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Béla Kerékjártó

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Born
1 October 1898
Budapest, Hungary
Died
26 June 1946
Gyöngyos, Hungary
Summary
Béla Kerékjártó was a Hungarian mathematician who worked in topology.
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Biography

Béla Kerékjártó is sometimes called Béla von Kerékjártó or, in French, Béla de Kerékjártó. We note that, until 10 December 1937, he used the spelling Kerékgyártó. He was the son of Károly Kerékgyártó (1853-1907) and Berta Holczer (1868-1959). Károly Kerékgyártó had attended Sárospatak Reformed College and Miskolc Reformed Gymnasium before studying at the University of Budapest. He studied mathematics and engineering and was then employed as an engineer and later as an accounting officer in the Mayor's Office in Budapest. He was 41 years old when he married 26 year old Berta Holczer in Budapest on 22 October 1894. Berta, the daughter of a coffee shop owner, was Jewish but she was baptised into the Christian Reformed Church three weeks before the wedding. Károly and Berta had three children: Gizella Kerékgyártó (born 1895); Béla Kerékgyártó (born 1898), the subject of this biography; and Margit Kerékgyártó (born 1901), who became a dentist. Béla's father died of pneumonia on 9 November 1907 when Béla was nine years old. From that time on, Berta took care of the upbringing and education of her three children by herself.

In 1908 Béla's mother enrolled him in the Budapest Royal High School which [16]:-
... opened in 1905, where he was granted tuition-free education based on his good performance. He had a particularly good aptitude for learning languages. In addition to German, which he learned primarily from his mother, he began learning several languages ​​as a high school student. His thorough knowledge of Latin helped him master French and Italian. Later, he also learned English and Spanish. In the upper grades of high school, he began studying higher mathematics from university notes in German. He was still a high school student when he sent one of his papers to Luitzen Egbertus Jan Brouwer ... In addition to learning languages ​​and developing an interest in mathematics, he learned to play the violin and then the piano. ... When he was an eighth-grade student, on 15 March 1916, Béla Kerékjártó gave the ceremonial speech at the school commemoration. On 26 June 1916, he graduated with honours.
He entered the Budapest University of Science and Technology in 1916 and there he attended mathematics lectures by Manó Beke (1862-1946), Lajos Dávid, Lipót Fejér and others. Among his physics lecturers we mention Loránd Eötvös. From April 1918 to February 1920 he was employed in Eötvös's geophysical laboratory. He was awarded his doctorate on 14 February 1920 with mathematics as his major subject and theoretical and experimental physics as minor subjects. His thesis was A kör és a gömbfelület átalakulásairól és véges csoportjairól Ⓣ. He already had six papers in print in 1919 before the award of his doctorate. Three were consecutive papers in the first part of volume 80 of Mathematische Annalen, these are: Über die Brouwerschen Fixpunktsätze Ⓣ; Über Transformationen des ebenen Kreisringes Ⓣ; and Über die periodischen Transformationen der Kreisscheibe und der Kugelfläche Ⓣ. In the third of these he writes:-
I express my most sincere thanks to Mr Brouwer for his kind instructions, to which I owe the simple presentation of my method of proof given here.
After the award of his doctorate he was awarded a scholarship and he travelled to Berlin arriving on 3 March 1920. There he attended lectures and seminars before travelling on to Göttingen on 17 September. He met William Henry Young who employed Kerékjártó as his assistant. At this time William Young and his wife Grace Chisholm Young were living in Lausanne and Kerékjártó lived with the Young family in Lausanne from October 1920 to March 1921. While living with the Youngs he continued his research and prepared lectures on set-theoretic topology, the theory of algebraic and Abelian functions, and group theory. He returned to Budapest on 21 March 1921 and applied for a position in the Faculty of Mathematics and Natural Sciences of the Ferenc József University. Let us explain what had happened to this university.

The Ferenc József University had been founded in 1872 in Kolozsvár, a city in Hungary which is now named Cluj-Napoca and is in Romania. After World War I Kolozsvár was taken over by Romania and became Cluj. This was formally recognised by the Treaty of Trianon in June 1920. The Ferenc József University moved to Budapest as a temporary measure in 1919 then, on 10 October 1921 formally opened in Szeged.

