Finite Groups of Mapping Classes of Surfaces
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875 Heiner Zieschang
Finite Groups of Mapping Classes of Surfaces
Springer-Verlag Berlin Heidelberg New York 1981
Author
Heiner Zieschang Institut fur Mathematik, Ruhr-Universitat Bochum UniversitatsstraBe 150,4630 Bochum 1 Federal Republic of Germany
AMS Subject Classifications (1980): 20 F xx, 20 H 10,20 H 15,20 J 05, 32 G 15,51 M 10,57 M xx, 57 N 05,57 N 10 ISBN 3-540-10857-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10857-2 Springer-Verlag New York Heidelberg Berlin
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© by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
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INTRODUCTION
These Notes center around the Nielsen Realization Problem,
[Nielsen 1942]
i.e. the question whether a finite group of mapping classes of a surface can be realized by a finite groups of mappings. This was first answered positively for the case of cyclic groups in where Nielsen used geometrical arguments developed in the other famous Acta papers. Using FricJ 0 disjoint disks in S2 by Moebius strips and punch out m holes
we get the non-orientable surfaces N The holes may be punctures or may be g,m bounded by closed curves. The number g is called the genus. For the consideration of manifolds one must, in general, specify the category in which one is
working. However, in dimension 2 the situation is much more pleasant
as the Hauptvermutung has been proved: each topclogical surface can be triangulated, and any two triangulations of the same surface have .i.sorrorpn.ic subdivisions [Rado 1924]. This allows a topclogical C= PL) classification of 2-manifolds: a surface S with a finitely generated first hOllDlogy group H is homeomcrphic to one of 1CS) the surfaces F or N ; for a proof and literature see [ZVC 1980, sections 3. g,m g,m 1-2, 8]. A consequence is that all surfaces ;;assess differentiable, even real analytic structures; mcreover, two SIlDOth surfaces are d.i.ff eorror-ph.ic
if they are
bomeonor-ph.ic . Although all orientable surfaces allow complex analytic structures,
the situation with respect to classification is quite different. Here there arises the rrodul.ar- problem, comp. chapter 2 and 3. 11.1 Definition. A surface S has finite type
if one of the fo'l Lcwing equivalent
conditions holds. (a )
The surface S can be embedded into a compact surface S', and S',S consists of a finite number of disks or pcints.
Cb) n CS) is finitely generated. 1 Cc) H is finitely generated. 1CS) By an application of the Van Kampen-Seifert theorem there follows 11.2 Theorem. Let S be a surface of genus g with m holes. Then n 1 CS)
=
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