Pseudo-periodic Maps and Degeneration of Riemann Surfaces

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan B. Teissier, Paris For further volumes: http://www.springer...

Author: Yukio Matsumoto | José María Montesinos-Amilibia

59 downloads 615 Views 2MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan B. Teissier, Paris

For further volumes: http://www.springer.com/series/304

2030

•

Yukio Matsumoto José Mar´ıa Montesinos-Amilibia

Pseudo-periodic Maps and Degeneration of Riemann Surfaces

123

Yukio Matsumoto Gakushuin University Department of Mathematics Mejiro 1-5-1 171-8588 Toshima-ku Tokyo Japan [email protected]

José Mar´ıa Montesinos-Amilibia Universidad Complutense Facultad de Matemáticas Departamento de Geometr´ıa y Topolog´ıa Plaza de las Ciencias 3 28040 Madrid Spain [email protected]

ISBN 978-3-642-22533-8 e-ISBN 978-3-642-22534-5 DOI 10.1007/978-3-642-22534-5 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2011934808 Mathematics Subject Classification (2010): 14-XX, 57-XX c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Dedicated with respect and affection to the memory of Professor Itiro Tamura (1926–1991)

•

Preface

In 1944, Nielsen introduced a certain type of mapping classes of a surface which were called by him surface transformation classes of algebraically finite type, [53]. He introduced this type of mapping classes as a generalization of the mapping classes of finite order. By the celebrated Nielsen Theorem [52], the latter classes contain surface homeomorphisms of finite order (For a generalization, see Kerckhoff [30]). A mapping class of algebraically finite type does not necessarily contain a homeomorphism of finite order, but using Nielsen’s theorem [52], one can show that it contains a homeomorphism f satisfying the following conditions (in what follows f will be an orientation-preserving homeomorphism of a closed, connected, oriented surface of genus g, ˙g ): 1. There exists a disjoint union of simple closed curves (which will be called cut curves) C D C 1 [ C2 [ [ C r on ˙g such that f .C / D C , and 2. the restriction of f to the complement of C , f j.˙g C / W ˙g C ! ˙g C is isotopic to a periodic map, namely a homeomorphism of finite order. (Cf. [53, Sect. 14], [22]). In the present memoir, such a homeomorphism (and also a homeomorphism which is isotopic to such a homeomorphism) will be called a pseudo-periodic map. A periodic map is a special case of a pseudo-periodic map. In recent terminology, a homeomorphism f is pseudo-periodic if and only if either it is of finite order or its mapping class [f ] is reducible and all the component mapping classes are of finite order. (See [12,16,22,24,63]). A surface transformation class of algebraically finite type is nothing but a mapping class of a pseudo-periodic map. Nielsen [53] studied these classes extensively and defined several important invariants, for instance, the screw number of f about a cut curve Ci which measures

vii

viii

Preface

the amount of the (fractional) Dehn-twist performed by a certain power f ˛ of f sending Ci to itself; or the character of Ci : whether it is “amphidrome” or not. Here Ci is amphidrome if there is an integer such that ! ! f .Ci / D Ci : He asserted in [53] that his invariants were a complete set of conjugacy invariants, meaning that if two pseudo-periodic maps f1 W ˙g.1/ ! ˙g.1/ and f2 W ˙g.2/ ! ˙g.2/ have these same invariants, then their mapping classes Œf1 and Œf2 are equivalent under a certain homeomorphism h W ˙g.1/ ! ˙g.2/ ; i.e. Œf1 D Œh1 f2 h. (For an exact formulation, see [22, Theorem 13.4]). However, his proof of this assertion was rather vague, and we need an invariant (the action of monodromy on the partition graph) which he did not mention explicitly. See Examples 6.3 and 6.4 in Chap. 6. A pseudo-periodic map f is said to be of negative twist if the screw numbers about a certain system of cut curves are all negative (Chap. 3). The purpose of Part I of the present memoir is to construct a complete set of conjugacy invariants for a pseudo-periodic map f of negative twist. We have added to Nielsen’s invariants one more: the action of f on the “partition graph”, which is the action, induced by f , on the configuration of the partition of ˙g obtained by cutting ˙g along a certain system of cut curves fCi griD1. The main result of Part I is roughly stated as follows (see Theorem 6.1 and 6.3 for precise statements): .1/

.1/

.2/

.2/

Theorem 0.1. Let f1 W ˙g !˙g and f2 W ˙g ! ˙g be pseudo-periodic maps of negative twist. Suppose that they have the same values in Nielsen’s invariants and that their actions on the respective partition graphs are equivariantly isomorphic. .1/ .2/ Then there exists an orientation preserving homeomorphism h W ˙g ! ˙g such 1 that Œf1 D Œh f2 h. In the course of the proof, we develop (in Chaps. 3–5) the theory of generalized quotients, which are naturally associated with pseudo-periodic maps, just as ordinary quotient spaces are associated with periodic maps. This makes our proof of Theorem 0.1 unexpectedly long, but the generalized quotients will play an essential role also in the study of the degeneration of Riemann surfaces (in Part II). This was the main reason of our investigation, which therefore concentrated in the study

Preface

ix

of generalized quotients. As a matter of fact, Theorem 0.1 above is just a (non immediate) corollary of our research. The organization of Part I is as follows: In Chap. 1, we review some basic results of Nielsen from [51, 53]. In Chap. 2, we define the “standard form” of a pseudo-periodic map f . Nielsen [53, Sect. 14] constructed a special homeomorphism which served as a standard form, but our standard form is slightly different from his. We prove the existence and the essential uniqueness of the homeomorphism in standard form which is isotopic to a given pseudo-periodic map (Theorem 2.1). In Chap. 3, we introduce the notion of generalized quotients, and in particular, of minimal quotients which are the special case of generalized quotients that satisfy a certain “minimality condition”. According to the definition given in Chap. 3, in order to have a generalized quotient, a pseudo-periodic map f must be in a very special form which we would like to call “superstandard form”. It will be proved that any pseudo-periodic map f of negative twist is isotopic to a pseudo-periodic map in superstandard form having a minimal quotient (Theorem 3.1). In Chap. 4, the following essential uniqueness will be proved (Theorem 4.1): suppose f1 and f2 W ˙g ! ˙g are pseudo-periodic maps of negative twist, both in superstandard form. If they are homotopic, then their respective minimal quotients are isomorphic. By the above existence and uniqueness theorems, we can generalize the definition of minimal quotients to cover any pseudo-periodic map of negative twist not necessarily in superstandard form, i.e., the minimal quotient of a pseudo-periodic map f of negative twist is constructed by first isotoping f to the superstandard form f 0 and then taking the minimal quotient of f 0 , which is declared to be the minimal quotient of f . The minimal quotient captures all of the Nielsen invariants constructed in [53]. Moreover, it will be proved in Part II that the “base space” of the minimal quotient of a pseudo-periodic map f of negative twist is homeomorphic to a (normally minimal) singular fiber of a one-parameter family of Riemann surfaces of genus g around which the topological monodromy is equivalent to [f ]. In Chap. 5, we prove a theorem in elementary number theory, which is basic to the arguments in Chaps. 3 and 4. In Chap. 6, we consider the partition graph and the action of f on it. This action, together with the minimal quotient, determines the conjugacy class of [f ] in Mg (Theorem 6.1). This result is further reformulated in terms of certain cohomology of “weighted graphs” (Theorem 6.3). In Appendix A, we will give a proof of the following theorem: let f and f 0 be (orientation- preserving) periodic maps of a compact surface ˙ each component of which has negative Euler characteristic. Suppose f and f 0 W .˙; @˙/ ! .˙; @˙/ are homotopic as maps of pairs. Then there exists a homeomorphism h W ˙ ! ˙ isotopic to the identity, such that f D h1 f 0 h. This theorem is used in the proof of Theorem 2.1. Among specialists, this theorem seems folklore. A. Edmonds informed, in a letter to the second named

x

Preface

author, that C. Frohman had proved a stronger result which implied the above theorem. Unfortunately, the authors could not find any reference giving an explicit proof, so we decided to write this appendix. Pseudo-periodic maps of negative twist are closely related to the degeneration of Riemann surfaces. In fact, the topological monodromy around a singular fiber in a one-parameter family of Riemann surfaces is a pseudo-periodic map of negative twist (see [19], also [26, 58]). In Part II of this memoir, we will apply the results in Part I to the topology of degeneration of Riemann surfaces. The main result of Part II is roughly summarized as follows: Theorem 0.2. The topological types of minimal degenerating families of Riemann surfaces of genus g 2, over a disk, which are nonsingular outside the origin, are in a bijective correspondence with the conjugacy classes in the mapping class group Mg represented by pseudo-periodic maps of negative twist. The correspondence is given by the topological monodromy. In the case of g D 1, the validity of Theorem 0.2 is reduced by half: By Kodaira’s classification [32] of singular fibers for genus 1, we see that every pseudo-periodic mapping class (of negative twist) of a torus can be realized as the topological monodromy of a singular fiber. Thus the correspondence is “surjective”, but it is not “injective”. For example, all the multiple fibers of type m I0 (in Kodaira’s notation) have the identity mapping class as their topological monodromy. The assumption g 2 is used almost everywhere in our proof: The existence of an admissible system of cut curves subordinate to a pseudo-periodic map (Lemma 1.1) is essential to the definition of various invariants of the map, and the proof of the existence requires g 2. Also “homotopy implies conjugacy” theorem for periodic maps assumes g 2, because in the proof we apply the hyperbolic geometry (see Appendix A). This theorem is indispensable in the proof of the uniqueness of the standard form (see Theorem 2.1 (ii)). Our arguments in later chapters depend on this uniqueness theorem. We have tried to make the memoir as self-contained as possible, except for the two quotations from [51,53]. (Theorems 1.1 and 1.2 of the present memoir). All the other arguments are elementary. The authors are grateful to Allan Edmonds, Takayuki Oda and Hiroshige Shiga for their useful information and comments. This work started during the first named author’s first visit to Spain (1988) and was completed during his second visit (1991). The first named author would like to express his warmest thanks to the members and staffs of Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, for their kind hospitality. Finally but not least at all, the authors deeply thank Srta. Mar´ıa Angeles Bringas for her benevolence, patience, and excellent skill shown in typing this memoir, without which it could have never been published.

Preface

xi

ADDED:SEPTEMBER,20091 The main body of the manuscript of the present memoir was completed in December 1991, and some remaining additional parts in January, 1992. Since then we have not found any occasion to publish this work, for several reasons; the length certainly was one. Another reason, but probably the main one, was the authors’ inability to use Tex. After a long delay of almost two decades, the authors find some unsatisfactory points in the manuscript, for example, it contains too many details, which might be a hindrance for readers who want to get a quick view, but on the other hand, it might help them to understand the details of the argument. Anyway the authors needed to compile these long (elementary and sometimes seemingly trivial) arguments to complete the proof of our theorems. Therefore, we have decided to keep the manuscript in its original form, except for the numbering of the chapters, theorems, propositions, figures, etc. A change we have made is the unification of the two different bibliographies, which were separately attached to each part, into one bibliography at the end. Also we added some references that were published after the completion of our manuscript and some more that we had missed involuntarily or were unknown to us due to our limitations. Unfortunately, the authors cannot be sure even now of the completeness of the augmented bibliography. A pseudo-periodic map would well be called chiral if either it is periodic or all of its screw numbers are of the same sign. A chiral pseudo-periodic map is a pseudo-periodic map of negative twist or of positive twist. If a pseudo-periodic map has both positive and negative screw numbers, it will be called achiral. In Part I of this memoir, we confined ourselves to chiral pseudo-periodic maps (of negative twist). From the viewpoint of surface topology, it would be more natural to treat not only chiral pseudo-periodic maps but also achiral ones, of course. We tried such a general treatment for some time. However, to construct a generalized quotient for an achiral pseudo-periodic map, we are forced to adopt an artificial convention on signs of intersections between the components consisting of a tail-part of the generalized quotient, and we lose the natural uniqueness of the generalized quotient of a pseudoperiodic map. Moreover, our construction of generalized quotients is intended to be applied to topology of degeneration of Riemann surfaces in Part II, and for that purpose, we only need chiral pseudo-periodic maps. For these reasons, we gave up our trial to generalize the construction of generalized quotients to achiral maps. As is immediately seen from the title, the main objects of this memoir are (chiral) pseudo-periodic homeomorphisms and degeneration of Riemann surfaces. Our main point is that these two objects are topologically classified by the same objects, i.e. certain types of “numerical chorizo spaces” together with a cohomology class in the weighted cohomology of their decomposition graphs. This type of chorizo spaces appear as “minimal quotients”of pseudo-periodic homeomorphisms of negative twist, and exactly the same type of chorizo spaces appear also as “normaly minimal

