Pseudo-periodic Maps and Degeneration of Riemann SurfacesThe 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. |
Contents
Part II The Topology of Degeneration of Riemann Surfaces | 170 |
Appendix A Periodic Maps Which Are Homotopic | 221 |
233 | |
Other editions - View all
Pseudo-periodic Maps and Degeneration of Riemann Surfaces Yukio Matsumoto,José María Montesinos-Amilibia No preview available - 2011 |
Common terms and phrases
1.ARCHı A₁ amphidrome annuli ARCH b₁ bijective boundary components boundary curves Ch(D Chap chorizo space closed nodal neighborhood completes the proof condition conjugacy connected component construct coordinates Corollary cut curves cyclically define definition degenerating family Dehn twist denote disjoint union disk edge Euler characteristic f and f genus g homeomorphism homotopic intersection irreducible component isomorphism isotopic jz2j linear twist map f map of negative mapping class minimal quotient monodromy multiplicity n₁ negative twist ni+1 Nielsen node numerical homeomorphism parametrizations partition graph periodic map permutes pinched covering positive integer proof of Lemma proof of Theorem pseudo-periodic map quotient space resp Riemann surfaces rotation S[fi satisfies screw number Sect self-intersection sequence special twist superstandard form Suppose system of cut Theorem 5.1 topological monodromy union of annuli valency vertex