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© 2011

Pseudo-periodic Maps and Degeneration of Riemann Surfaces

  • The publication of the manuscript which has been circulated among specialists for 20 years

  • The first complete study of pseudo-periodic maps after J. Nielsen's original work

  • The first application of the pseudo-periodic maps to the topology of degeneration of Riemann surfaces

  • This book is self-contained and is readable by advanced students (master level)

Book

Part of the Lecture Notes in Mathematics book series (LNM, volume 2030)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Conjugacy Classification of Pseudo-periodic Mapping Classes

    1. Front Matter
      Pages 1-1
    2. Yukio Matsumoto, José María Montesinos-Amilibia
      Pages 3-15
    3. Yukio Matsumoto, José María Montesinos-Amilibia
      Pages 17-52
    4. Yukio Matsumoto, José María Montesinos-Amilibia
      Pages 53-92
    5. Yukio Matsumoto, José María Montesinos-Amilibia
      Pages 93-129
    6. Yukio Matsumoto, José María Montesinos-Amilibia
      Pages 131-144
    7. Yukio Matsumoto, José María Montesinos-Amilibia
      Pages 145-169
  3. The Topology of Degeneration of Riemann Surfaces

    1. Front Matter
      Pages 171-171
    2. Yukio Matsumoto, José María Montesinos-Amilibia
      Pages 173-188
    3. Yukio Matsumoto, José María Montesinos-Amilibia
      Pages 189-198
    4. Yukio Matsumoto, José María Montesinos-Amilibia
      Pages 199-220
  4. Back Matter
    Pages 221-238

About this book

Introduction

The first part of the book studies pseudo-periodic maps of a closed surface of genus greater than or equal to two. This class of homeomorphisms was originally introduced by J. Nielsen in 1944 as an extension of periodic maps. In this book, the conjugacy classes of the (chiral) pseudo-periodic mapping classes are completely classified, and Nielsen’s incomplete classification is corrected. The second part applies the results of the first part to the topology of degeneration of Riemann surfaces. It is shown that the set of topological types of all the singular fibers appearing in one-parameter holomorphic families of Riemann surfaces is in a bijective correspondence with the set of conjugacy classes of the pseudo-periodic maps of negative twists. The correspondence is given by the topological monodromy.

Keywords

14-XX; 57-XX chorizo spaces degeneration of Riemann surface pseudo-periodic maps topological monodromy

Authors and affiliations

  1. 1.Department of MathematicsGakushuin UniversityToshima-kuJapan
  2. 2.Facultad de Matemáticas, Departamento de Geometría y TopologíaUniversidad ComplutenseMadridSpain

Bibliographic information

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Reviews

From the reviews:

“The present monograph consists of two parts. In the first part a class of pseudo-periodic maps resp. mapping classes of closed surfaces is studied and classified up to conjugation (those of ‘negative twist’). This is motivated also by the second part on the topology of degenerating families of Riemann surfaces over a disk … . exposition is self-contained, elementary and written with great care for details … this makes arguments and proofs quite long, but certainly easy and pleasant to read.” (Bruno Zimmermann, Zentralblatt MATH, Vol. 1239, 2012)

“Surface mapping classes of algebraically finite type were introduced by Nielsen in 1944. … Nielsen’s arguments are sometimes considered as too vague, and one of the main objects of this book is to make Nielsen’s assertions precise. … The results in this book are given with detailed proofs. The book is well written and it should be useful for low-dimensional topologists and algebraic geometers.” (Athanase Papadopoulos, Mathematical Reviews, Issue 2012 h)