# Milnor Fiber Boundary of a Non-isolated Surface Singularity [electronic resource] / by András Némethi, Ágnes Szilárd.

- Publication:
- Berlin, Heidelberg : Springer Berlin Heidelberg, 2012.
- Series:
- Lecture Notes in Mathematics, 0075-8434 ; 2037

Lecture Notes in Mathematics, 0075-8434 ; 2037 - Format/Description:
- Book

1 online resource (XII, 240 pages) - Subjects:
- Differential equations, partial.

Geometry, algebraic.

Algebraic topology. - Local subjects:
- Several Complex Variables and Analytic Spaces. (search)

Algebraic Geometry. (search)

Algebraic Topology. (search) - System Details:
- text file PDF
- Summary:
- 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.
- Contents:
- 1 Introduction

2 The topology of a hypersurface germ f in three variables Milnor fiber

3 The topology of a pair (f ; g)

4 Plumbing graphs and oriented plumbed 3-manifolds

5 Cyclic coverings of graphs

6 The graph GC of a pair (f ; g). The definition

7 The graph GC . Properties

8 Examples. Homogeneous singularities

9 Examples. Families associated with plane curve singularities

10 The Main Algorithm

11 Proof of the Main Algorithm

12 The Collapsing Main Algorithm

13 Vertical/horizontal monodromies

14 The algebraic monodromy of H1(¶ F). Starting point

15 The ranks of H1(¶ F) and H1(¶ F nVg) via plumbing

16 The characteristic polynomial of ¶ F via P# and P#

18 The mixed Hodge structure of H1(¶ F)

19 Homogeneous singularities

20 Cylinders of plane curve singularities: f = f 0(x;y)

21 Germs f of type z f 0(x;y)

22 The T;;-family

23 Germs f of type ˜ f (xayb; z). Suspensions

24 Peculiar structures on ¶ F. Topics for future research

25 List of examples

26 List of notations. - Contributor:
- Szilárd, Ágnes. author., Author,

SpringerLink (Online service) - Contained In:
- Springer eBooks
- Other format:
- Printed edition:

Printed edition: - ISBN:
- 9783642236471
- Publisher Number:
- 10.1007/978-3-642-23647-1 doi
- Access Restriction:
- Restricted for use by site license.
- Online:
- Connect to full text

http://hdl.library.upenn.edu/1017.12/2329202 -
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