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Lecture Notes in Mathematics

Milnor Fiber Boundary of a Non-isolated Surface Singularity

Authors: Némethi, András, Szilárd, Ágnes

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  • 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
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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)

Table of contents (24 chapters)

Table of contents (24 chapters)
  • Introduction

    Pages 1-7

    Némethi, András (et al.)

  • The Topology of a Hypersurface Germ f in Three Variables

    Pages 11-15

    Némethi, András (et al.)

  • The Topology of a Pair (f,g)

    Pages 17-23

    Némethi, András (et al.)

  • Plumbing Graphs and Oriented Plumbed 3-Manifolds

    Pages 25-43

    Némethi, András (et al.)

  • Cyclic Coverings of Graphs

    Pages 45-54

    Némethi, András (et al.)

Buy this book

eBook $44.99
price for USA in USD
  • ISBN 978-3-642-23647-1
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $59.99
price for USA in USD
Rent the eBook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Milnor Fiber Boundary of a Non-isolated Surface Singularity
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
2037
Copyright
2012
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-23647-1
DOI
10.1007/978-3-642-23647-1
Softcover ISBN
978-3-642-23646-4
Series ISSN
0075-8434
Edition Number
1
Number of Pages
XII, 240
Topics