Supported by Alfréd Haár and Frigyes Riesz, Kerékjártó habilitated at the University of Szeged in December 1921 becoming a privatdozent. He delivered his inaugural lecture On the Set-Theoretic and Topological Foundations of Analysis and Geometry on 15 December. He then took up a scholarship which supported his study abroad. In the summer semester of 1922, at the invitation of the University of Göttingen, he gave a course on topology and the following semester a course entitled Mathematische Betrachtungen zur Kosmologie Ⓣ. The first of these courses was enlarged into a book Vorlesungen über Topologie Ⓣ which appeared in the series Grundlehren der Mathematischen Wissenschaften Ⓣ in 1923. Kerékjártó writes in the Preface (dated August 1923) [14]:-
The present work originated from the lectures I gave at the invitation of the University of Göttingen during the summer semester of 1922. H Kneser, J Nielsen, and K Reidemeister provided me with valuable advice during the reading and revision of the manuscript; I hereby express my sincere thanks to them. Furthermore, I am grateful to Professor R Courant for his kind invitation to publish this book in his collection. Finally, I am thankful to the publishing house, which accommodated my wishes and excellently resolved the technically difficult issue of the book's design.
This work was the first of its kind and inspired much later research in this new branch of geometry.

Solomon Lefschetz was at that time writing his own famous monograph on topology in 1924 entitled L'analysis situs et la géométrie algébrique Ⓣ and he wrote a review (published in 1925) of Kerékjártó's book for the Bulletin of the American Mathematical Society [17]:-
This production, from the pen of a young Hungarian mathematician, who is beginning to be known for his contributions to analysis situs, is welcome for several reasons. There are altogether too few books on the subject and one more is decidedly in place. It also gives under one cover a fairly complete treatment of the results obtained by Brouwer and his school on two dimensional topology, a useful thing indeed. The many and well chosen examples and figures are another good feature. For the sake of the average reader at least, we wish that the author had better amalgamated his material and introduced greater unity in his presentation. The 'Topologie' will be especially useful to the reader familiar with point sets and wishing to learn more about their geometric applications, and also, say, in connection with Veblen's Colloquium Lectures.

The material in the book may be essentially classified into three groups: (a) Topology of the plane and its curves, centring around the Jordan curve theorems and including such questions as invariance of dimensionality and regionality, structure of regions and their boundaries, the general closed curve, etc. (b) Combinatorial analysis situs of two-dimensional manifolds. The treatment of this part is less felicitous than Veblen's in Chapter II of his Lectures.
This is a pretty positive review from a leading expert. However, others had much more negative views. Hans Freudenthal writes [8]:-
... everyone knew that Kerékjártó's 'Vorlesungen über Topologie' Ⓣ was not a good book and, therefore, nobody read it. The present author has held this opinion for many years and now feels obliged to make a closer examination of Kerékjártó's works. The book opens with a proof which is unintelligible and probably wrong. This, indeed, is the worst possible beginning, but it continues in the same way. The greater part of Kerékjártó's own contributions are hardly intelligible and most apparently wrong. The work of others is often taken over almost literally or in a way which proves that Kerékjártó had not really assimilated the material. The level of the book is far below that of topology at that time, and the organisation is chaotic. When referring to a concept, a notation, or an argument, he often quotes a proof, a page, or an entire chapter - but often the material quoted is not found where he cites it ...
When considering the importance of the book, I [EFR] would tend to put more weight on Lefschetz's review written shortly after it was published than that of Freudenthal written fifty years later. Two specific comments relating to Freudenthal's opinions. First, Hermann Weyl wrote that Kerékjártó's book completely changed his views on topology. Second, the index contains a reference to the mathematician Erich Bessel-Hagen. Turning to the indicated page, there is no mention of Bessel-Hagen. However, there is a diagram of a torus with large handles attached on the sides looking a bit like a face with oversized ears. In fact Bessel-Hagen was renowned for having large ears that stuck out of his head. Perhaps Freudenthal missed the jokes!

After Kerékjártó's visit to Göttingen in 1922, in the following year he gave courses at the University of Barcelona entitled Geometry and The theory of functions. On his visits to Göttingen he had met Lisette Kronbauer (1906-1998), the daughter of Richard Friedrich Kronbauer who had a bookstore in Göttingen. When he went from Budapest to Barcelona in 1923, Kerékjártó had travelled via Göttingen where he became engaged to Lisette on 28 April. They married in Göttingen on 6 December 1924. They had two children: Margit Kerékjártó (1926-2009), who became a professor of medical psychology in Hamburg; and Béla Kerékjártó (born 1929), who became a chemist who worked in the pharmaceutical industry.

