    4. András Némethi, Ágnes Szilárd
      Pages 131-138
    5. András Némethi, Ágnes Szilárd
      Pages 139-151
    6. András Némethi, Ágnes Szilárd
      Pages 153-156
    7. András Némethi, Ágnes Szilárd
      Pages 157-160
    8. András Némethi, Ágnes Szilárd
      Pages 161-166
    9. András Némethi, Ágnes Szilárd
      Pages 167-172

About this book

Introduction

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

Keywords

32Sxx, 14J17, 14B05, 14P15, 57M27 monodromy non-isolated singularity plumbed 3-manifolds resolution graphs

Authors and affiliations

  1. 1.Algebraic Geometry and Differential TopoRényi Institute of MathematicsBudapestHungary
  2. 2.Algebraic Geometry and Differential TopoRényi Institute of MathematicsBudapestHungary

Bibliographic information

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)