1

Revised in February 2011.

xii

Preface

singular fibers” in one-parameter families of Riemann surfaces. The former objects come from surface topology, while the latter objects come from complex analysis. The authors think interesting that numerical chorizo spaces lie at the common basis of the objects from two different disciplines. The appearance of pseudo-periodic homeomorphisms of negative twist in degenerating families of Riemann surfaces was clarified through the work of Imayoshi [26], Shiga–H. Tanigawa [58], and finally by Earle and Sipe [19]. We should mention here, however, that the pseudo-periodic nature of the monodromy had been observed for the Milnor fibering [44] at an isolated singular point of a complex hypersurface. Brieskorn [14] showed, in general dimensions, that the eigenvalues of the (co)homological monodromy are roots of unity. Lê [34] showed, in the case of curves, that the homological monodromy is periodic if the curve is irreducible at the singular point. A’Campo [1] proved that it is not the case if the curve is not irreducible. Also he showed that the geometric (i.e. topological) monodromy is not necessarily periodic, even if the curve is irreducible. A’Campo [2] and Eisenbud – Neumann [20] gave a description of geometric pseudo-periodic monodromies. Finally Lê – Michel – Weber [35] proved that the geometric monodromy is pseudo-periodic (“quasi-finie” in their terminology). Michel – Weber [43] gave a detailed description of the negative twist and showed that the geometric monodromy associated to a complex polynomial map from C2 to C (affine case) is also pseudoperiodic of negative twist. During the two decades, after the completion of our manuscript, several related papers have appeared. The most related one is, of course, the anouncement of this memoir, which was published in Bull. A.M.S. in 1994 [42]. This might serve as an introduction to this memoir (see also [40]). Pichon [55] used the pseudo-periodicity of the geometric monodromy to characterize the 3-manifolds that appear as the boundary manifolds of degenerating families of Riemann surfaces over a disk. In both of the papers of Pichon [55] and Lê – Michel – Weber [35], Waldhausen’s graph manifolds [66, 67] play an important role. The first authors that put the present memoir to good use were Ashikaga and Ishizaka [7] who gave a complete list of singular fibers in degenerating families of genus 3 (they were more than sixteen hundred!). They very explicitely exploited the algorithm, implicitly contained in the present memoir. It should be noted that the numerical classification of genus 3 singular fibers had been accomplished by Uematsu [64] in 1993 independently of our work. Xiao and Reid [56] proposed the problem of determining all the “atomic” singular fibers, which are defined as such singular fibers that cannot be “split” by any perturbation of the degenerating families. This problem is very interesting from the viewpoint of the present memoir. By our main result, the topological types of singular fibers are classified by the corresponding topological monodromies around them. Then a natural question to be settled would be if all atomic singular fibers (except for “multiple fibers”) correspond to the full .1/-Dehn twist about a certain simple closed curve. Examples of this geometrical situation are contained

Preface

xiii

in [3, 29, 38, 39]. (For recent related results, see [5, 6, 8].) Following these trend of ideas, S.Takamura [59, 60] is undertaking a project in solving this problem. The authors would like to thank Professors D. T. Lê, J.-P. Brasselet, and M. Oka who showed interest in our work, and especially Professor Lê for his explanation on the related results of his own and others. We are also grateful to Professor T. Ashikaga for taking our work seriously and for actively developing related subjects in algebraic geometry and topology, which encouraged us very much. Thanks are also due to Professor Y. Imayoshi for his interest in our results, and for his very benevolent review of our work [28]. In November of 2000 we met in Oberwolfach Professors A’Campo, Weber and Pichon, who encouraged us very strongly to publish our results. We owe to them the final impulse that we needed to conclude the typing of this memoir that has eventually lead to its publication. Tokyo and Madrid, September 2009

Yukio Matsumoto José Mar´ıa Montesinos-Amilibia

Acknowledgement The authors are very grateful to the referees for their careful reading, valuable comments and suggestions. The first named author has been supported by Grantin-Aid for Scientific Research (No. 20340014), J.S.P.S. The second named author has been supported by MEC, MTM2009-07030.

•

Contents

Part I

Conjugacy Classification of Pseudo-periodic Mapping Classes

1

Pseudo-periodic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Nielsen’s Results on Pseudo-periodic Maps . . . . . .. . . . . . . . . . . . . . . . . . . .

3 3 6

2

Standard Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Definitions and Main Theorem of Chap. 2 . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Periodic Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Non-amphidrome Annuli . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Amphidrome Annuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Proof of Theorem 2.1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

17 17 20 20 35 47

3

Generalized Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Definitions and Main Theorem of Chap. 3 . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Proof of Theorem 3.1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Quotient of .B 0 ; f jB 0 / . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Re-normalization of a Rotation . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Re-normalization of a Linear Twist . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6 Re-normalization of a Special Twist. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7 Completion of Theorem 3.1. (Existence) .. . . . . . . . .. . . . . . . . . . . . . . . . . . . .

53 53 60 61 61 71 83 91

4

Uniqueness of Minimal Quotient . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Main Theorem of Chap. 4 .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Structure of 1 .arch/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Structure of 1 .tail/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Structure of 1 .body/ . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Completion of the Proof of Theorem 4.1. (Uniqueness) . . . . . . . . . . . . . 4.6 General Definition of Minimal Quotient . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7 Conjugacy Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

93 93 97 103 107 113 118 119

xv

xvi

Contents

5

A Theorem in Elementary Number Theory . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 131 5.1 Proof of Theorem 5.1. (Uniqueness) . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 132 5.2 Proof of Theorem 5.1. (Existence).. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 135

6

Conjugacy Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Partition Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Weighted Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Completion of the Proof of Theorem 6.1 .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Weighted Cohomology .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Part II

145 149 151 162 164

The Topology of Degeneration of Riemann Surfaces

7

Topological Monodromy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Proof of Theorem 7.1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Construction of and fNi gsiD1 . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 The Decomposition F D A [ B . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Construction of a Monodromy Homeomorphism .. . . . . . . . . . . . . . . . . . . . 7.5 Negativity of Screw Numbers.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.6 Completion of the Proof of Theorem 7.1 .. . . . . . . . .. . . . . . . . . . . . . . . . . . . .

8

Blowing Down Is a Topological Operation . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 189

9

Singular Open-Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Completion of the Proof of Theorem 7.2 .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Characterization of the Triples .S; Y; c/ That Come from Pseudo-periodic Maps. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Completion of the Proof of Theorem 9.2 .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 Concluding Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

A

173 175 176 177 177 181 185

199 211 213 218 220

Periodic Maps Which Are Homotopic . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 221

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 233 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 237

Part I

•

Chapter 1

Pseudo-periodic Maps

1.1 Basic Definitions In this Chapter, we will review some basic results from Nielsen [51, 53]. We will begin with the definition of a pseudo-periodic map. Hereafter, all surfaces will be oriented, and all homeomorphisms between them will be orientationpreserving. Let ˙g be a closed connected surface of genus g 1. Definition 1.1 (Compare [53, Sect. 8]). A homeomorphism f W ˙g ! ˙g is a pseudo-periodic map if f is isotopic to a homeomorphism f 0 W ˙g ! ˙g which satisfies the following conditions: (i) there exists a disjoint union of simple closed curves C D C 1 [ C2 [ [ C r on ˙g such that f 0 .C / D C , and (ii) the restriction f 0 j ˙g C W ˙g C ! ˙g C of f 0 to the complement of C , is isotopic to a periodic map, i.e. a mapping of finite order. We call fCi griD1 a system of cut curves subordinate to f . It may be empty if f is isotopic to a periodic map. It is easy to see that a homeomorphism of a torus f W T2 ! T2

Y. Matsumoto and J.M. Montesinos-Amilibia, Pseudo-periodic Maps and Degeneration of Riemann Surfaces, Lecture Notes in Mathematics 2030, DOI 10.1007/978-3-642-22534-5 1, © Springer-Verlag Berlin Heidelberg 2011

3

4

1 Pseudo-periodic Maps

is pseudo-periodic if and only if jTrace .f /j 2, where f W H1 .T 2 I Z/ ! H1 .T 2 I Z/ is the induced homomorphism. In what follows, we will assume g 2 unless otherwise stated. Lemma 1.1. Let f W ˙g ! ˙g be a pseudo-periodic map. Then there exists a system of cut curves fCi griD1 subordinate to f such that (i) Ci does not bound a disk on ˙g ; i D 1; 2; : : : ; r; and (ii) Ci is not parallel to Cj if i ¤ j . Proof. Let fCj 0 gsj D1 be any system of cut curves subordinate to f . Isotoping f , we may assume 1 0 [ [ f @ Cj 0 A D Cj 0 : j

j

0

If a cut curve Cj bounds a disk, we simply remove Cj 0 . Suppose that Ci 0 and Cj 0 bounds an annulus A. Since g 2, A is uniquely determined by Ci 0 and Cj 0 . Let m be the smallest positive integer such that f m .A/ D A. We equivariantly isotop f on A [ f .A/ [ [ f m1 .A/ so that f m maps the “center line” CA of A onto itself. Then we omit Ci 0 and Cj 0 and also their images under the iteration of f , and add instead, CA together with its images. Two curves Ci 0 and Cj 0 are replaced by a curve CA . Proceeding in this way, we arrive at a system of cut curves fCi griD1 satisfying (i) and (ii) of the lemma. t u We call a system of cut curves which satisfies conditions (i) and (ii) of Lemma 1.1 an admissible system ([12, 22]). Let fCi griD1 be an admissible system of cut curves subordinate to a pseudoperiodic map f W ˙g ! ˙g : Then Nielsen [53, Sect. 12] introduced the “screw number” s.Ci / for each curve Ci in the system. The definition is as follows: We may assume ! [ [ Ci D Ci : f i

Let Cj be a fixed cut curve in the the system.

i

1.1 Basic Definitions

5

Let ˛ be the smallest positive integer ˛ such that ! ! f ˛. C j / D C j ! where C j denotes Cj with an orientation assigned, so f ˛ preserves Let b and b 0 be the connected components of ˙g C such that Cj belongs to their adherence in ˙g , where r [ C D Ci : i D1 0

0

It is possible that b D b . Let ˇ (resp. ˇ ) be the smallest positive integer such that 0 f ˇ .b/ D b (resp. f ˇ .b 0 / D b 0 ). Clearly ˛ is a common multiple of ˇ and ˇ 0 . Since f j ˙g C is isotopic to a periodic map, there exists a positive integer nb such that .f ˇ jb/nb Š idb . We choose the smallest number among such integers and denote it by nb again. Likewise we choose nb0 .> 0/ for b 0 . Let L be the least common multiple of nb ˇ and nb 0 ˇ0 . Then f L is isotopic to the identity on b and b 0 . Thus, on the union b [ Cj 0 [ b 0 ; f L is isotopic to the result of a number of full Dehn twists performed about Cj . Let e.2 Z/ be this number of full twists. We adopt the following convention for the sign of e: Convention (). The sign convention of e is depicted in Fig. 1.1. (Compare [13, p. 166], where the orientation of the surface is opposite to ours; and compare [46, Fig. 3 and p. 158]). Since the system fCi griD1 is admissible, neither b nor b 0 is an annulus or a disk. Nielsen [53] proved that the number e is well-defined. Definition 1.2 ([53, Sect. 12]). The rational number e˛=L is called the screw number of f about Cj and is denoted by s.Cj /. It measures the “amount of Dehntwist” performed by f ˛ about Cj . If s.Cj / D 0, we can equivariantly isotop f on b [ Cj [ b 0 and on its images under the iteration of f so that the result of the isotopy is a periodic map of b [ Cj [ b 0 : Thus we can omit Cj , together with its images under the iteration of f , keeping the remaining system admissible.

6


Fig. 1.1 Positive and negative Dehn twists

Iterating this omission, we arrive at a cut system in which all cut curves have non-zero screw number. Definition 1.3 (Cf. [12,22]). An admissible system of cut curves fCi griD1 is precise if s.Ci / ¤ 0 for each Ci . Nielsen did not use the terminology “precise system”; he talked, instead, about “division of the surface into complete kernels” ([53, Sect. 13]) meaning the decomposition of the surface by a precise system. Definition 1.4 ([53, Sect. 10]). An essential simple closed curve C on ˙g is said to be amphidrome with respect to a homeomorphism f W ˙g ! ˙g ! ! ! if there is an integer such that f . C / is freely homotopic to C , where C and ! C denote the same C with the opposite directions assigned.