Kerékjártó had perfected his knowledge of English while living with the Youngs, and he submitted his first paper written in English Note on continuous transformations to the London Mathematical Society in November 1922; it was published in 1924. The paper extends results published by William Young in 1919.

Kerékjártó spent the year 1923-24 on a Rockefeller scholarship as a visiting lecturer at Princeton where he gave courses on topology and on continuous groups. While in the United States he visited New York University four times, and also visited Buffalo, Chicago, Washington and Baltimore. He left the United States, returning to Europe where he had an invitation to visit the Sorbonne in Paris. With his Lessons on topology and its applications he gained the respect of the most distinguished French mathematicians. His friendship with foreign mathematicians continued to deepen for the rest of his life. After Paris he returned to Budapest in August 1924 via Göttingen. He was back in Göttingen in December when he was married and, after a honeymoon in Goslar in Lower Saxony, Kerékjártó and his wife spent Christmas in Göttingen. He then returned to Princeton when he was given the title of honorary professor. Returning to Europe in June 1925 he spent the summer at the University of Hamburg but also spent time in Göttingen, Milan and Venice.

On 14 October 1925, Kerékjártó was appointed to the professorship of the Chair of Geometry and Descriptive Geometry at the University of Szeged. The Department of Mathematics at that time consisted of the Mathematical Seminary and the Institute of Descriptive Geometry. Before Kerékjártó's appointment, there were four members of staff in mathematics: Frigyes Riesz, Alfréd Haar, Rudolf Ortvay (1885-1945), who held the Chair of Mathematical Physics, and Tibor Radó, Haar's assistant and the only assistant in the Department. Examples of papers Kerékjártó published during this time are On a geometrical theory of continuous groups (1925) and On a geometrical theory of continuous groups. II. Euclidean and hyperbolic groups of three-dimensional space (1927) both in the Annals of Mathematics. Kerékjártó remained in Szeged until 1938 and we have this nice description from Wilfred Kaplan who spent the academic year 1936-37 abroad. Kaplan writes that he went to Hungary to see "Professor Bela von Kerékjártó." In a letter written on 18 June 1937 he wrote [11]:-
... I went to the Hotel Gellert (by street-car) and found Professor Kerékjártó who greeted me in a very friendly fashion. He looks quite young, is tall, resembles Gary Cooper. ... We went to the Institute of Physics in the University of Budapest. Here I was introduced to a number of professors of mathematics and physics, gathered together for their monthly congress. ... Sunday morning I took a train to Gyöngyös and there transferred to a bus (of very odd primitive construction) by which I was transported to Matrafüred. Here I was greeted by Professor Kerékjártó, who had arrived the night before. ... We walked together along a path through the woods which led to the pension at which I was to stay. ... I give a picture of a full typical day. I awake at 7.30, wash with cold water and go into the dining room for breakfast. ... At 9 I stroll over to Professor Kerékjártó's home. He has a little house here where he lives with his wife, son (9) and daughter (11). The family greets me and we talk a bit in German. ... they are charming, informal people. Then Professor Kerékjártó and I have a long discussion about some mathematical problems (these have been quite fruitful). ...
In 1938, Kerékjártó was appointed full professor at the University of Budapest. He was succeeded in Szeged by Gyula Szõkefalvi-Nagy (1887-1953), a distinguished geometer, founder of the theory of curves with maximal index, formerly professor of the Teacher's Training College in Szeged. The proposal for Kerékjártó's appointment to Budapest was made by Lipót Fejér who wrote:-
It is clear that he is a constant and intensive individual immersed in scientific research, who rendered an excellent service to science primarily with his original geometric research, but also with his high-level monographic works. ... with his talent he earned not only for himself, but also for his university, an excellent recognition in the scientific world, worthy of the old traditions of Kolozsvár.
Kerékjártó spent the rest of his career in Budapest which was sadly much shorter than it might have been since he died in 1946 at the age of 47. Kerékjártó's mother was Jewish so one would have expected him to be in difficulties, particularly when Budapest was occupied by the Nazis during World War II. His mother had been baptised into the Christian faith before her marriage so, since he had no problems, it appears that the German authorities were unaware of his Jewish connections.

