Pseudo-periodic Maps and Degeneration of Riemann Surfaces [Paperback]

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 appliesthe 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.Part I: Conjugacy Classification of Pseudo-periodic Mapping Classes.- 1 Pseudo-periodic Maps.- 2 Standard Form.- 3 Generalized Quotient.- 4 Uniqueness of Minimal Quotient.- 5 A Theorem in Elementary Number Theory.- 6 Conjugacy Invariants.- Part II: The Topology of Degeneration of Riemann Surfaces.- 7 Topological Monodromy.- 8 Blowing Down Is a Topological Operation.- 9 Singular Open-Book.

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. & Nielsens arguments are sometimes considered as too vague, and one of the main objects oflós