1.2 Nielsen’s Results on Pseudo-periodic Maps The main result of Nielsen’s paper [53] is the following: Theorem 1.1 ([53, Sect. 15], cf. [22, p. 90]). Let f W ˙g ! ˙g

1.2 Nielsen’s Results on Pseudo-periodic Maps

and

7

f 0 W ˙g ! ˙g

be pseudo-periodic maps, and let fCi griD1 and fCj 0 gsj D1 be precise systems of cut curves subordinate to f and f 0 respectively. Suppose f is homotopic to f 0 . Then (i) there is a homeomorphism h W ˙g ! ˙g which is isotopic to the identity and such that h.C / D C 0 , where C D

r [

Ci

i D1

and C0 D

s [

Cj 0 :

i D1

In particular, r D s. (ii) For each cut curve Ci in fCi griD10 the number ˛ and the screw number s.Ci / are equal to the corresponding numbers for h.Ci /. (iii) For each cut curve Ci in fCi griD10 its character of being amphidrome or not (with respect to f ) coincides with the same character of h.Ci / (with respect to f 0 ). (iv) For each connected component b of ˙g C , the integers ˇ and nb are equal to the corresponding integers for h.b/. (v) For each connected component b of ˙g C , the periodic map of b which is isotopic to f ˇ jb is conjugate to the periodic map of h.b/ which is isotopic to .f 0 /ˇ jh.b/: (Here we are assuming f .C / D C and f 0 .C 0 / D C , which can be achieved by isotoping f and f 0 if necessary). Nielsen [53, Sect. 15] also asserted that conversely these invariants determine the conjugacy class of the mapping class to which f belongs. This, however, is not the case. In fact, the converse as formulated as Theorem 13.4 of [22] has counter examples. See Sect. 7, Examples, 3 and 4, of the present paper. If in the statement of Theorem 1.1 we add the following condition: (vi) the actions of f and f 0 on the respective partition graphs are conjugate (Here the partition graph of .˙g ; f / is a graph whose vertices and edges correspond to connected components of ˙g C and cut curves fCi griD10 , respectively. Similarly for .˙g ; f 0 /),

8


then Nielsen’s assertion is correct at least for pseudo-periodic maps of negative twists (see Sect. 4 for the definition). The conditions (i)–(vi) become sufficient. See Sect. 7 for details. The part of Nielsen’s result stated above is certainly correct, because he developed his theory so that it depends only on the homotopy class of f . For the construction of h in assertion (i), consult, for example, Casson’s lecture notes ([16, Lemma 2.4]). Note that by the above theorem we can speak of the precise system of cut curves subordinate to f , provided that we identify two isotopic systems up to indexing of the curves. As is obvious from assertion (v), we must study periodic maps as a prerequisite to Theorem 1.1. Nielsen [51] studied periodic maps of surfaces and established a complete set of conjugacy invariants, which we will now describe. Let ˙ be a compact surface with or without boundary, f W˙ !˙ a periodic map of order n > 1. Let p be a point of ˙. Then there is a positive integer m D m.p/ such that the points p; f .p/; : : : f m1 .p/ are distinct and f m .p/ D p. If m D n, the point p is called a simple point, while if 0 < m < n, p is called a multiple points. In particular, if m D 1, p is a fixed point. Since we are assuming that f is orientation-preserving, a multiple point is an isolated, interior point of ˙. For later use, we consider a more general situation than is needed here. Let ! ! ! ! C D C1 [ C2 [[ C s be a set of oriented and disjoint simple closed curves in a surface ˙, and let g be ! ! ! a map g W ˙ ! ˙ such that g. C / D C and gj C is periodic. We will define the ! ! notion of the valency of a curve C 2 C with respect to g. ! ! Let m be the smallest positive integer such that gm . C / D C . The restriction ! ! gm j C is a periodic map of C of order, say, > 0. Then m D n (D the order of ! ! gj C ). Let q be any point on C , and suppose that the images of q under the iteration ! of g m are situated on C in this order. fq; g m .q/; g 2 m .q/; : : : ; g .1/ m .q/g ! when viewed in the direction of C . Here, by Convention (), the integer satisfies 0 < and gcd.; / D 1, so D 0 iff D 1. Define an integer ı as follows:

1.2 Nielsen’s Results on Pseudo-periodic Maps

ı 1 .mod /;

9

0ı jz1 j jz2 j:

Here the " in Chap. 7 is replaced again by 1, and the two banks of A0 are denoted by D10 and D20 ( 0 jAı W Aı ! A0 is a pinched covering). If we identify these banks with fz j jzj < 1g through the parametrizations z 7! .z; 0/ and z 7! .0; z/, then we have . 0 jAı / f 0 D T1 . 0 jAı / on . 0 /1 .D10 / \ Aı and

. 0 jAı / f 0 D T2 . 0 jAı /

on . 0 /1 .D20 / \ Aı :

(See the proof of Theorem 7.4). Therefore, there exist homeomorphisms H j@Wp .0/ W @Wp .0/ ! Aı and H jNp W Np ! A0 satisfying 1. 0 .H j@Wp .0/ / D .H jNp / , 2. f 0 .H j@Wp .0/ / D .H j@Wp .0/ / f , 3. Ti .H jNp / D .H jNp / Ti ;

and

i D 1; 2.

Next, for each r with 0 r 1, let r0 W Aı ! Arı be the projection which satisfies the following conditions: .r/ .r/ If .z1 ; z2 / 2 Aı and r0 .z1 ; z2 / D .z1 ; z2 /, then .r/

.r/

(a) t.jz1 j; jz2 j/ D t.jz1 j; j z2 j/, .r/

.r/

(b) j.z1 /m2 .z2 /m1 j D rı, (c) arg.z1 / D

.r/ arg.z1 /,

and .r/

arg.z2 / D arg.z2 /.

9 Singular Open-Book

205

Note that these conditions uniquely determine the projection r0 W Aı ! Arı : If 0 < r 1,

r0 W Aı ! Arı

is a homeomorphism (in particular, 10 D identity of Aı ), and if r D 0, 00 W Aı ! A0 coincides with

0 jAı W Aı ! A0 :

It can be easily checked that . 0 jArı / r0 D . 0 jAı /: We set V D f.z1 ; z2 / 2 Int j j.z1 /m2 .z2 /m1 j ıg and define V .0 / by V .0 / D f.z1 ; z2 / 2 V j arg.z1 /m2 .z2 /m1 D 0 g [ A0 : Let us extend .H j@Wp .0/ / [ .H jNp / W @Wp .0/ [ Np ! Aı [ A0 to H jWp .0/ W Wp .0/ ! V .0/ by

H.Œ0; y; r/ WD r0 .H.Œ0; y; 1//;

where H.Œ0; y; 1/ D .H j@Wp .0/ /.y/. This H jWp .0/ is a homeomorphism: Wp .0/ ! V .0/: Let

.t; r/ W Œ0; 1 .0; 1 ! Œ0; 1 be a real-valued function defined by

.t; r/ D

8 ˆ ˆ 0, and if .z1 ; z2 / 2 A0 , set hQ .z1 ; z2 / D .z1 ; z2 /: It is not difficult to see that hQ is in fact a homeomorphism. We are now in a position to define the desired homeomorphism H W Wp ! V which extends H jWp .0/ W W p .0/ ! V .0/ and gives complex coordinates to Wp .D Wp @Wp /. The definition is as follows: H.Œ; y; r/ WD hQ H.Œ0; y; r/: To check the well-definedness of H , we will use a lemma, in which we identify each bank Di0 (i D 1 or 2) of A0 with D D fz j jzj < 1g, and let Ir W D ! D denote the map Ir .z/ D rz. Lemma 9.1. Let .z1 ; z2 / be a point of Arı , and put z D 0 .z1 ; z2 / 2 A0 D D10 [D20 . If z 2 Di0 (i D 1 or 2), then ( z Q h2 .z1 ; z2 / D Ir Ti Ir1 .z/ 0

if

r jzj 1;

if

0 jzj r:

Proof. Suppose z 2 D10 . Then t D t.jz1 j; jz2 j/ 1=2, and by the definition of 0 .z1 ; z2 / we can check jzj D 2t 1. We have hQ 2 .z1 ; z2 / D z e 0

2 .1 .t; r// m2


D

8 ˆ 0. Therefore, the identity ( ) implies in fact that H.Œ2; y; r/ D H.Œ0; fr .y/; r/ for r .0 < r 1/. This proves (ii). (iii) H.Œ; y; 0/ D H.Œ0; y; 0/. This is trivial because both sides are equal to .H jNp / .y/. (See the proof of (i)). This completes the proof of well-definedness of H W Wp ! V . The proof that H is a homeomorphism is left to the reader. By the definition of

W Wp ! C, .Œ; y; r/ D r e./:

On the other hand, if H.Œ; y; r/ D .z1 ; z2 /; then j.z1 /m2 .z2 /m1 j D rı because H.Œ; y; r/ 2 Arı , and also we have arg.z1 /m2 .z2 /m1 D because H.Œ; y; r/ D hQ H.Œ0; y; r/ 2 V ./. Thus if we define z01 D ı 0 z1 and z02 D ı 0 z2 , where ı 0 D .ı/1=.m1 Cm2 / , then with these coordinates .z01 ; z02 / on , we have the commutative diagram: H

Wp ! Int ? ? ? ?.z0 /m2 .z0 /m1 y y 1 2 C ! C: D


209

The complex coordinates .z01 ; z02 / pulled back to Wp by H are the desired ones. This completes the (somewhat lengthy) consideration of Case A in the proof of Theorem 9.1. Next, we consider Case B where p 2 S is a generic point of S . Suppose p is on an irreducible component 1 whose multiplicity is m1 1. Let U be a sufficiently small open disk-neighborhood of p in 1 . We may assume that U does not contain any node of S and intersects at most one closed nodal neighborhood Np0 in S . For a small positive real r0 > 0, define Wp Wp

.r0 /

.r0 /

. M / by

D fŒ; y; r j .y/ 2 U; r r0 g:

.r0 /

.r0 /

j r D r0 g/ consists of m1 copies The boundary @Wp .D fŒ; y; r 2 Wp of U , denoted by UQ 1 , UQ 2 , : : :, UQ m , and f W ˙g ! ˙g permutes them cyclically. .r0 /

If r0 is sufficiently small, Wp is disjoint from the “twisting region” of Wp0 and f m1 jUQ˛ W UQ ˛ ! UQ ˛ is the identity (˛ D 1; 2; : : : ; m). See Fig. 9.1. From this, it .r /

.r0 /

.r0 /

follows that Wp 0 .D Wp @Wp / is homeomorphic to an open 4-disk if r0 is small enough. In Case A we constructed complex coordinates .z01 ; z02 / in Wp0 such that jWp0 D 0 m2 0 m1 .z1 / .z2 / . (We assume that the bank of Np0 on the irreducible component 1 is given by z02 D 0.) Since 1 is a closed oriented surface, we can put a complex structure on 1 , and may assume z01 gives the complex coordinate on Wp0 \ 1 . Let z001 be a complex coordinate on U that is compatible with the complex structure of 1 . The coordinate change

Fig. 9.1 Construction of complex coordinates

210


z001 D z001 .z01 /

on

U \ Wp 0

is bi-holomorphic. .r / Let us define complex coordinates .z001 ; z002 / on Wp 0 as follows: .r / If q 2 .Wp 0 Wp0 /, then put .z001 .q/; z002 .q// D .z001 ..q//;

.q/1=m1 /;

.r /

and if q 2 Wp0 \ Wp 0 , then put .z001 .q/; z002 .q// D .z001 .z01 ; z02 /;

.q/1=m1 /;

where z001 .z01 ; z02 / stands for the composition q D .z01 ; z02 / 7! .z01 ; 0/ 7! z01 7! z001 .z01 /

( )

where the first arrow stands for ”projection” and the third arrow stands for “coordinate change”. Since the composition ( ) converges to the projection W q 7! .q/, as .q/ .2 Np0 \ 1 / approaches the boundary of the bank Np0 \ 1 , the above two .r / cases continuously match up to give complex coordinates .z001 ; z002 / on Wp 0 . The .r0 / 0 0 00 00 coordinate change between .z1 ; z2 / (on Wp0 ) and .z1 ; z2 / (on Wp ) is easily seen to be biholomorphic. The function jWp is given by D .z002 /m1 in terms of the coordinates .z001 ; z002 /. Case B is completed. In this way we have constructed complex coordinates on a neighborhood of each point of S . The coordinate changes among them are easily checked to be biholomorphic. Thus we have constructed a complex structure on an open neighborhood W of S in M with which jW W W ! C is holomorphic. Evidently, there is a (shrinking) embedding h W M ! W into W such that the diagram commutes (with a small real > 0): hW,!