While in Budapest, he seems to have had a difference of opinion with his colleague Lipót Fejér who once remarked:-
What Kerékjártó says is only topologically equivalent to the truth.
An overview of Kerékjártó's contributions is given in [20]. We provide a fairly free translation. His most important scientific results were in the area of "classical" topology founded by Poincaré and Brouwer and in the theory of continuous groups. After the problem of classifying surfaces up to homeomorphism had been solved (and that for open surfaces resolved by Kerékjártó) it became possible to study more deeply the structure of the transformations of such surfaces. At the beginning of Kerékjártó's research there were two classic results in this area. The first was "Poincaré's last geometric theorem", later proved by G D Birkhoff, and the second was "Brouwer's translation theorem". Kerékjártó had earlier shown, in his book, the close relationship between these two theorems, and in an article in Acta Universitatis szegediansis, Acta scientiarum mathematicarum Ⓣ in 1928, he showed how they could have a common proof. This demonstration of the simultaneous proof of the two results united some widely separated areas of mathematics. In fact, while the first theorem played an important role in Poincaré's research in dynamics, the second was applied by Brouwer in the theory of continuous groups. Also in his later work he preferred to work in topological problems which were closely connected with problems of classical geometry, theory of functions, etc. The famous speech by Hilbert at the International Congress in Paris had been of great importance to the development of topology. It had also given impetus to the development of the theory of continuous groups. The most beautiful results of this theory, in the case of dimension 2, are due to Kerékjártó. It is enough to mention the topological characterisation of the homographic representations of the sphere and of the affine group of the plane, the foundations of complex projective geometry and theorems on the transitive groups of the line. It was these methods which also led to fundamental results on topology and Euclidean and hyperbolic geometry in 3 dimensions. He also studied the regular transformations of surfaces, which are closely connected to the above problems and which lead to interesting applications in dynamics. His expertise in this new branch of geometry, topology, was recognised in, among other ways, his being asked to write the chapter on 'Topology' in the Encyclopédie Francaise.

His final work was intended to be a series of five books, only two of which were written before his death. The first was written in Hungarian and published in 1937. However a French translation was published in 1955 under the title Les fondements de la géométrie Tome I. La construction élémentaire de la géométrie euclidienne Ⓣ. It receives high praise from H Busemann in the review [4]:-
From the very modest introduction one would never guess that this is one of the richest works on the foundations of geometry. The introduction states that the author essentially follows Hilbert; it lists as major deviations only that angle does not appear as primitive concept, but is defined, and that the existence of motions (based on congruence of segments) is postulated instead of the first congruence theorem for triangles. Actually, the book contains a wealth of unusual material. ... The exposition is meticulous and quite easily readable ...
The second volume The Foundations of Geometry. Volume Two. Projective Geometry was published in Hungarian in 1944. E Lukacs begins his review by writing [18]:-
This is the second volume of a treatise on the foundations of geometry; the preceding volume dealt with Euclidean geometry. The classical projective geometry is developed in great detail so that this book can also be used as a text book. The author's aim is to give a foundation of projective geometry on which it is possible to build either Euclidean, hyperbolic or elliptic geometry. The greater part of the book is confined to the discussion of real projective geometry. The author avoids imaginary elements because, on the chosen axiomatic basis, their use could only mean a change in terminology. An analytical discussion of complex projective geometry is given separately.
In May 1944 Kerékjártó took his wife, two children and his mother to Mátrafüred, where his sister and her husband worked as doctors, to avoid the bomb threat in Budapest. He returned to Budapest to complete his lectures before joining his family in Mátrafüred. He returned to Budapest in March 1945 to take up his university position and his family joined him there in June. Life was not easy since his apartment on Ráth György Street had become uninhabitable due to artillery fire during the fighting, and most of its furnishings had been destroyed. The University provided the family with temporary accommodation. In September 1945 he became ill with pulmonary tuberculosis and in November of that year he entered the sanatorium in Mátraháza where his sister and her husband worked. He underwent an operation on 11 February 1946 but did not live long after it. He was buried in the sanatorium cemetery two days after his death. In 1957 his ashes were exhumed and buried next to his father in the famous Megyeri cemetery in Budapest.

