Table Of ContentContemporary Mathematicians
Gian-Carlo Rota
Editor
. Birkhauser
Boston · Basel · Stuttgart
Jakob Nielsen
Constantinople, 1916
Ja ·kNoibe lsen:
ColleMcattehde maPtaipcearls
Volu1m e
(1913-1932)
Edifotret dh e
DaniMsaht hematical Society
by
VagLnu ndsgHaaanrsde n
Birkhauser
Bost·Bo ans Setlu.t·t gart
1986
Editor
VagLnu ndsHgaasnaner d
MathemIantsitciatlu te
ThTee chUnniicvaelor Dfse intmya rk
DK-2L8y0n0g Cboyp,e nhDaegnemna,r k
LibroaCfro yn gress iCPnau tballiDocagatitani go n
NielJsaek1no8,b9 ,0 -1959.
Collmeacttheedm paatpiecrasl.
(Contemmpaotrhaermya ticians)
Bibliopg.r aphy:
1.Groups,o fT-hCeoolwrloyer cktse.d
2.S urfaces-CwoolrlkIe.s Hc.at nesdVae gn n,
LundsgaIaLDr ands.mk a temfoarteinsiIknI. TgI ti.el .
IVS.e ries.
QA171.N159781653 21' .22
ISBN3- 7643-(Sw3ti1zr4el0da-v:.n21 )
ISB3-N76 34-31(S5w1ti-z8e rvl 2.a)nd:
CIP-KurztidteeDrle auutfnsBacihhbmeleni othek
NielJsaekno.b :
Collepcatp/eeJ dra sk Noibe lEsdefo.nr t. h e
DaniMsaht heSmoacbt.yV. ag nL undsHgaanasredn
Bost;Bo ans ;eS lt ut;tB giarrkth auser
(Contemmpaotrhaermya ticians)
ISB3N- 7643(-B3a1.s5.e)2.l - 6
ISB0N- 8176(-B3o1s5t2o-n6)
NEN:i elJsaeknSo,ab m alung
Vo1l( .1 986)
ISB3N- 7643(-B3a1.s4.e)0.l - 2
ISB0N- 8176(-B3o1s4t0o-n2)
Alrli grhetsseN ropv aeordtft.h piusb limcaabyter i eopnr oduced,
stoiraner de trsiyesvotartel rm a,n simanin tfoytr emod rb, ya ny
meaenlse,c tmreocnhiacn,i cal, phootroo tchoeprywiinsge,, recording
without proitfoh rce o ppyerrimgihsts ioonw ner.
© BirkhBaousst1eo9rn8 ,6
PrinitUneS dA
ISB0N- 8176(-s3e1t5)2 I-S6B 3N- 7643(-s3e1t5)2 -6
ISB0N- 8176(-V3o11l)4. 0 I-S2B 3N- 7643(-V3o11l)4. 0 -2
ISB0N- 8176(-V3o21l)5. 1 I-S8B 3N- 7643(-V3o21l)5. 1 -8
With this publication, the Danish Mathematical
Society wishes to honour the memory of Jakob
Nielsen, who was an outstanding member of the
Society from 1921 to his death in 1959. We are
happy, indeed proud, to present to the mathemati
cal community witness of the creativity and fore
sight of Jakob Nielsen.
The Society expresses its deep gratitude to the many
scholars who have worked on the project. Special
thanks go to Vagn Lundsgaard Hansen who re
sumed the job as editor.
The publication has been made possible by gener
ous donations from the Carlsberg Foundation and
from the Danish Natural Science Research Council.
On behalf of the
Danish Mathematical Society
' Flemming Tops0e
Preface
One of the central figures in Danish mathematics in the first half of this century was Jakob
Nielsen (1890-1959), who won international recognition through his pioneering work in
group theory and the topology of surfaces. For some time the Danish Mathematical Society
has had the desire to publish the collected papers of Jakob Nielsen, but for various reasons it
has not been possible to realize the project until now.
The work of Nielsen in group theory won early recognition, and an important theorem on
free groups carries his name: the Nielsen-Schreier theorem. His work on surface transforma
tions, to which he devoted much time throughout his life, was highly respected by his con
temporaries, but it was not in the main stream of the then burgeoning field of algebraic
topology. The more striking of his results in topology were well known and often applied,
although in many cases alternative proofs were found. For many years the seemingly com
plicated methods of Jakob Nielsen were therefore studied seriously only by very few mathe
maticians. Recently, however, there has been a growing interest in his methods, and this has
created a renewed strong interest in his original papers. It is an obstacle to many potential
readers that the most important of Nielsen's papers are written in a rather old-fashioned, but
perfect, German.
For these reasons, in the Spring of 1980, the Danish Mathematical Society decided to
investigate the possibility of publishing the collected papers of Jakob Nielsen, with many of
the important papers translated into English. The plans were strongly motivated by Hans
J0rgen Munkholm. From the outset Werner Fenchel and Heiner Zieschang were important
scientific advisors, and during a visit to Denmark in December 1981 Joan Birman also offered
valuable advice. We were very lucky to find an exceptionally competent and effective trans
lator in John Stillwell.