M ! ? ? y

W ? ?. y

jW /

D ! D:

Pulling back the complex structure of W to M , we can make M a complex manifold of complex dimension two. The conditions (i), (ii) and (iii) in the statement of Theorem 9.1 are readily verified. This completes the proof of Theorem 9.1. u t

9.1 Completion of the Proof of Theorem 7.2

211

9.1 Completion of the Proof of Theorem 7.2 We will define a map

O W Pg ! SOg ;

which is to be the inverse to the monodromy correspondence O W SOg ! Pg : Let hf i be any element of Pg . By Theorem 3.1, f W ˙g ! ˙g is isotopic to an

f 0 W ˙g ! ˙g

in superstandard form. Let W ˙g ! S be its minimal quotient. By the singular open-book construction starting from f 0 and , we obtain a degenerating family .Mf ; Df ; f / whose monodromy coincides with f up to isotopy and conjugation (Theorem 9.1). This .Mf ; Df ; f / is normally minimal because W ˙g ! S is the minimal quotient (Corollary 8.1). Suppose hf1 i D hf2 i, i.e. f1 coincides with f2 up to isotopy and conjugation. Then by Theorem 4.2, there exist a homeomorphism h W ˙g ! ˙g and a numerical homeomorphism H W S Œf1 ! S Œf2 such that the diagram commutes: ˙g ? ? 1 y

h

! ˙g ? ? y 2

S Œf1 ! S Œf2 H

In other words, their minimal quotients 1 W ˙g ! S Œf1

212


and 2 W ˙g ! S Œf2 are topologically equivalent. If f10 and f20 are the corresponding superstandard forms, then by Theorem 4.2 (iii), f10 D h1 f20 h: Since the construction of a singular open-book made in this chapter is purely topological and has no topological ambiguity once the initial data f 0 W ˙g ! ˙g (in superstandard form) and W ˙g ! S are given, we conclude that the degenerating families .M1 ; D1 ; 1 / and .M2 ; D2 ; 2 / resulting from f10 and f20 respectively are topologically equivalent. Thus sending hf i 2 Pg to the topological equivalence class ŒMf ; Df ; f of .Mf ; Df ; f /, we obtain a well-defined map O W Pg ! SO g : Lemma 9.2. O O D id. This is clear because the topological monodromy of .Mf ; Df ; hf i (Theorem 9.1). To prove the bijectivity of , O it remains to show

f

/ is equal to

Lemma 9.3. O O D id. Proof. Let ŒM; D; be any element of SOg . By Theorem 7.4, the monodromy homeomorphism f W Fı ! Fı ; constructed in Chap. 7, is a pseudo-periodic map in superstandard form and there exist a pinched covering W Fı ! F0 which is a generalized quotient of f . This W Fı ! F0 is actually the minimal quotient because F0 is normally minimal (cf. Corollary 8.1.). By essentially the same argument as was done in the proof of Theorem 9.1 (especially in Case A), M is shown to have the structure of a singular open-book (minus the boundary) which is constructed starting from the data f W Fı ! Fı and W Fı ! F0 :

9.2 Characterization of the Triples .S; Y; c/ That Come from Pseudo-periodic Maps

213

Therefore, ŒM; D; coincides with ŒMf ; Df ;

f

D O .hf i/ D O .ŒM; O D;

This proves O O D id.

/: t u

As we remarked at the beginning of Chap. 9, the bijectivity of O W SOg ! Pg (just proved) implies the bijectivity of W Sg ! Pg . Thus Theorem 7.2 is completely proved. t u Recall the diagram b SO g ! Pg ? ? ? ?D ˇy y Sg ! Pg

from Chap. 8. Now we have proved that O and are bijective maps. This readily implies Proposition 9.1. The map ˇ W SOg ! Sg is bijective.

9.2 Characterization of the Triples .S; Y; c/ That Come from Pseudo-periodic Maps Let S be a connected numerical chorizo space which satisfies the minimality condition (Chap. 4); Y the decomposition diagram of S (Chap. 6), namely, a weighted graph whose vertices (resp. edges) are in one-to-one correspondence to the bodies (resp. arches) of S . Let c 2 HW1 .Y / be a class in the weighted cohomology group of Y (Chap. 6). In this paragraph, we will give a necessary and sufficient condition under which the triple .S; Y; c/ comes from a pseudo-periodic map f W ˙g ! ˙g of negative twist, of a surface of a given genus g 2. There are several necessary conditions for this that we have already proved: 1. Let 0 be any irreducible component of S , fp1 ; p2 ; : : : ; pk g the set of the intersection points among 0 and the other irreducible components. Let mi be the multiplicity of the irreducible component which intersects 0 at pi .i D 1; 2; : : : ; k/, and let m0 be the multiplicity of 0 . Then m0 divides m1 C m2 C C mk (see Proposition 3.1). 2. Let e be an edge of Y . Then the weight of e is equal to the gcd of the successive multiplicities on the arch corresponding to e (see Corollary 6.2. (i)). 3. Let v be a vertex of Y . Suppose the core part P0 of the body corresponding to v is contained in an irreducible component 0 . Let m0 be the multiplicity of 0 . Let m1 ; m2 ; : : : ; mk be the multiplicities that have the same meaning as in (1). If P0 has genus 0, then the weight of v is equal to gcd .m0 ; m1 ; : : : ; mk /. If P0 has genus 1, then the weight of v divides gcd .m0 ; m1 ; : : : ; mk / (see Corollary 6.2 (ii) and (iii).)

214


4. Let W0 be the gcd of the weights of all vertices of Y . Then the homomorphism c W H1 .Y W Z/ ! Z=W0 determined by c 2 HW1 .Y / is onto. (See Lemma 6.6.) 5. Let S D m1 1 Cm2 2 C Cms s be the expression of S as a (formal) divisor. Let Pi be a part of i (cf. Chap. 3). Then 2 2g D

s X

mi .Pi /

.< 0/

i D1

where .Pi / is the Euler characteristic of Pi . Theorem 9.2. Given a triple .S; Y; c/ satisfying the above conditions .1/ .5/, there exists a pseudo-periodic map f W ˙g ! ˙g of negative twist whose minimal quotient S Œf and its decomposition diagram are topologically equivalent to S and Y in such a way that the cohomology classes cŒf and c naturally correspond. More precisely, let f W S Œf ! Y Œf and W S ! Y denote the collapsing maps. Then there exist a numerical homeomorphism H W S Œf ! S and a weighted isomorphism W Y Œf ! Y such that the diagram f

S Œf ! Y Œf ? ? ? ? Hy y S

!

Y

commutes and .c/ D cŒf . The proof is indirect in the sense that it applies our results of Part II on degenerating families of Riemann surfaces of genus g. We will start with Winters’ existence theorem restated in our context. Theorem 9.3 (Winters [70, Corollary 4.3]). Let S D m1 1 C m2 2 C C ms s be a numerical chorizo space satisfying the above condition (1). Then there exist a complex surface M and a proper holomorphic map WM !D onto an open unit disk D such that

1

.0/ has normal crossings and such that

m1 1 C m2 2 C C ms s is the divisor expression of

1

.0/.


215

Outline of the proof. From topological viewpoint, Winters’ argument may be interpreted as follows: First one embeds i (considered as a Riemann surface) in a certain complex surface Ni . In case i has no self-intersection points, the description of Ni is easy; Ni is a holomorphic bundle over i whose fiber is biholomorphically homeomorphic to C and which admits a cross section (this is the embedded image of i ). The self-intersection number of i in Ni is 0 1 1 @X mj j i A ; mi j ¤i

where j i denotes the number of the nodes of S which are contained in i \j . The above number is an integer because of Condition (1). Also this condition assures the existence of such an Ni that admits a holomorphic map i W Ni ! C whose zero-set i1 .0/ (as a divisor) is expressed as mi i C

r.i / X

.i / .i /

mk ck ;

kD1 .i / where c1.i / , c2.i / , : : :, cr.i / are the fibers of the bundle Ni ! i over the intersection .i /

.i /

.i /

.i /

points, p1 , p2 , : : :, pr.i / , of i with the other irreducible components, mk .i /

being the multiplicity of the irreducible component which meets i at pk (k D 1; 2; : : : ; r.i /). See Fig. 9.2. In case i has self-intersection points, Ni is an immersed image of a holomorphic C-bundle NQ i over the normalized Q i (D i with the self-intersections separated). NQ i admits a cross section whose self-intersection number is equal to

(i)

C1

(i) C2 . . .

(i )

Cr (i)

Θi

Fig. 9.2 Other components intersect i transversely

Ni

216


Fig. 9.3 i has self-intersections

0 1 1 @X mj j i A 2Ki ; mi j ¤i

Ki denoting the number of the self-intersection points of i . The reason for the appearance of 2Ki is that each (transverse) self-intersection point of i contributes 2 to the self-intersection number i i in Ni . (See Fig. 9.3) Similarly to the case of a non-singular i , one can adjust the construction of the Ni so that it has a holomorphic map i W Ni ! C whose divisor i1 .0/ consists of .i / .i / .i / .i / .i / .i / mi i and m1 c1 C C mr.i /cr.i / , where c1 , : : :, cr.i / have the same meaning as before. At a self-intersection point of i , i is locally expressed as i

D .z1 z2 /mi

with appropiate coordinates .z1 ; z2 /. Now we have obtained the parts (one for each i ) to construct W M ! D. To get M , one has only to do “plumbing” with these parts N1 ; N2 ; : : : ; Ns ; i.e. glue Ni and Nj at each intersection point of i and j (in S ) so that the fiber and the base coordinates are interchanged at the time of gluing. At an intersection point of i and j , i

W Ni ! C

is locally expressed as i

D .z1 /mj .z2 /mi

with appropiate coordinates .z1 ; z2 /. Thus one can make the plumbing so that i ’s are patched together to give a holomorphic map W M ! C. We can easily modify


217

M and to obtain a surjective, proper and holomorphic map W M ! D whose fiber over 0, 1 .0/, coincides with the numerical chorizo space S (as a divisor). This completes the outline of the proof of Theorem 9.3. t u Addendum 1 In the above construction, we may assume that the number of the connected components of the general fiber of i W Ni ! C is equal to .i /

.i /

.i /

gcd .mi ; m1 ; m2 ; : : : ; mr.i / /; for each i D 1; 2; : : : ; s. N.B. The general fiber of

i

W Ni ! C is non-compact if r.i / 1.

