**William Threlfall**'s parents were Helene Elizabeth Clara Koch and William Threlfall (1863-1888). William Senior was a son of Richard Threlfall of Hollowforth, near Preston in Lancashire, England. He was a botanist at the University of Cambridge and had married Helene Koch from Dresden. Sadly William, the subject of this biography, never knew his father who drowned in the Diala river in India in March 1888 while on a research expedition. Helene Threlfall returned to her home town of Dresden where her son was born and there she lived with her two unmarried brothers, Hugo and Max Koch. William was brought up in Dresden and he attended the Humanistischen Königlichen Gymnasium in Dresden-Neustadt, graduating with his Abitur in 1907.

After graduating from the high school, Threlfall entered the University of Jena where his main interest was in chemistry but he also studied mathematics courses. He graduated with a degree in chemistry in 1910 and then went to the University of Göttingen to study mathematics in the year 1911-12. However, with the outbreak of World War I in 1914, Threlfall was interned because of his British citizenship. He was released and served in the German army in 1915. After military service, from 1916 he lived on his uncle's estate near Dresden. There he lived as a gentleman farmer but mainly occuppied himself in mathematical studies. During the academic year 1921-22 he attended courses at the University of Leipzig and he was awarded his doctorate on 3 February 1926 for his thesis *Über regelmässige Flächenteilung* Ⓣ. His thesis advisors at the University of Leipzig had been Friedrich Wilhelm Daniel Levi (1888-1966) and Otto Ludwig Hölder.

In 1927 Threlfall submitted his habilitation thesis to the Technische Hochschule in Dresden and began lecturing there as a Privatdocent, particularly on his research topic of topology. The first course he gave in 1927 was attended by the young student Herbert Seifert. Not only did Threlfall turn Seifert into an enthusiastic student of topology, but far more than that, they became firm friends and mathematical collaborators. We should note in passing the considerable age difference between the two with Threlfall being twenty years older than his friend and student. Threlfall's topology course, attended by Seifert as a student, became the basis of their famous book* Lehrbuch der Topologie* Ⓣ (1934). We give more details about this book below but, at this point, let us quote a sentence originally written by Threlfall for the Preface:-

However, Seifert was unhappy to receive this much credit so, after some discussion, the two friends settled on the following compromise which appears in the Preface of the published work:-This textbook arose from a course which one of us gave to the other at the Technische Hochschule in Dresden. But soon the student contributed new ideas to such an extent and changed the presentation so fundamentally that it would be more justifiable to omit on the title page the name of the original author than his.

Threlfall was appointed as an assistant in the Institute of Engineering Mechanics of the Technical University of Dresden in 1929. In January 1930 Seifert moved into the spacious house in Nordstrasse where Threlfall was still living with his mother and two uncles. They began keeping a diary which is discussed in [3] and continued to do so until Threlfall's death, by which time they had filled twenty-two volumes. On 6 June 1933, he was appointed as an untenured extraordinary professor at the University of Halle-Wittenberg. He was promoted to a tenured extraordinary professor of analytical geometry, calculus of variations, analysis and function theory on 1 April 1936. From 1 January 1938 he was appointed as a full professor at the University of Frankfurt, also becoming Director of Mathematical Seminars there in October. His appointment to Frankfurt was to fill the chair left vacant when Carl Siegel resigned in 1937 and accepted a chair at Göttingen. Frankfurt had been severely affected by the anti-Jewish legislation brought in by the Nazis (this was one of the reasons that Siegel left) and when Threlfall arrived there were several leading mathematicians, such as Paul Epstein, Ernst Hellinger and Max Dehn, who were still in Frankfurt but not allowed to teach.The first step towards writing this textbook was a course which one of us(Threlfall)taught at the Technische Hochschule in Dresden. But only part of the course was included in the book. The main part of its contents originated later from daily discussions between the two authors.

Let us now look at some of Threlfall's publications. He published the book *Gruppenbilder* Ⓣ in 1932 which was reviewed by Donald Coxeter in [1]:-

Much of Threlfall's best-known work was done in collaboration with Herbert Seifert. The most famous is their classic topology bookMost of these group-pictures consist of partitions of a multiply-connected surface into polygons. The operations of a group are represented as automorphisms of a suitably chosen surface; each polygon, or pair of polygons(one "white" and one "black"), is a fundamental region. The case considered most thoroughly is when the polygons are triangles, which combine in sets to form the faces of a "regular polyhedron". ... The treatment is clear and systematic.

