Publication: Pseudo-periodic homeomorphisms and degeneration of Riemann surfaces
Loading...
Full text at PDC
Publication Date
1994
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
American Mathematical Society
Citations
Exportar
Abstract
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.
Description
UCM subjects
Unesco subjects
Keywords
Citation
L. Bers, Spaces of degenerating Riemann surfaces, Ann. of Math. Stud., vol. 79, Princeton Univ. Press, Princeton, NJ, 1975, pp. 43-55.
C. J. Earle and P. L. Sipe, Families of Riemann surfaces over the punctured disk, Pacific J. Math. 150 (1991), 79-96.
J. Gilman, On the Nielsen type and the classification for the mapping class group, Adv. Math. 40 (1981), 68-96.
Y. Imayoshi, Holomorphic families of Riemann surfaces and Teichmüller spaces, Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, NJ, 1981, pp. 277-300.
K. Kodaira, On compact analytic surfaces. II, Ann. of Math. (2) 77 (1963), 563-626.
Y. Matsumoto and J. M. Montesinos-Amilibia, Singular fibers and pseudo-periodic surface automorphisms, Conference Report for Knots 90, Osaka, 1990, pp. 50-51.
Y. Matsumoto and J. M. Montesinos-Amilibia, Pseudo-periodic maps and degeneration of Riemann surfaces. I, II, preprint, Univ. of Tokyo and Univ. Complutense de Madrid, 1991/1992.
Y. Namikawa and K. Ueno, The complete classification of fibers in pencils of curves of genus two, Manuscripta Math. 9 (1973), 143-186.
J. Nielsen, Die structur periodischer transformationen von Flächen, Mat.-Fys. Medd. Danske Vid. Selsk 15 (1937); English transl. by J. Stillwell, The structure of periodic surface transformations, Collected Papers 2, Birkhäuser, 1986.
J. Nielsen, Surface transformation classes of algebraically finite type, Mat.-Fys. Medd. Danske Vid. Selsk. 21 (1944), Collected Papers 2, Birkhäuser, 1986.
T. Oda, A note on ramification of the Galois representation on the fundamental group of an algebraic curve. II, J. Number Theory (to appear). cf.
H. Shiga and H. Tanigawa, On the Maskit coordinates of Teichmüller spaces and modular transformations, Kodai Math. J. 12 (1989), 437-443.
I. Tamura, Foliations and spinnable structures on manifolds, Ann. Inst. Fourier (Grenoble) 23 (1973), 197-214.
W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), 417-431.
H. E. Winkelnkemper, Manifolds as open books, Bull. Amer. Math. Soc. (N.S.) 79 (1973), 45-51.