Proof. Let 0 be any irreducible component of S ; m0 its multiplicity. Suppose 0 intersects the other irreducible components in r points p1 ; p2 ; : : : ; pr : Let m 1 ; m 2 ; : : : ; mr be the multiplicities of the irreducible components which intersect 0 in p1 ; p 2 ; : : : ; p r ; respectively. First suppose 0 has no self-intersection points, in which case, N0 can be constructed as follows. Take an open disk D0 on 0 having coordinate z1 and containing p1 ; p2 ; : : : ; pr : The values of the z1 -coordinate for p1 ; p2 ; : : : ; pr in D0 are supposed to be a1 ; a2 ; : : : ; ar : Let D1 be another open disk on 0 such that D1 D0 . Suppose C has coordinate z2 . The bundle N0 is constructed by holomorphically pasting D1 C and .0 D0 / C along .D1 D0 / C so that the holomorphic maps 0

D .z1 a1 /m1 .z1 a2 /m2 .z1 ar /mr .z2 /m0 W D1 C ! C

218

and


00

D .z2 /m0 W .0 D0 / C ! C

coincide on .D1 D0 / C. (The condition that m0 divides m 1 C m 2 C C mr assures that the fibers of

0

W D1 C ! C

make a trivial foliation in .D1 D0 / C, each leaf covering D1 D0 just once.) If N0 is so constructed, the general fiber of the resulting holomorphic map 0

.D

0

[

00

/ W N0 ! C

has gcd .m0 ; m1 ; : : : ; mr / number of components. (cf. Proposition 4.2.) This completes the proof when 0 has no self-intersection points. If it has selfintersection points, the proof is essentially the same. t u

9.3 Completion of the Proof of Theorem 9.2 Note that the general fiber of W M ! D constructed by Theorem 9.3 is not necessarily connected. But just as in the proof of Theorem 7.4, we can construct a pinched covering W Fı ! F0 D S; from the general fiber Fı D 1 .ı/ to the central fiber F0 D 1 .0/ (identified with S ), where ı .2 D/ is a small positive real. Let ARCH 0 be an arch of S . Then by the argument before Lemma 4.2, 1 .ARCH 0 / consists of m annuli, m being the gcd of the multiplicities of the successive irreducible components on ARCH 0 . By Condition (2), this number m is equal to the weight of the edge .ARCH 0 / of Y . Let BDY 0 be a body of S . Let P0 be the core part of BDY 0 , 0 the irreducible component containing P0 . Let m0 ; m 1 ; : : : ; m k have the same meaning as in Condition (3). By Addendum 1, we may assume that the number of the connected components of 1 .BDY 0 / is equal to gcd .m0 ; m1 ; : : : ; mk /: If P0 has genus 0, this number coincides with the weight of the vertex .BDY 0 / of Y . (Condition (3).) If P0 has genus 1, the weight of the vertex .BDY 0 / divides

9.3 Completion of the Proof of Theorem 9.2

219

gcd .m0 ; m1 ; : : : ; mk /: We will change M and so that the number of the connected components of 1 .BDY 0 / is equal to the weight, say w, of .BDY 0 /: We can take a pair of simple closed curves C1 and C2 on P0 cutting each other transversely once, because P0 has genus 1. Cut open N0 . M / along C1 C (which is the bundle over C1 induced from the C-bundle N0 ! 0 ) then reglue the two copies .C1 C/0 and .C1 C/1 by the following rotation of the fibers: p .z1 ; z2 / 7! .z1 ; exp.2w 1=m0 /z2 /; Since w divides m0 and is locally expressed as holomorphic map

.z1 ; z2 / 2 C1 C:

WM !D D .z2 /

m0

at a generic point of 0 , 0

naturally induces a

W M 0 ! D0

of the modified complex surface M 0 . (NB. M and M 0 are diffeomorphic.) It is easy to see that, in M 0 , 1 .BDY 0 / has w number of connected components. Proceeding in this way on each body of S , we may assume that the number of connected components of 1 .BDY / is equal to the weight of the vertex .BDY / of Y , for each . Number the connected components of 1 .BDY / cyclically for each BDY . Let ARCH 0 be any arch of S . Let BDY 0 and BDY 1 be two bodies which ARCH 0 connects (Possibly BDY 0 D BDY 1 ). Now a cochain c representing the cohomology class c 2 HW1 .Y / indicates which connected components of 1 .BDY 0 / should be joined to which ones of 1 .BDY 1 / by the annuli over ARCH 0 . (Cf. Lemma 6.5). We can realize this joining by modifying M and as follows: Let l be a simple arc on ARCH 0 joining the two end-circles of ARCH 0 , and let C1 be a simple closed curve on an irreducible component of ARCH 0 (or on an attaching bank) which cuts l transversely once. Then we do the same process as before of cutting open M along C1 C and re-glue the two copies .C1 C/0 and .C1 C/1 through appropiate rotation of the “fibers” C. Proceeding in this way on each arch in S , we will get the modified complex surface, again denoted by M , and a proper, surjective, and holomorphic map W M ! D having the desired aspect of the joints among the connected components of 1 .ARCH / and 1 .ARCH / indicated by the cochain c. The general fiber Fı is now connected because of Condition (4). (See Lemma 6.6). The fiber Fı has genus g because of Condition (5). Let f W Fı ! Fı be the monodromy homeomorphism associated to the degenerating family .M; D; /. Then f is a pseudo-periodic map of negative twist with the desired properties. This completes the proof of Theorem 9.2. t u

220


9.4 Concluding Remark Let .S; Y; c/ be a triple consisting of a compact, connected, and numerical chorizo space S satisfying the minimality condition; a weighted graph Y (with a collapsing map W S ! Y ); and a weighted cohomology class c 2 HW1 .Y /. Two such triples .S1 ; Y1 ; c1 / and .S2 ; Y2 ; c2 / are equivalent if there exist a numerical homeomorphism H W S1 ! S2 and a weighted isomorphism W Y1 ! Y2 such that the diagram 1 S1 ! Y1 ? ? ? ? Hy y S2 ! Y2 2

commutes and such that .c2 / D c1 . Let Tg denote the set of all equivalence classes of such triples .S; Y; c/ that satisfy Conditions (1)(5) stated before Theorem 9.2. By sending each pseudo-periodic map f W ˙g ! ˙g .g 2/ of negative twist to the triple .S Œf ; Y Œf ; cŒf /, defined in Chaps. 4 and 6, we obtain a map

W Pg ! Tg : Theorems 4.2 and 6.3 ensure well-definedness and injectivity of this map, and Theorem 9.2 ensures its surjectivity (thus bijectivity). Therefore, we have proved that every map in the following diagram is bijective (if g 2): ! SOg Tg

b & Î b ˇ Pg ! #

Tg % Î ! Sg Tg

Map ˇ: Theorem 8.1 and Proposition 9.1. Map b (with inverse b ): Lemmas 9.2 and 9.3. Map : Theorem 7.2. Map : Theorems 4.2, 6.3 and 9.2.

Appendix A

Periodic Maps Which Are Homotopic

The purpose of this appendix is to give a proof of Theorem 2.2. We will state the theorem again. Theorem A.1. Let f and f 0 be (orientation-preserving) periodic maps of a compact surface ˙ each component of which has negative Euler characteristic. Suppose f and f 0 W .˙; @˙/ ! .˙; @˙/ are homotopic as maps of pairs. Then there exists a homeomorphism h W ˙ ! ˙ isotopic to the identity, such that f D h1 f 0 h. Proof. First we consider the case when ˙ is connected. Note that the quotient space M D ˙=f is an orbifold with negative orbifold-Euler characteristic, (cf. [62]). Case 1. The underlying space jM j of M is not S 2 nor D 2 . Let DM D M [ M be the double of M . (If @M D ;, set DM D M .) We put a hyperbolic metric on DM so that DM admits a decomposition by a finite number of simple closed geodesics G 1 ; G 2 ; : : : ; Gl which satisfy the following conditions: 1. No Gi passes through a cone point (of course, this can never happen, because the cone angle is less than 2), 2. the boundary curves of M are closed geodesics, and are members of fG1 ; G2 ; : : : ; Gl g; 3. the intersections of G1 ; G2 ; : : : ; Gl are only double points, S 4. each component of DM li D1 Gi is an open cell whose closure is a polygon with more than 3 edges, and Y. Matsumoto and J.M. Montesinos-Amilibia, Pseudo-periodic Maps and Degeneration of Riemann Surfaces, Lecture Notes in Mathematics 2030, DOI 10.1007/978-3-642-22534-5, © Springer-Verlag Berlin Heidelberg 2011

221

222

A Periodic Maps Which Are Homotopic

Fig. A.1 Hyperbolic parts (“annulus with a cone” and “pants”)

Fig. A.2 Decomposing the double DM by simple closed geodesics

S 5. each component of DM li D1 Gi contains at most one cone point. Such a decomposition is certainly possible. In fact, one can start with two kinds of hyperbolic parts; annuli with one cone point and pants, both having geodesic boundaries of length 1. See Fig. A.1. Glue them along the boundaries and construct a hyperbolic orbifold homeomorphic to DM . Add further simple closed geodesics to get a desired decomposition. See Fig. A.2. Let D˙ D ˙ [ ˙ denote the double of ˙. The periodic map f W ˙ ! ˙ symmetrically extends to a periodic map D˙ ! D˙, which will be denoted by f again. Lift the metric on DM just constructed, to D˙ to make the latter a (smooth) hyperbolic surface. The periodic map f W D˙ ! D˙ preserves this metric . Let 1 ; 2 ; : : : ; m be the lifts to D˙ of the simple closed geodesics G1 ; G2 ; : : : ; Gl : The total number m of the i ’s might be different from the total number l of the Gi0 s. The simple closed geodesics 1 ; 2 ; : : : ; m


223

satisfy the following conditions: (i) No i passes a multiple point of f , (ii) the boundary curves of ˙ are members of f1 ; 2 ; : : : ; m g; (iii) the intersections of 1 ; 2 ; : : : ; m are only double points, S (iv) each component of D˙ m i Di i is an open cell whose closure is a polygon with more than 3 edges, and S (v) each component of D˙ m i D1 i contains at most one cone point. Moreover, we have the following: (vi) No pair i , j .i ¤ j / are freely homotopic (because they are distinct closed geodesics. Cf. [16].) (vii) i and j .i ¤ j / have minimal intersection. Cf. [16]. (viii) f preserves the configuration 1 [ 2 [ [ m I f.

m [

i / D

i D1

m [

i

i D1

due to the construction of 1 ; 2 ; : : : ; m : So far we have only considered the periodic map f . Now we consider the other f 0 W ˙ ! ˙: Extend f 0 symmetrically to the double D˙ and denote the resulting periodic map by f 0 W D˙ ! D˙ also. Put a hyperbolic metric 0 on D˙ which is invariant under f 0 and such that the boundary curves of ˙ are closed geodesics. The simple closed curves 1 ; 2 ; : : : ; m are no longer geodesics with respect to 0 , in general. But they are freely homotopic to simple closed geodesics 10 ; 20 ; : : : ; m0 with respect to 0 . [16, Lemma 2.3.] These curves are distinct thanks to property (vi) of the i ’s. We will construct an isotopy g W D˙ ! D˙;

0 1;

224


such that (a) g0 D idD˙ , (b) if i D i0 (for instance, a boundary curve of ˙), then g .i / D i0 , and (c) g1 .i / D i0 , i D 1; 2; : : : ; m. The construction is only a mimic of Lemmmas 2.4, 2.5 of Casson’s lecture notes, [16], where the case m D 2 is treated, and is done essentially by an innermost arc argument. We proceed by induction. Suppose i D i0 .i D 1; 2; : : : ; k/ for some k < m. We will find an isotopy which starts with the identity, setwise preserves 0 i .D i0 / for i D 1; 2; : : : ; k, and at the final stage sends kC1 to kC1 . (It will be helpful to consider 10 ; 20 ; : : : ; m0 as patterns drawn on a floor, that remain fixed, and 1 ; 2 ; : : : ; m 0 as loops laid on it, that can move.) First we isotop kC1 to separate it from kC1 , 0 0 and then using the annulus between kC1 and kC1 move kC1 onto kC1 , as we will sketch now. A typical move is performed through the shaded disk in Fig. A.3. The geodesics 10 ; : : : ; k0 .1 ; : : : ; k / 0 (cf. [16, Lemma 2.5]). Also they have minimal intersection with the geodesic kC1 have minimal intersection with kC1 because of property (vii) of i ’s. Thus as in Casson’s Lemma 2.5, i0 \ .the shaded disk/ is a family of arcs passing “through” the disk from “top to bottom”, for i D 1; 2; : : : ; k. But if there were a situation as shown in Fig. A.4, we would have an obstacle; we could not move kC1 “along” 10 , : : :, k0 . This “bad” situation is, however, prohibited by property (iv) of the i ’s. Therefore, we have the situation of Fig. A.3, and can move kC1 through the shaded disk “along” 10 ; 20 ; : : : ; k0 .D 1 ; 2 ; : : : ; k /:

This completes the inductive step, and we have obtained an isotopy g W D˙ ! D˙;

0 1

satisfying conditions (a), (b), (c) stated above. Γ k+1

Fig. A.3 A move of kC1

Γ′k+1

Γ′1 ,

...,

Γ′k

A Periodic Maps Which Are Homotopic Fig. A.4 A situation in which the move of kC1 is obstructed

225 Γ k+1

Γ′k+1

Γ′1 , . . . ,

Γ′k+1 (= Γ 1 , . . . , Γ k )

Let g W D˙ ! D˙ be the final stage g1 of the isotopy. Then g.1 [ 2 [ [ m / D 10 [ 20 [ [ m0 : Lemma A.1. f 0 .10 [ 20 [ [ m0 / D 10 [ 20 [ [ m0 . Proof. First note that f 0 .i0 / ' f 0 .i /

