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El. knyga: Pseudo-periodic Maps and Degeneration of Riemann Surfaces

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  • Formatas: PDF+DRM
  • Serija: Lecture Notes in Mathematics 2030
  • Išleidimo metai: 17-Aug-2011
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Kalba: eng
  • ISBN-13: 9783642225345
Kitos knygos pagal šią temą:
Pseudo-periodic Maps and Degeneration of Riemann SurfacesDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Lecture Notes in Mathematics 2030
  • Išleidimo metai: 17-Aug-2011
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Kalba: eng
  • ISBN-13: 9783642225345
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The first part of the book studies pseudo-periodic maps of a closed surface of genus greater than or equal to two. This class of homeomorphisms was originally introduced by J. Nielsen in 1944 as an extension of periodic maps. In this book, the conjugacy classes of the (chiral) pseudo-periodic mapping classes are completely classified, and Nielsen's incomplete classification is corrected. The second part applies the results of the first part to the topology of degeneration of Riemann surfaces. It is shown that the set of topological types of all the singular fibers appearing in one parameter holomorphic families of Riemann surfaces is in a bijective correspondence with the set of conjugacy classes of the pseudo-periodic maps of negative twists. The correspondence is given by the topological monodromy.

Recenzijos

From the reviews:

The present monograph consists of two parts. In the first part a class of pseudo-periodic maps resp. mapping classes of closed surfaces is studied and classified up to conjugation (those of negative twist). This is motivated also by the second part on the topology of degenerating families of Riemann surfaces over a disk . exposition is self-contained, elementary and written with great care for details this makes arguments and proofs quite long, but certainly easy and pleasant to read. (Bruno Zimmermann, Zentralblatt MATH, Vol. 1239, 2012)

Surface mapping classes of algebraically finite type were introduced by Nielsen in 1944. Nielsens arguments are sometimes considered as too vague, and one of the main objects of this book is to make Nielsens assertions precise. The results in this book are given with detailed proofs. The book is well written and it should be useful for low-dimensional topologists and algebraic geometers. (Athanase Papadopoulos, Mathematical Reviews, Issue 2012 h)

Part I Conjugacy Classification of Pseudo-periodic Mapping Classes
1 Pseudo-periodic Maps
3(14)
1.1 Basic Definitions
3(3)
1.2 Nielsen's Results on Pseudo-periodic Maps
6(11)
2 Standard Form
17(36)
2.1 Definitions and Main Theorem of Chap. 2
17(3)
2.2 Periodic Part
20(1)
2.3 Non-amphidrome Annuli
20(15)
2.4 Amphidrome Annuli
35(12)
2.5 Proof of Theorem 2.1
47(6)
3 Generalized Quotient
53(40)
3.1 Definitions and Main Theorem of Chap. 3
53(7)
3.2 Proof of Theorem 3.1
60(1)
3.3 Quotient of (B', f|B')
61(1)
3.4 Re-normalization of a Rotation
61(10)
3.5 Re-normalization of a Linear Twist
71(12)
3.6 Re-normalization of a Special Twist
83(8)
3.7 Completion of Theorem 3.1. (Existence)
91(2)
4 Uniqueness of Minimal Quotient
93(38)
4.1 Main Theorem of Chap. 4
93(4)
4.2 Structure of π-1(arch)
97(6)
4.3 Structure of π-1(tail)
103(4)
4.4 Structure of π-1(body)
107(6)
4.5 Completion of the Proof of Theorem 4.1. (Uniqueness)
113(5)
4.6 General Definition of Minimal Quotient
118(1)
4.7 Conjugacy Invariance
119(12)
5 A Theorem in Elementary Number Theory
131(14)
5.1 Proof of Theorem 5.1. (Uniqueness)
132(3)
5.2 Proof of Theorem 5.1. (Existence)
135(10)
6 Conjugacy Invariants
145(28)
6.1 Partition Graphs
149(2)
6.2 Weighted Graphs
151(11)
6.3 Completion of the Proof of Theorem 6.1
162(2)
6.4 Weighted Cohomology
164(9)
Part II The Topology of Degeneration of Riemann Surfaces
7 Topological Monodromy
173(16)
7.1 Proof of Theorem 7.1
175(1)
7.2 Construction of Δ and {Ni}si=1
176(1)
7.3 The Decomposition F = A U B
177(1)
7.4 Construction of a Monodromy Homeomorphism
177(4)
7.5 Negativity of Screw Numbers
181(4)
7.6 Completion of the Proof of Theorem 7.1
185(4)
8 Blowing Down Is a Topological Operation
189(10)
9 Singular Open-Book
199(22)
9.1 Completion of the Proof of Theorem 7.2
211(2)
9.2 Characterization of the Triples (S, Y, c) That Come from Pseudo-periodic Maps
213(5)
9.3 Completion of the Proof of Theorem 9.2
218(2)
9.4 Concluding Remark
220(1)
A Periodic Maps Which Are Homotopic 221(12)
References 233(4)
Index 237
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