We have included almost all of Nielsen's published papers in these two volumes, and
also a set of hitherto unpublished notes from lectures in Breslau in 1921. Not included are his
textbooks on mechanics and aerodynamics, two papers dealing with problems in mechanics,
a few papers of purely didactic or expository character, and two obituaries. A photo
graph of Jakob Nielsen and a complete list of his publications can be found in the memorial
paper by Werner Fenchel, which opens Volume 1. At the end of Volume 2 are various essays
describing the work of Jakob Nielsen and later developments. Bruce Chandler and Wilhelm
Magnus comment on Nielsen's work in group theory. In two essays Joan Birman and Heiner
Zieschang describe his investigations of surface mapping classes. Robert F. Brown comments
on Nielsen fixed point theory. Finally, Werner Fenchel describes Nielsen's contributions to the
theory of discontinuous groups of isometries of the hyperbolic plane.
The publication of Jakob Nielsen's collected papers has involved the collaboration of
many individuals in Denmark and abroad as well as encouragement from the publishers. As
editor appointed by the Danish Mathematical Society, I wish to express my sincere thanks to
all of those who have contributed to make it possible. Special thanks go to Werner Fenchel for
his never-failing enthusiasm and help.
Lyngby, June 1986 Vagn Lundsgaard Hansen
vii
Note Concerning References: References to Nielsen's own papers are given in brackets by an
N followed by the number which the paper carries in the list of Nielsen's publications in
Fenchel's memorial paper. The numbers are also listed in the Table of Contents together with
relevant information about each paper. In the essays, other references will be listed alphabeti
cally at the end of each essay and referred to by a number in brackets.
viii Preface
Table.of Contents
Volume 1
Preface vii
Jakob Nielsen in Memoriam, by Werner Fenchel [Acta Math., 103 (1960), VII-XIX.] 1
[N-1] Kurvennetze auf Flachen. [Inaugural-Dissertation, Kiel 1913, 58 pp.] 14
[N-2] Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden.
[Math. Ann., 78 (1917), 385-397.] 77
[N-3] Uber die Isomorphismen unendlicher Gruppen ohne Relation. [Math. Ann., 79
(1918), 269-272.] 91
[N-6] Uber fixpunktfreie topologische Abbildungen geschlossener Flachen. [Math. Ann.,
81 (1920), 94-96.] 96
[N-7] Uber die Minimalzahl der Fixpunkte bei den Abbildµngstypen der Ringflachen.
[Math. Ann., 82 (1920), 83-93.] 99
[N-7a] The MappingTypes of Closed Surfaces and their Relationship with Infinite Groups.
Translation by John Stillwell of: Die Abbildungstypen gescblossener Flachen und
ihre Beziehungen zu unendlichen Gruppen. [2 Vortrage. gehalten in Breslau am 9.
und 11. Marz 1921 von J. Nielsen.] · · 110
[N-8] On calculation with non-commutative factors and its application to group theory.
[The Mathematical Scientist, 6 (1981), 73-85.J Translation by Anne W. Neumann,
n
with assistance from Walter Neumann, of: Om Regning med ikke-kommutative
Faktorer og dens Anvendelse i Gruppeteorien. [Mat. Tidsskr. B (1921), 77-94.] 117
[N-9] Diskontinuerte Flytningsgrupper. [Mat. Tidsskr. B (1923), 53-61.] For a summary
of this paper see §1 in the essay by Werner Fenchel, Volume 2, 427-440.
[N-10] The Torus and the Plane. Translation by John Stillwell ~f: Ringfladen og Planen.
[Mat. Tidsskr. B (1924), 1-22;] 130
[N-11] Die Gruppe der dreidimensionalen Gittertransformationen. [Math. -fys. Medd.
Danske Vid. Selsk. 5, nr. 12 (1924), 3-29.] 147
[N-12] The Isomorphism Group ofa Free Group. Translation by John Stillwell of: Die
Isomorphismengruppe der freien Gruppen. [Math. Ann., 91 (1924), 169-209.] 174
[N-14] Om geodredtiske Linier i lukkede Mangfoldigheder med konstant negativ
Krumning. [Mat. Tidsskr. B (1925), 37-44.] For a summary of this paper see §2 in
the essay by Werner Fenchel, Volume 2, 427-440 ..
[N-15] Zur Topologie der geschlossenen Flachen. [Den sjette skandinav. Matematiker-
kongres, K0benhavn 1926, 263-276.] 209
[N-18] Investigations in the Topology of Closed Orientable Surfaces, I. Translation by John
Stillwell of: Untersuchungen zur Topologie der gesGhloss.enen zweiseitigen Flachen.