.because i0 ' i /

' f .i / .because f 0 ' f / D j

.for some j , because of property (viii) of the i ´s/

' j0

.because j ' j0 /;

where “'” denotes “is freely homotopic to”. The metric 0 on D˙ is invariant under f 0 , so f 0 .i0 / is a simple closed geodesic as well as i ’s. In a free homotopy class of simple closed curves, there is only one simple closed geodesic, (cf. [16, Lemma 2.4]). Therefore, f 0 .i0 / D j0 . t u Let H2 be the hyperbolic plane which is the universal covering of .D˙; 0 /. Let D D H [ S1 be its compactification to the unit disk. Let gQ W H2 ! H2 ;

0 1

be the lifted isotopy of g starting from gQ 0 D idH2 . Then gQ WD gQ 1 W H2 ! H2 is a lift of g W D˙ ! D˙: By Nielsen [52], the isotopy gQ W H2 ! H2

226


extends to an isotopy .gQ /O W D ! D; and being a lift of an isotopy of a compact surface D˙, the restriction of .gQ /O to S1 is constant because gQ varies equivariantly with respect to the group of superpositions of the covering H2 ! D˙, and the variation is determined by the restriction of gQ to a compact set (because D˙ is compact). Since the euclidean size of a fundamental domain for D˙ tends to zero if it goes to infinity, then gQ at S1 does not depend on . Thus .gQ /OjS1 D id jS1 . Let fQ W H2 ! H2 be a lift of f W D˙ ! D˙; and

.fQ/O W D ! D

its extention. Since

f ' f 0 W D˙ ! D˙;

fQ is homotopic, through a lifted homotopy, to a lift fQ0 of f 0 . For the same reason as above, we have .fQ/OjS1 D .fQ0 /OjS1 . Now let us regard 1 [ 2 [ [ m and

10 [ 20 [ [ m0

as finite graphs and 0 , respectively, drawn on D˙. They decompose D˙ into finite cell complexes. Consider their lifts Q , Q 0 in H2 . Q consists of lifts of the i ’s, and Q 0 of lifts of the i ’s. (Each lift of i0 is a geodesic line.) Q and Q 0 decompose H2 into locally finite cell complexes. gQ W H2 ! H2 preserves these patterns: g. Q Q / D Q 0 . .0/ Let Q and Q 0.0/ be the set of vertices of Q and Q 0 , respectively. Lemma A.2. gQ j Q .0/ W Q .0/ ! Q 0.0/ is equivariant with respect to fQ j Q .0/ and fQ0 j Q 0.0/ . Proof. Take a vertex vQ 2 Q .0/ . vQ is an intersection point of two infinite curves Qi and Qj , which are lifts of i and j . By property (iii) of the i ’s, the pair fQi ; Qj g is uniquely determined by vQ . We have gQ fQ.Qi / D fQ0 g. Q Qi /

and

gQ fQ.Qj / D fQ0 g. Q Qj /:

(Proof. Let inf.Qi / denote the pair of the two “infinite” points of Qi in S1 . Then Q O.fQ/O.inf.Qi // D .fQ0 / O.g/O.inf. Q Qi // D inf.fQ0 g. Q Qi // inf.gQ fQ.Qi // D .g/


227

because .g/O Q j S1 D id and .fQ/O j S1 D .fQ0 /O j S1: Q Qi / are geodesic lines in H2 with the same pair of infinite points. gQ fQ.Qi / and fQ0 g. Then they coincide. Similarly, gQ fQ.Qj / D fQ0 g. Q Qj /.) Therefore, fgQ fQ.Qv/g D gQ fQ.Qi / \ gQ fQ.Qj / D fQ0 g. Q Qi / \ fQ0 g. Q Qj / D ffQ0 g.Q Q v/g; Q v/. which proves gQ fQ.Qv/ D fQ0 g.Q

t u

Remember that the Qi ’s are geodesic lines with respect to the lifted metric Q of . Using Lemma A.2, we can find an isotopy gQ .1/ W H2 ! H2 ;

0 1;

.1/ .1/ .1/ such that gQ 0 D g, Q gQ .Q / D Q 0 , gQ is equivariant with respect to the group of .1/ covering translations of H2 ! D˙, and the final stage gQ 1 is “linear” from each edge of Q to an edge of Q 0 with respect to Q and Q 0 (the lifted metric of 0 ). Then .1/ gQ 1 jQ W Q ! Q 0

is equivariant with respect to fQjQ and fQ0 jQ 0 . .1/ Finally by the Alexander trick, we can deform gQ 1 within each cell and obtain an isotopy gQ .2/ W H2 ! H2 ; 0 1; .2/ .1/ .2/ .1/ .2/ such that gQ 0 D gQ 1 , gQ jQ D gQ jQ , gQ is equivariant with respect to the group of covering translations of H2 ! D˙;

and the final stage .2/

gQ 1 W H2 ! H2 is equivariant with respect to fQ and fQ0 . This homeomorphism gQ 1 projects down to a homeomorphism h W D˙ ! D˙: .2/

The restricted homeomorphism hj˙ W ˙ ! ˙ is the one whose existence is asserted by Theorem 2.2. This completes the proof of Case 1.

228


Case 2. The underlying space jM j of M is S 2 . The proof will be accompanied by several lemmas. Lemma A.3. The orders of f and f 0 W ˙ ! ˙ are equal. Proof. Let L.g/ denote the Lefschetz number of a homeomorphism g W ˙ ! ˙. Then the order of f is equal to the smallest positive integer n such that L.f n / < 0 because if f n ¤ id˙ , L.f n / is equal to the number of the fixed points of f n (see [23, pp. 130, 121]) which is non-negative, while L.id˙ / D .˙/ < 0. Thus our assumption f ' f 0 implies that their orders coincide. t u Lemma A.4. There is a bijective correspondence between the set of multiple points of f and the same set of f 0 which preserves the valencies. Proof. We impose a hyperbolic metric on ˙ and identify the universal covering ˙Q with H2 , which is compactified to D D H2 [ S1 as before. Let F .f / denote the set of fixed points of f . Take a point p0 2 F .f / and lift it to pQ0 2 H2 . Let fQ W H2 ! H2 be a lift of f which fixes pQ0 . By Nielsen [50, Sect. 2], fQ extends to a homeomorphism .fQ/O W D ! D: The lift

fQ W H2 ! H2

is a periodic map because it is a lift of a periodic map and it has a fixed point. Then .fQ/O W D ! D is also a periodic map. By [31], an orientation-preserving periodic map of a 2-disk D is conjugate to a rotation. In particular, .fQ/OjS1 has no fixed points. By our assumption f ' f 0 W ˙ ! ˙; fQ is homotopic, through a lifted homotopy, to a lift fQ0 of f 0 . Then by the same argument as in Case 1, .fQ/OjS1 D .fQ0 /OjS1 . By Brouwer’s fixed point theorem, .fQ0 /O W D ! D has a fixed point. But .fQ0 /jS1 .D .fQ/OjS1 / has no fixed points, so the fixed point of fQ0 is in H2 . Then the same argument as for fQ can apply to fQ0 , and .fQ0 /O W D ! D is conjugate to a rotation. Thus the fixed point pQ00 of fQ0 is uniquely determined in H2 , which projects to a fixed point p00 of f 0 . The correspondence


229

' W F .f / ! F .f 0 / is defined by sending p0 to p00 . ' is independent of the choice of the lift pQ0 , but it might depend on the homotopy between f and f 0 . We will fix the homotopy throughout the argument. Clearly ' is bijective because the roles of f and f 0 are symmetric. The valency of p0 (resp. p00 ) with respect to f (resp. f 0 ) is the same as the valency of pQ0 (resp. pQ00 ) with respect to fQ (resp. fQ0 ), which can be read off from the action of .fQ/O (resp. .fQ0 /O) on S1 . But .fQ/OjS1 D .fQ0 /OjS1 . Thus the valency of p0 is equal to the valency of p00 D '.p0 /. Similarly for each factor m of the order of f we can construct a bijective correspondence between F .f m / and F ..f 0 /m /. This correspondence preserves the valencies exactly as above. This completes the proof of Lemma A.4. t u Lemmmas A.3, A.4 and Nielsen’s theorem (Theorem 1.2) imply that f W ˙ ! ˙ and f 0 W ˙ ! ˙ are conjugate. (Remember that we are considering a closed ˙ in Case 2.) In particular, we have Lemma A.5. M D ˙=f and M 0 D ˙ 0 =f 0 are homeomorphic as orbifolds. Let p 1 ; p2 ; : : : ; p s ;

s 3;

be the cone points of M with valencies .m1 ; 1 ; 1 /; .m2 ; 2 ; 2 /; : : : ; .ms ; s ; s /; respectively. Consider a polygon P (s-gon) in H2 whose angles are =1 ; =2 ; : : : ; =s : See Fig. A.5. (Such a P exists because orb .M / < 0.) M can be considered as a hyperbolic “bi-hedron” having two faces, each congruent with P .

Fig. A.5 A hyperbolic polygon

230


Let be the hyperbolic metric on ˙ obtained by lifting the bi-hedron metric on M through the projection ˙ ! M . We lift this metric farther to the universal covering ˙Q to make it a hyperbolic plane H2 ./. The plane H2 ./ is tessellated by tiles, each congruent with P . The fundamental region for the action of 1orb .M / ([62, Sect. 13]) is F D P [ P , where P is a flipped P having an edge in common with it. Let ri be the rotation (in H2 ./) of angle 2=i centered at vertex pi .2 P /, i D 1; 2; : : : ; s. It is known that the rotations r1 , r2 , : : :, rs generate the group 1orb .M /, the orientation-preserving automorphism group of the tessellation, (cf. Milnor [45] or [41]). Moreover, from our construction, ri is a lift of f i mi W ˙ ! ˙;

i D 1; 2; : : : ; s:

Let f W ˙ ! ˙;

0 1;

0

be a homotopy between f and f W f0 D f , f1 D f 0 . Then, through a lifted homotpy .fi mi /Q W H2 ./ ! H2 ./; ri is homotopic to a homeomorphism ri0 W H2 ./ ! H2 ./: This ri0 is a lift of .f 0 /i mi , satisfies .ri0 /OjS1 D .ri /OjS1 , and is topologically equivalent to a rotation. (See Proof of Lemma A.4.) Let pi0 be the center of the “rotation” ri0 . We denote the cone point of M 0 to which pi0 projects by the same notation pi0 . Thus we have obtained the following correspondence: M 3 pi

! pi0 2 M 0 ;

i D 1; 2; : : : ; s:

By Lemma A.4, this correspondence preserves the valencies. We impose a 0 structure of a hyperbolic bi-hedron on M 0 whose faces P 0 , P are congruent with P , P preserving the above correspondence of the vertices. Let 0 be the hyperbolic metric on ˙ obtained by lifting the bi-hedron metric on M 0 through the projection ˙ ! M 0 . We lift 0 to ˙Q to make it a hyperbolic plane H2 . 0 /, in which the polygon P 0 is inscribed with vertices p10 ; p20 ; : : : ; ps0 : In H2 ./, the topological rotation ri0 is a genuine rotation with center pi0 of angle 2=i , i D 1, 2, : : :, s. The plane H2 . 0 / is tessellated by tiles, each congruent with P 0 . The rotations r10 , r20 , : : :, rs0 generate the group 1orb .M 0 /, the orientationpreserving automorphism group of the tessellation. Since P 0 is congruent with P , the group 1orb .M 0 / is isomorphic to 1orb .M / via the correspondence


ri

! ri0 ;

231

i D 1; 2; : : : ; s:

Note that H2 ./ and H2 . 0 / are merely different “pictures” on the same space ˙Q , Q the group so 1orb .M / and 1orb .M 0 / are considered as subgroups of Homeo.˙/, Q Q of all self-homeomorphisms of ˙ . If we fix the action of 1 .˙/ on ˙ , then 1 .˙/ Q contained in orb .M / \ orb .M 0 /. Moreover, is also a subgroup of Homeo.˙/, 1 1 the action of 1 .˙/ is isometric, with respect to H2 ./ and at the same time with respect to H2 . 0 /. Lemma A.6. The isomorphism between 1orb .M / and 1orb .M 0 / given by the correspondence ri ! ri0 .i D 1; 2; : : : ; s/ restricts to the identity on 1 .˙/. Proof. Take an element g 2 1 .˙/. Since 1 .˙/ < 1orb .M 0 /, g can be written as a product of r10 ; r20 ; : : : ; rs0 : gD

.r10 ; r20 ; : : : ; rs0 /:

gD

.r1 ; r2 ; : : : ; rs /:

We will show that For this, compactify H ./ to D D H2 ./ [ S1 as before. Then 2

gjS O 1D Since both g and

.r10 ; r20 ; : : : ; rs0 /OjS1 D

.r1 ; r2 ; : : : ; rs /OjS1 :

.r1 ; r2 ; : : : ; rs / are isometries of H2 ./, we have gD

.r1 ; r2 ; : : : ; rs /: t u

Let us construct a homeomorphism hQ W ˙Q ! ˙Q which is equivariant with respect to the actions of 1orb .M / and 1orb .M 0 /. The construction is obvious. First, map the fundamental region F D P [ P “isometrically” to the fundamental region 0 F 0 D P [ P . Then extend it equivariantly to the whole space. By Lemma A.6, we have hQ g D g hQ for all g 2 1 .˙/. Thus hQ projects to a homeomorphism h W ˙ ! ˙. Lemma A.7. h W ˙ ! ˙ satisfies f D h1 f 0 h. Proof. Let fQ W ˙Q ! ˙ be a lift of f W ˙ ! ˙. Since fQ preserves the tessellation, fQ is written as a product of r1 , r2 , : : :, rs : fQ D '.r1 ; r2 ; : : : ; rs /: Then

'.r10 ; r20 ; : : : ; rs0 /

232


is a lift of f 0 ; denote it by fQ0 . Since hQ is equivariant with respect to the actions of 1orb .M / and 1orb .M 0 /, we have Q hQ fQ D hQ '.r1 ; r2 ; : : : ; rs / D '.r10 ; r20 ; : : : ; rs0 / hQ D fQ0 h; so h f D f 0 h as asserted.

t u

Lemma A.8. h W ˙ ! ˙ is isotopic to the identity. Q Proof. It will be sufficient to prove .h/OjS 1 D id:, because then h W ˙ ! ˙ preserves the free homotopy class of every simple closed curve. Let g be any element of 1 .˙/ different from 1. Then g W H2 ! H2 is a hyperbolic transformation, and for any point x 2 H2 , gn .x/ converges (in D) to a definite point Vg 2 S1 as n ! C1. Also g n .x/ converges to another definite point Ug 2 S1 as n ! 1. Nielsen [50, Sect. 1] calles Ug , Vg the negative and the positive fundamental points of g. Then, in D, we have n n Q Q Q n Q .h/O.V g / D .h/O. lim g .x// D lim .h/.g .x// D lim g .h/.x/ D Vg : n!1

n!1

n!1

Q Also .h/O.U g / D Ug . But the set of fundamental points fUg j g 2 1 .˙/g [ fVg j g 2 1 .˙/g Q is dense is S1 ([53, Sect. 1 Case b)]). This proves .h/OjS 1 D id .

t u

The proof is completed in Case 2. Case 3. The underlying space jM j of M is D 2 . Making the double DM , the proof is reduced to Case 2. Now we have completed the proof in the case when ˙ is connected. In the general case when ˙ is not necessarily connected, divide the set of the components of ˙ into cycles under the permutation caused by f . Since f 0 is homotopic to f , f 0 causes the same permutation. Then in each cycle we can argue just as in the proof of Theorem 2.3(ii). (Beware that we need here the “homotopy implies isotopy” theorem, [10, 21].) This completes the proof of Theorem A.1. u t

References

1. A’Campo, N.: Sur la monodromie des singularités isolées d’hypersurfaces complexes. Invent. Math. 20, 147–169 (1973) 2. A’Campo, N.: La fonction zêta d’une monodromie. Comment. Math. Helvetici 50, 233–248 (1975) 3. Arakawa, T., Ashikaga, T.: Local splitting families of hyperelliptic pencils I. Tohoku Math. J. 53, 369–394 (2001): II. Nagoya Math. J. 175, 103–124 (2004) 4. Asada, M., Matsumoto, M., Oda, T.: Local monodromy on the fundamental groups of algebraic curves along a degenerate stable curve. J. Pure Appl. Algebra. 103, 235–283 (1995) 5. Ashikaga, T.: Local signature defect of fibered complex surfaces via monodromy and stable reduction. Comment. Math. Helv. 85, 417–461 (2010) 6. Ashikaga, T., Konno, K.: Global and local properties of pencils of algebraic curves. In: Usui, S., et al. (eds.) Algebraic Geometry 2000 Azumino Adv. Studies in Pure Math. 36, 1–49 (2002) 7. Ashikaga, T., Ishizaka, M.: Classification of degenerations of curves of genus three via Matsumoto-Montesinos’ theorem. Tohoku Math. J. 54, 195–226 (2002) 8. Ashikaga, T., Endo, H.: Various aspects of degenerate families of Riemann surfaces, Sugaku Expositions 19, Amer. Math. Soc. 171–196 (2006) 9. Ashikaga, T., Ishizaka, M.: A geometric proof of the reciprocity law of Dedekind sum. Unpublished note (2009) 10. Bear, R.: Isotopie von Kurven auf orientierbaren, geschlossenen Flächen und ihr Zusammenhang mit der topologischen Deformation der Flächen. J. Reine Angew. Math. 159, 101–111 (1928) 11. Bers, L.: Space of degenerating Riemann surfaces. Annals of Mathematics Studies, vol. 79, pp. 43–55. Princeton U.P., Princeton, New Jersey (1975) 12. Bers, L.: An extremal problem for quasiconformal mappings and a theorem of Thurston. Acta Math. 141, 73–98 (1978) 13. Birman, J.S.: Braids, Links, and Mapping Class Groups. Annals of Mathematics Studies, vol. 82. Princeton U.P., Princeton, New Jersey (1974) 14. Brieskorn, E.: Die Monodromie der isolierten Singularitäten von Hyperflächen. Manuscritpta math. 2, 103–161 (1970) 15. Burde, G., Zieschang, H.: Knots, de Gruyter Studies in Mathematics, vol. 5. Walter de Gruyter, Berlin, New York (1985) 16. Casson, A.J., Bleiler, S.A.: Automorphisms of Surfaces after Nielsen and Thurston. Cambridge U.P., Cambridge, New York, New Rochelle, Melbourne, Sydney (1988) 17. Clements, C.H.: Picard-Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities. Trans. Amer. Math. Soc. 136, 93–108 (1969) 18. Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. Math. I.H.E.S., 36 , 75–110 (1969) Y. Matsumoto and J.M. Montesinos-Amilibia, Pseudo-periodic Maps and Degeneration of Riemann Surfaces, Lecture Notes in Mathematics 2030, DOI 10.1007/978-3-642-22534-5, © Springer-Verlag Berlin Heidelberg 2011

233

234

References

19. Earle, C.J., Sipe, P.L.: Families of Riemann surfaces over the punctured disk. Pacific J. Math. 150, 79–86 (1991) 20. Eisenbud, D., Neumann, W.: Three-dimensional link theory and invariants of plane curve singularities. Annals of Mathematics Studies, vol. 110. Princeton U.P. Princeton, NJ (1985) 21. Epstein, D.B.A.: Curves on 2-manifolds and isotopies. Acta Math. 115, 83–107 (1966) 22. Gilman, J.: On the Nielsen type and the classification for the mapping class group. Adv. Math. 40, 68–96 (1981) 23. Guillemin, V., Pollack, A.: Differential Topology. Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1974) 24. Handel, M., Thurston, W.: New Proofs of Some results of Nielsen. Adv. Math. 56, 173–191 (1985) 25. Holzapfel, R.-P.: Chern number relations for locally abelian Galois coverings of algebraic surfaces. Math. Nachr. 138, 263–292 (1988) 26. Imayoshi, Y.: Holomorphic families of Riemann surfaces and Teichmüller spaces. In: Riemann Surfaces and Related Topics. Annals of Mathematics Studies, vol. 97, pp. 277–300 (1981) 27. Imayoshi, Y.: A construction of holomorphic families of Riemann surfacs over the punctured disk with given monodromy. In: Papadopoulos, A. (ed.) A Handbook of Teichmüller Theory, vol. II, pp. 93–130. European Mathematical Society, Geneva (2009) 28. Imayoshi, Y.: Math. Rev. MR1217354(94h:30057) 29. Ito, T.: Splitting of singular fibers in certain holomorphic fibrations. J. Math. Sci. Univ. Tokyo 9, 425–486 (2002) 30. Kerckhoff, S.P.: The Nielsen realization problem. Ann. Math. 117(2), 235–265 (1983) 31. de Kerekjarto, B.: Sur les groupes compacts de transformations topologiques des surfaces. Acta Math. 74, 129–173 (1941) 32. Kodaira, K.: On compact analytic surfaces II. Ann. Math. 77, 563–626 (1963) 33. Laufer, H.B.: Normal two-dimensional singularities. Annals of Mathematical Studies, vol. 71. Princeton, U.P., Princeton, New Jersey (1971) 34. Lê D.T.: Sur les noeuds algébriques. Compositio Math. 25, 281–321 (1972) 35. Lê D.T., Michel F., Weber C.: Courbes polaires et topologie des courbes planes. Ann. Sci. ´ Ecole Norm. Sup. 24(4), 141–169 (1991) 36. Magnus, W., Karras, A., Solitar, D.: Combinatorial Group Theory, 2-nd edn. Dover Publication, Inc., New York (1976) 37. Matsumoto, Y.: Good torus fibrations. Contemp. Math. 35, 375–397 (1984) 38. Matsumoto, Y.: Lefschez fibrations of genus two – A topological approach –. In: Kojima, S., et al. (eds.) Topology and Teichmüller spaces, Proceedings of the 37-th Taniguchi Symposium held in Finland, pp. 123–148. World Scientific (1996) 39. Matsumoto, Y.: Splitting of certain singular fibers of genus two. Bolet´ın de la Soc. Mat. Mexicana, Special issue, 10(3), 331–355 (2004) 40. Matsumoto, Y.: Topology of degeneration of Riemann surfaces. In: Brasselet, J.-P., et al. (ed.) Singularities in Geometry and Topology, Proceedings of the Trieste Singularity Summer School and Workshop, pp. 388–393. World Scientific (2007) 41. Matsumoto, Y., Montesinos-Amilibia, J.M.: A proof of Thurston’s uniformization theorem of geometric orbifolds. Tokyo J. Math. 14, 181–196 (1991) 42. Matsumoto, Y., Montesinos-Amilibia, J.M.: Pseudo-periodic homeomorphisms and degeneration of Riemann surfaces. Bull. Am. Math. Soc. (N.S.) 30, 70–75 (1994) 43. Michel, F., Weber, C.: On the monodromies of a polynomial map from C2 to C. Topology 40, 1217–1240 (2001) 44. Milnor, J.: Singular points of complex hypersurfaces. Ann. Math. Studies 61 (1968) 45. Milnor, J.: On the 3-dimensional Brieskorn manifolds M.p; q; r/. In: Neuwirth, L.P. (ed.) Knots, Groups, and 3-Manifolds, Annals of Mathematical Studies, vol. 84, pp. 175–225. Princeton, U.P., Princeton, New Jersey (1975), Papers Dedicated to the Memory of R.H. Fox 46. Montesinos-Amilibia, J.M.: Lectures on 3-fold simple coverings and 3-manifolds. Contemp. Math., 44, 157–177 (1985) 47. Myerson, G.: On semi-regular finite continued fractions. Arch. Math. 48, 420–425 (1987)