*Lehrbuch der Topologie*Ⓣ (1934) which we have already mentioned above. Henry Whitehead reviewed the book in [4] and it is interesting to note that topology was such a new topic at the time that he felt it necessary to begin his review by explaining what the subject was:-

When Henry Whitehead begins to comment on the book itself he is full of praise [4]:-In our conception of space a simple-minded idea of continuity comes before everything else. Ideas of distance, angle, straightness and so on are much more elaborate. Topology is the mathematical expression of the former. If a map were drawn on a rubber membrane its topological character would not be affected however drastically the membrane were distorted. Two islands would still be recognisable as such, but not square or circular countries.

He ends his review as follows [4]:-This book opens with a chapter on the intuitive background of the subject. It is eminently successful in showing the extent to which topological problems pervade mathematics and how fascinating they are. There is no branch of mathematics which is more tantalizing to the imagination.

Ross Geoghegan, reviewing the English translation which was published in 1980, writes:-In all, the material is presented clearly enough and with sufficient imagination to provide a first-class introduction to the subject. Moreover, the scope of the book is such that research workers will find it a stimulus and to many of them it will be a source of new knowledge

In 1938 Seifert and Threlfall published the bookThe great strength of this book is the geometric insight it has given to generations of readers.

*Variationsrechnung im grossen*(

*Morsesche Theorie*) Ⓣ. Again this book was reviewed by Henry Whitehead who writes [5]:-

There is a story we should recount about this book but, before doing so we should note that Seifert was appointed to the University of Heidelberg in 1935. Threlfall continued to collaborate with Seifert by correspondence and they spent holidays together working on mathematics. Both were unhappy about Nazi policies and they put a Latin epigraph by Johannes Kepler intoThis book is a first-class introduction to the topological part of Marston Morse's book, 'The Calculus of Variations in the Large' ... the book contains a self-contained account of critical values of a function, whose argument may be a curve in a given manifold, their type numbers and their relation to the topology of the space over which the function is defined. This theory is applied in detail to the study of a typical problem in the calculus of variations in the large. The exposition is brilliantly clear and the book is altogether pleasing.

*Variationsrechnung im grossen*Ⓣ which, in translation, reads:-

The book was accepted by Wilhelm Blaschke for the Hamburg monograph series but, since Blaschke went along with the Nazi ideas, he objected to the Kepler quote on the grounds that it looked like a political statement - of course this is exactly what it was meant to be. It is remarkable that Threlfall and Seifert risked their positions by insisting that the epigram remain. They won their case and the epigram appeared in the book when published. Blaschke expressed fury that the quote had not been deleted.Today it is very hard to write mathematical books.

During the war, both Seifert and Threlfall worked at the Institut für Gasdynamik in Braunschweig, a research centre attached to the German Air Force. While Threlfall undertook war work there, he was given leave of absence from Frankfurt where Wolfgang Franz took over his lecturing duties. Certainly Threlfall had his problems during these years. For example, on 26 November 1942 he was ordered to the State Police Office in Braunschweig where he was accused of an offence under the "Treachery Act". On 10 February 1943, following an investigation, it was recorded by the Secret Police that no proof of "subversive utterances of a criminal nature" could be found.

Threlfall and Seifert had made efforts to obtain jobs at the same university from the time that they were separated in 1935. Except for the time they spent at the Institut für Gasdynamik, they did not achieve this until 1946 when Threlfall, supported by Seifert, was appointed to a professorship at the University of Heidelberg on 4 May. However, towards the end of the war a mathematics research centre had been established at Oberwolfach, in the Black Forest, under the directorship of Wilhelm Süss. Mathematicians from military establishments were sent there, including Threlfall and Seifert. They remained there, suffering considerable hardship, until the war ended. Threlfall was at Mathematics Research Centre at Oberwolfach when he died and he was buried at Oberwolfach. However, in 1980 Seifert had Threlfall's body exhumed and transferred to Heidelberg. Threlfall's body is now buried in the Bergfriedhof in Heidelberg next to the graves of Seifert and his wife.

Finally, let us mention some of Threlfall's last publications. He published the paper *Stationäre Punkte auf geschlossenen Mannigfaltigkeiten* Ⓣ in 1941, which is based on a lecture he gave on the theory of critical points on closed manifolds. In 1942 he published *Le calcul des variations global* Ⓣ which describes in an elementary manner the ideas behind the theory of critical points of functions in the large. After a joint paper *Topologie* (1948) written with Seifert, he published the single-authored paper *Knotengruppe und Homologieinvarianten* Ⓣ in 1949. After Threlfall's death, another joint paper with Seifert entitled *Old and new results on knots* (1950) was published. Ralph Fox writes in a review:-

A carefully reasoned development of the central part of knot theory(the group of a knot, the Alexander polynomial, the homology groups and linking invariants of the cyclic coverings)together with explanation of new results by Graeub, Schubert, Seifert and Threlfall, this survey should furnish an excellent guide to the non-specialist.

**Article by:** *J J O'Connor* and *E F Robertson*

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