Pseudo-periodic homeomorphisms and degeneration of Riemann surfaces
Pseudo-periodic homeomorphisms and degeneration of Riemann surfaces
复制标题
黎曼曲面的伪周期同胚和简并
DOI：
10.1090/s0273-0979-1994-00437-9
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发表时间：
1994
影响因子：
1.3
通讯作者：
J. M. Montesinos
中科院分区：
文献类型：
--
作者：
Yukio Matsumoto;J. M. Montesinos
作者对单位圆盘上亏格g≥2的闭Riemann曲面的全纯族中出现的退化中心纤维的所有拓扑类型进行了分类。一个亏格g的退化族是一个三元组（M，D，φ），它由一个2维复流形M，一个复平面上的开单位圆盘D和一个满射真全纯映射φ组成，使得φ的所有纤维连通，且φ| φ−1（D）：φ−1（D）→D是一个光滑纤维丛，其纤维为g，其中g是亏格g的定向闭曲面，D =D−{0}。（M，D，φ）的单值同胚f：f_g→ f_g在合痕和共轭的情况下照常确定。已知f是一个负扭的伪周期同胚，即它的映射类[f]要么是有限阶的，要么是可约的，在后一种情况下，所有的分支映射类都是有限阶的，并且它的螺旋数都是负的.一个族被称为极小族，如果它没有（-1）-曲线。两个族（Mi，D，φi），i= 1，2是拓扑等价的，如果存在满足h（0）=0和h <$φ1=φ2 <$H的同胚H：M1→M2和h：D→D.设Sg={亏格g的极小退化族}模拓扑等价.用P−g表示所有负扭的伪周期映射类的集合。那么我们就有了一个定义明确的地图
单值ρ：Sg→ P-g。
主要结果是以下定理：对于g≥2，ρ：Sg→P−g是双射的。证明该定理的关键部分是构造ρ的逆映射，即对给定的负扭伪周期同胚f，构造一个亏格g的退化族（M，D，φ）与单值同胚f.本文的第二部分给出了负扭伪周期同胚的共轭不变量的完备集，这说明Nielsen不变量集是不完备的。
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,φ) consisting of a 2-dimensional complex manifold M, an open unit disk D in the complex plane, and a surjective proper holomorphic map φ such that all fibers of φ are connected and φ|φ−1(D∗): φ−1(D∗)→D∗ is a smooth fiber bundle with fiber Σg, where Σg is an oriented closed surface of genus g and D∗=D−{0}. The monodromy homeomorphism f: Σg→Σg of (M,D,φ) 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,φi), i=1,2, are topologically equivalent if there exist homeomorphisms H:M1→M2 and h:D→D satisfying h(0)=0 and h∘φ1=φ2∘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 Σg. Then we have a well-defined map
monodromy ρ:Sg→P−g.
The main result is the following theorem: For g≥2, ρ:Sg→P−g is bijective. The most essential part of the proof of this theorem is to construct the inverse map of ρ, that is, for a given pseudo-periodic homeomorphism f of negative twist the authors construct a degenerating family (M,D,φ) 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.