References

235

48. Namikawa, Y.: Studies on degeneration. In: Classification of Algebraic Varieties and Compact Complex Manifolds, Lecture Notes in Mathematics, vol. 412. Springer-Verlag, Berlin, Heidelberg, New York (1974) 49. Namikawa, Y., Ueno, K.: The complete classification of curves in pencils of curves of genus two. Manuscripta Math. 9, 143–186 (1973) 50. Nielsen, J.: Investigations in the topology of closed orientable surfaces, I. Translation by John Stillwell of: Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, [Acta Math., 50, 189–358 (1927)], [Collected Math. Papers, 1, Birkhäuser, Boston, Basel, Stuttgart (1986)] 51. Nielsen, J.: The structure of periodic surface transformations. Translation by John Stillwell of : Die Struktur periodischer Transformationen von Flächen. [Math.-fys. Medd. Danske Vid.Selsk, 15, nr.1 (1937) 77p.], [Collected Math. Papers, 2, Birkhäuser, Boston, Basel, Stuttgart (1986)] 52. Nielsen, J.: Mapping classes of finite order. Translation by John Stillwell of: Abbildungsklassen endlicher Ordnung. [Acta Math., 75, 23–115 (1942)], [Collected Math. Papers, 2, Birkhäuser, Boston, Basel, Stuttgart (1986)] 53. Nielsen, J.: Surface transformation classes of algebraically finite type. Mat.-fys. Medd. Danske Vid.Selsk., 21 nr. 2 (1944), [Collected Math. Papers, 2, Birkhäuser, Boston, Basel, Stuttgart (1986)] 54. Papakyriakopoulos, C.D.: On Dehn’s lemma and the asphericity of knots. Ann. Math. 66, 1–26 (1957) 55. Pichon, A.: Fibrations sur le cercle et surfaces complexes. Ann. Inst. Fourier (Grenoble) 51(2), 337–374 (2001) 56. Reid, M.: Problems on pencils of small genus. unpublished note (1990) in his home page http:// www.warwick.ac.uk/masda/, click Surfaces, then Problem in geography of surfaces. 57. Satake, I.: On a generalization of the notion of manifolds. Proc. Nat. Acad. Sci. USA, 42, 359–363 (1956) 58. Shiga, H., Tanigawa, H.: On the Maskit coordinates of Teichmüller spaces and modular transformations. Kodai Math. J. 12, 437–443 (1989) 59. Takamura, S.: Towards the classification of atoms of degenerations. I. Splitting criteria via configurations of singular fibers. J. Math. Soc. Jpn. 56(1), 115–145 (2004) 60. Takamura, S.: Towards the classification of atoms of degenerations. III. Splitting Deformations of Degenerations of Complex Curves. Lecture Notes in Math. 1886, Springer Verlag, Berlin, Heidelberg, New York (2006). (See also his series of preprints, II and IV, under the same title.) 61. Tamura, I.: Foliations and spinnable structures on manifolds. In: Actes du Colloque International du C.N.R.S. de Strasbourg. Ann. Inst. Fourier, 23, 197–214 (1973) 62. Thurston, W.: The geometry and topology of 3-Manifolds. Preprint, Princeton University Press, Princeton, NJ (1977–79) 63. Thurston, W.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. AMS. 19, 417–431 (1988) 64. Uematsu, K.: Numerical classification of singular fibers in genus 3 pencils. J. Math. Kyoto Univ. 39(4), 763–782 (1999) 65. Van der Waerden, B.L.: Einfürung in die algebraische Geometrie, 2nd edn. Springer-Verlag, Berlin, Heidelberg, New York (1973) [Japanese Translation: Daisu-Kikagaku Nyumon, (Translation by H. Maeda) Springer-Verlag-Tokyo (1991)] 66. Waldhausen, F.: Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II. Invent. Math. 3, 308–333 (1967) 67. Waldhausen, F.: Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II. Invent. Math. 4, 87–117 (1967) 68. Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math. 87, 56–88 (1968) 69. Winkelnkemper, H.E.: Manifolds as open books: Bull. Amer. Math. Soc. 79, 45–51 (1973) 70. Winters, G.B.: On the existence of certain families of curves. Amer. J. Math. 96, 215–228 (1974)

•

Index

-sequence (asterisk sequence), 136 i -boundary, 179 j -boundary, 179 s-number, 137

general fiber, 173 generalized quotient, viii, 55

irreducible component, 53, 175 achiral, xi admissible coordinates, 175 admissible neighborhood, 195 admissible system, 4 amphidrome, viii, 6 amphidrome annulus, 35 arch, 96 atomic singular fiber, xii

bank, 54 binding, 200 black annulus, 63 blowing down, 137 body, 96

chain diagram, 136 chiral, xi chorizo space, xi, 53 closed part, 54 collapsing map, 149, 154 cone point, 9 conjugate, 9 contraction, 137 cut curve, vii, 3

deformation, 54 degenerating family, x, 173

linear twist, 17 linearization, 20, 30

mapping cylinder, 200 Milnor fiber, 193 minimal, 173 minimal quotient, ix, 60, 119 minimality conditions, 60 monodromy exponent, 11 monodromy homeomorphism, 173 multiple point, 8 multiplicity, 54, 70, 82, 90, 175

negative twist, viii, 53 nodal neighborhood, 54 node, 53 non-amphidrome annulus, 20 normal crossing, 175 normally minimal, 189 numerical chorizo space, 55 numerical homeomorphism, 93

open-book, 200 orbifold, 9 ordinary arch, 113 ordinary body, 113

Y. Matsumoto and J.M. Montesinos-Amilibia, Pseudo-periodic Maps and Degeneration of Riemann Surfaces, Lecture Notes in Mathematics 2030, DOI 10.1007/978-3-642-22534-5, © Springer-Verlag Berlin Heidelberg 2011

237

238 page, 200 part, 53 partition graph, viii, 7, 149 periodic map, 8 periodic part, 20 pinched covering, 54 plumbing, 216 precise system, 6 pseudo-periodic map, vii, 3 refined partition graph, 151 renormalization of a linear twist, 71 renormalization of a rotation, 61 renormalization of a special twist, 83 Riemann surface with nodes, 53 screw number, vii, 5, 22 simple point, 8 singular fiber, 173 singular open-book, 200 special body, 113 special tail, 113

Index special twist, 18 specialization, 35, 44 spinnable structure, 200 standard form, 19 subchorizo space, 96 superstandard form, 65, 77 superstandard parametrization, 101, 105, 112

tail, 96 topological monodromy, 174 topologically equivalent, 174

valency, 8 valency of a boundary curve, 9 valency of a multiple point, 9

weighted cohomology, 165 weighted graph, 152 weighted isomorphism, 152 white annulus, 63

LECTURE NOTES IN MATHEMATICS

123

Edited by J.-M. Morel, B. Teissier; P.K. Maini Editorial Policy (for the publication of monographs) 1. Lecture Notes aim to report new developments in all areas of mathematics and their applications - quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. Monograph manuscripts should be reasonably self-contained and rounded off. Thus they may, and often will, present not only results of the author but also related work by other people. They may be based on specialised lecture courses. Furthermore, the manuscripts should provide sufficient motivation, examples and applications. This clearly distinguishes Lecture Notes from journal articles or technical reports which normally are very concise. Articles intended for a journal but too long to be accepted by most journals, usually do not have this “lecture notes” character. For similar reasons it is unusual for doctoral theses to be accepted for the Lecture Notes series, though habilitation theses may be appropriate. 2. Manuscripts should be submitted either online at www.editorialmanager.com/lnm to Springer’s mathematics editorial in Heidelberg, or to one of the series editors. In general, manuscripts will be sent out to 2 external referees for evaluation. If a decision cannot yet be reached on the basis of the first 2 reports, further referees may be contacted: The author will be informed of this. A final decision to publish can be made only on the basis of the complete manuscript, however a refereeing process leading to a preliminary decision can be based on a pre-final or incomplete manuscript. The strict minimum amount of material that will be considered should include a detailed outline describing the planned contents of each chapter, a bibliography and several sample chapters. Authors should be aware that incomplete or insufficiently close to final manuscripts almost always result in longer refereeing times and nevertheless unclear referees’ recommendations, making further refereeing of a final draft necessary. Authors should also be aware that parallel submission of their manuscript to another publisher while under consideration for LNM will in general lead to immediate rejection. 3. Manuscripts should in general be submitted in English. Final manuscripts should contain at least 100 pages of mathematical text and should always include – a table of contents; – an informative introduction, with adequate motivation and perhaps some historical remarks: it should be accessible to a reader not intimately familiar with the topic treated; – a subject index: as a rule this is genuinely helpful for the reader. For evaluation purposes, manuscripts may be submitted in print or electronic form (print form is still preferred by most referees), in the latter case preferably as pdf- or zipped psfiles. Lecture Notes volumes are, as a rule, printed digitally from the authors’ files. To ensure best results, authors are asked to use the LaTeX2e style files available from Springer’s web-server at: ftp://ftp.springer.de/pub/tex/latex/svmonot1/ (for monographs) and ftp://ftp.springer.de/pub/tex/latex/svmultt1/ (for summer schools/tutorials).

Additional technical instructions, if necessary, are available on request from lnm@ springer.com. 4. Careful preparation of the manuscripts will help keep production time short besides ensuring satisfactory appearance of the finished book in print and online. After acceptance of the manuscript authors will be asked to prepare the final LaTeX source files and also the corresponding dvi-, pdf- or zipped ps-file. The LaTeX source files are essential for producing the full-text online version of the book (see http://www.springerlink.com/ openurl.asp?genre=journal&issn=0075-8434 for the existing online volumes of LNM). The actual production of a Lecture Notes volume takes approximately 12 weeks. 5. Authors receive a total of 50 free copies of their volume, but no royalties. They are entitled to a discount of 33.3 % on the price of Springer books purchased for their personal use, if ordering directly from Springer. 6. Commitment to publish is made by letter of intent rather than by signing a formal contract. Springer-Verlag secures the copyright for each volume. Authors are free to reuse material contained in their LNM volumes in later publications: a brief written (or e-mail) request for formal permission is sufficient. Addresses: Professor J.-M. Morel, CMLA, ´ Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan Cedex, France E-mail: [email protected] Professor B. Teissier, Institut Mathématique de Jussieu, ´ UMR 7586 du CNRS, Equipe “Géométrie et Dynamique”, 175 rue du Chevaleret 75013 Paris, France E-mail: [email protected] For the “Mathematical Biosciences Subseries” of LNM: Professor P. K. Maini, Center for Mathematical Biology, Mathematical Institute, 24-29 St Giles, Oxford OX1 3LP, UK E-mail : [email protected] Springer, Mathematics Editorial, Tiergartenstr. 17, 69121 Heidelberg, Germany, Tel.: +49 (6221) 487-8259 Fax: +49 (6221) 4876-8259 E-mail: [email protected]

Pseudo-periodic Maps and Degeneration of Riemann Surfaces

Riemann surfaces

Riemann surfaces

Riemann Surfaces

Riemann Surfaces

Extremal Polynomials and Riemann Surfaces

Ernst Equation and Riemann Surfaces

Riemann Surfaces of Infinite Genus

Compact Riemann surfaces

Lectures on Riemann surfaces

Symmetries of compact Riemann surfaces

Riemann surfaces, dynamics, and geometry

Riemann surfaces, dynamics, and geometry

Algebraic Curves and Riemann Surfaces

Algebraic Curves and Riemann Surfaces

Kuramochi Boundaries of Riemann Surfaces

Riemann Surfaces of Infinite Genus

Lectures on Riemann Surfaces

Riemann Surfaces (Mathematics Series)

Geometry of Riemann surfaces and Teichmüller spaces

Geometry of Riemann Surfaces and Teichmüller Spaces

Geometry and spectra of compact riemann surfaces

Geometry of Riemann Surfaces and Teichmüller Spaces

Geometry of Riemann surfaces and Teichmüller spaces

Conformal mapping on Riemann surfaces

Computational Approach to Riemann Surfaces

Conformal mapping on Riemann surfaces

Theta functions on Riemann surfaces

Theta Functions on Riemann Surfaces

Hardy classes on Riemann surfaces

Theta Functions On Riemann Surfaces

Pseudo-periodic Maps and Degeneration of Riemann Surfaces

Riemann surfaces

Riemann surfaces

Riemann Surfaces

Riemann Surfaces

Extremal Polynomials and Riemann Surfaces

Ernst Equation and Riemann Surfaces

Riemann Surfaces of Infinite Genus

Compact Riemann surfaces

Lectures on Riemann surfaces

Symmetries of compact Riemann surfaces

Riemann surfaces, dynamics, and geometry

Riemann surfaces, dynamics, and geometry

Algebraic Curves and Riemann Surfaces

Algebraic Curves and Riemann Surfaces

Kuramochi Boundaries of Riemann Surfaces

Riemann Surfaces of Infinite Genus

Lectures on Riemann Surfaces

Riemann Surfaces (Mathematics Series)

Geometry of Riemann surfaces and Teichmüller spaces

Geometry of Riemann Surfaces and Teichmüller Spaces

Geometry and spectra of compact riemann surfaces

Geometry of Riemann Surfaces and Teichmüller Spaces

Geometry of Riemann surfaces and Teichmüller spaces

Conformal mapping on Riemann surfaces

Computational Approach to Riemann Surfaces

Conformal mapping on Riemann surfaces

Theta functions on Riemann surfaces

Theta Functions on Riemann Surfaces

Hardy classes on Riemann surfaces

Theta Functions On Riemann Surfaces

Recommend Documents