[Acta Math., 50 (1927), 189-358.] 223
[N-19] Topological Mappings of a Jordan Curve onto Itself. Translation by John Stillwell
of: Om topologiske Afbildninger af en Jordankurve pa sig selv. [Mat. Tidsskr. B
(1928), 39-46.J 342
[N-20] Investigations in the Topology of Closed Orientable Surfaces, II. Translation by
John Stillwell of: Untersuchungen zur Topologie der geschlossenen Flachen, II.
[Acta Math;, 53(1929),1-76.] 348
[N-21] Investigations in the Topology of Closed Orientable Surfaces, III. Translation by
John Stillwell of: Untersuchungen zur Topologie der geschlossenen zweiseitigen
Flachen, III. [Acta Math., 58 (1932), 87-167.] 401
ix
Volume 2
[N-22] On Regular Riemann Surfaces. Translation by John Stillwell of: Om regulrere
Riemannske Flader. [Mat. Tidsskr. B (1932), 1-18.] 1
[N-23] On the Structure of a Vector Module with Finite Basis, with Application to
Diophantine Approximation. Translation by John Stillwell of: Om Strukturen
af en Vektormodul med endelig Basis. Med Anvendelse paa diofantiske
Approksimationer. [Mat. Tidsskr. B (1932), 29-42.J 13
[N-24] Einige Satze iiber topologische Flachenabbildungen. [Acta Litt. Scient. Szeged, 7
(1935), 200-205.J 24
[N-25] Topologie des transformations des surfaces. [Enseign. Math., 35 (1936), 269-287.] 31
[N-26] Topologischer Beweis eines Satzes von Wiman. [Mat. Tidsskr. B (1936), 11-24.] 51
[N-29] The Structure of Periodic Surface Transformations. Translation by John Stillwell
of: Die Struktur periodischer Transformationen von Flachen. [Math.-fys. Medd.
Danske Vid. Selsk. 15, nr. 1 (1937), 77 pp.] 65
[N-30] Einige Methoden und Ergebnisse aus der Topologie der Flachenabbildungen. [C.R.
Congr, Intern. Math. Oslo 1936, I (1937), 128-139.J 104
[N~31] The Symmetric and Alternating Groups. Translation by John Stillwell of: Den
symmetriske og den alternerende Gruppe. [Mat. Tidsskr. B. (1940), 7-18.J 116
[N-32] Uber Gruppen linearer Transformationen. [Mitt. Math. Gesellsch. Hamburg 8,
Festschrift, Teil II (1940), 82-104.] 124
[N-33] A Theorem from the Topology of Surface Mappings. Translation by Hans J.
Munkholm of: En Sretning fra Fladeafbildningernes Topologi. [Norsk Mat. Tidsskr.
23 (1941), 96-99.J 147
[N-34] Mapping Classes ofFinite Order. Translation by John Stillwell of: AbbildU:ngs-
klassen endlicher Ordnung. [Acta Math., 75 (1942), 23-115.J 150
[N-35] Fixed Point Free Mappings. Translation by John Stillwell of: Fixpunktfrie Afbild-
ninger. [Mat. Tidsskr. B (1942), 25-41.J 221
[N-36] Surface Transformation Classes of Algebraically Finite Type. [Mat.-fys. Medd.
Danske Vid, Selsk. 21, nr. 2 (1944), 87 pp.] 233
[N-37] Simplified Proofs of Some Theorems in Surface Topology. Translation by John
Stillwell of: Forenklede Beviser for nogle Sretninger i Fladetopologien. [Mat.
Tidsskr. B (1946), 1-16.J Joint paper with S. Bundgaard. 320
[N-38] On Discontinuous Groups oflsometric Transformations of the Non-Euclidean
Plane. [Studies and Essays presented to R. Courant, New York 1948, 117-128.]
Joint paper with W. Fenchel. 333
[N-39] The Commutator Subgroup of a Free Product of Cyclic Groups. Translation
by John Stillwell of: Kommutatorgruppen for det frie produkt af cykliske
grupper. [Mat. Tidsskr. B (1948), 49-56.J 345
[N-40] A Study Concerning the Congruence Subgroups of the Modular Group. [Mat.-fys.
Medd. Danske Vid. Selsk. 25, nr. 18 (1950), 32 pp.] 351
[N-43] On Normal Subgroups with Finite Index in F-groups. [Mat. Tidsskr. B (1951), 56-
58.J Joint paper with S. Bundgaard. 382
[N-44] Nogle grundlreggende begreber vedrnrende diskontinuerte grupper af liri.erere sub-
stitutioner i en komplex variabel. [C.R. 111eme Congr. Math. Scand. Trondheim 1949,
Trondheim 1952, 61-70J See the essay by Werner Fenchel, Volume 2, 427-440.
[N-46] A Basis for Subgroups ofFree Groups. [Math. Scand. 3 (1955), 31-43.J 385
X Table of Contents
