Título: | Pseudo-periodic homeomorphisms and degeneration of Riemann surfaces |
Autores: | Matsumoto, Yukio ; Montesinos Amilibia, José María |
Tipo de documento: | texto impreso |
Editorial: | American Mathematical Society, 1994 |
Dimensiones: | application/pdf |
Nota general: | info:eu-repo/semantics/restrictedAccess |
Idiomas: | |
Palabras clave: | Estado = Publicado , Materia = Ciencias: Matemáticas: Análisis matemático , Materia = Ciencias: Matemáticas: Topología , Tipo = Artículo |
Resumen: |
The authors classify all topological types of degenerate central fibers appearing in holomorphic families of closed Riemann surfaces of genus g?2 over the unit disc. A degenerating family of genus g is a triple (M,D,?) consisting of a 2-dimensional complex manifold M, an open unit disk D in the complex plane, and a surjective proper holomorphic map ? such that all fibers of ? are connected and ?|??1(D?): ??1(D?)?D? is a smooth fiber bundle with fiber ?g, where ?g is an oriented closed surface of genus g and D?=D?{0}. The monodromy homeomorphism f: ?g??g of (M,D,?) is determined as usual up to isotopy and conjugation. It is known that f is a pseudo-periodic homeomorphism of negative twist, that is, its mapping class [f] is either of finite order or reducible, and in the latter case, all component mapping classes are of finite order and its screw numbers are all negative. A family is said to be minimal if it is free of (?1)-curves. Two families (Mi,D,?i), i=1,2, are topologically equivalent if there exist homeomorphisms H:M1?M2 and h:D?D satisfying h(0)=0 and h??1=?2?H. Let Sg={minimal degenerating families of genus g} modulo topological equivalence. Denote by P?g the set of all pseudo-periodic mapping classes of negative twist of ?g. Then we have a well-defined map monodromy ?:Sg?P?g. The main result is the following theorem: For g?2, ?:Sg?P?g is bijective. The most essential part of the proof of this theorem is to construct the inverse map of ?, that is, for a given pseudo-periodic homeomorphism f of negative twist the authors construct a degenerating family (M,D,?) of genus g with monodromy homeomorphism f. In the second part of this paper the authors give a complete set of conjugacy invariants for the pseudo-periodic homeomorphisms of negative twist, which shows that Nielsen's set of invariants is not complete. |
En línea: | https://eprints.ucm.es/id/eprint/22174/1/montesinos52.pdf |
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