Pseudo-periodic Maps and Degeneration of Riemann Surfaces

1. Front Matter

   Pages i-xvi

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2. Conjugacy Classification of Pseudo-periodic Mapping Classes

   1. Front Matter

      Pages 1-1

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   2. Pseudo-periodic Maps

      • Yukio Matsumoto, José María Montesinos-Amilibia

      Pages 3-15

   3. Standard Form

      • Yukio Matsumoto, José María Montesinos-Amilibia

      Pages 17-52

   4. Generalized Quotient

      • Yukio Matsumoto, José María Montesinos-Amilibia

      Pages 53-92

   5. Uniqueness of Minimal Quotient

      • Yukio Matsumoto, José María Montesinos-Amilibia

      Pages 93-129

   6. A Theorem in Elementary Number Theory

      • Yukio Matsumoto, José María Montesinos-Amilibia

      Pages 131-144

   7. Conjugacy Invariants

      • Yukio Matsumoto, José María Montesinos-Amilibia

      Pages 145-169

3. The Topology of Degeneration of Riemann Surfaces

   1. Front Matter

      Pages 171-171

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   2. Topological Monodromy

      • Yukio Matsumoto, José María Montesinos-Amilibia

      Pages 173-188

   3. Blowing Down Is a Topological Operation

      • Yukio Matsumoto, José María Montesinos-Amilibia

      Pages 189-198

   4. Singular Open-Book

      • Yukio Matsumoto, José María Montesinos-Amilibia

      Pages 199-220

4. Back Matter

   Pages 221-238

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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.

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

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)

• Department of Mathematics, Gakushuin University, Toshima-ku, Japan

  Yukio Matsumoto

• Facultad de Matemáticas, Departamento de Geometría y Topología, Universidad Complutense, Madrid, Spain

  José María Montesinos-Amilibia

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