Kerékjártó was honoured by being elected a corresponding member of the Hungarian Academy of Sciences in 1934 and a full member of the Academy in 1945. The Academy awarded him their Zsigmond Kornfeld Prize in 1940. He was also an editor of Acta Universitatis szegediansis, Acta scientiarum mathematicarum Ⓣ from 1933 until his death.

The authors of [20] end their obituary with these words:-
Death has taken Béla de Kerékjártó when he was still at his most creative. We do not know what plans he had in mind, but certainly his death has deprived mathematics of some really irreplaceable works.
Other Mathematicians born in Hungary
A Poster of Béla Kerékjártó

References (show)

  1. Béla de Kerékjártó, Acta Scientiarum Mathematicarum 11 (1-2) (1946), 128.
  2. Béla Kerékjártó, Prabook (2026).
    https://prabook.com/web/bela.kerekjarto/2483119
  3. M Bognár, Béla Kerékjártó (1898-1946), in Ferenc Glatz (ed.): Memorial speeches on the deceased members of the Hungarian Academy of Sciences (MTA Budapest, 1998).
  4. H Busemann, Review: Les fondements de la géométrie Tome I. La construction élémentaire de la géométrie euclidienne, by Béla Kerékjártó, Mathematical Reviews MR0077126 (17,995a).
  5. A Csákány and A Varga, A szegedi egyetemi matematikai intézetek hetvenöt éve, in László Szentirmai (ed.), The past and present of the University of Szeged 1921-1998 (József Attila University, Szeged, 1999), 380-397.
  6. A Filipiak, Following Béla von Kerékjártó. The Journeys of a Hungarian Mathematician in the Post-war World, in Laurent Mazliak and Rossana Tazzioli (eds), Mathematical Communities in the Reconstruction after the Great War. 1918-1928. Trajectories and Institutions. Trends in the History of Science (Birkhäser, 2021), 277-306.
  7. J Fráter, A Magyar Tudományos Akadémia állandó bizottságai 1854-1949 (Publications of the Library of the Hungarian Academy of Sciences, Budapest, 1974).
  8. H Freudenthal, Béla Kerékjártó, Dictionary of Scientific Biography (New York 1970-1990).
  9. G Gáll, Béla Kerékjártó (A biographical sketch), Teaching Mathematics and Computer Science 2 (2) (2004), 231-263.
  10. G Gáll, Béla Kerékjártó, in Ferenc Nagy (ed.), Hungarians in the history of natural science and technology (National Technical Information Centre, Budapest), 266-267.
  11. W Kaplan, Recollections of a Year of Study in Europe, 1936-37, in Bettye Anne Case, A Century of Mathematical Meetings: Published in connection with the 100th annual meeting of the American Mathematical Society, held in Cincinnati, January 1994 (AMS Bookstore, 1996), 171-177.
  12. B Kerékjártó, Természeti törvény és mathematika, in Tibor Széki (ed.), Report on the operation of the Szeged Royal Ferencz József University in 1933-34 (Royal Ferencz József University, Szeged, 1933), 120-130.
  13. B Kerékjártó, Paul Valery láogatása Budapesten, Budapest Magazine 1 (2) (1945), 55-58.
  14. B Kerékjártó, Vorlesungen über Topologie (Verlag von Julius Springer, Berlin, 1923).
    https://archive.org/details/vorlesungenubert00kere/page/n7/mode/2up
  15. I G Kovács, Magyarországi zsidó és zsidó származású egyetemi tanárok
  16. I G Kovács and A Takács, Kerékjártó Béla (1898-1946) matematikus, in Az "összeilleszkedés" változatai. Az akkultur ációtól az asszimilációig (ELTE Eötvös Kiadó, 2022), 115-141.
    https://www.eltereader.hu/media/2023/06/Kovacs-I.-Gabor-Az-osszeilleszkedes-valtozatai-web.pdf
  17. S Lefschetz, Review: B von Kerékjártó, Vorlesungen über Topologie. I. Flächentopologie, Bull. Amer. Math. Soc. 31 (3-4) (1925), 176.
  18. E Lukacs, Review: A Geometria Alapjairól. Második Kötet. Projektív Geometria, by Béla Kerékjártó, Mathematical Reviews MR0023534 (9,369g).
  19. P Neerup, Review: Les fondements de la géométrie. I. La construction élémentaire de la géométrie euclidienne, by Béla Kerékjártó, Mathematica Scandinavica 5 (1) (1957), 116-118.
  20. Obituary of Béla Kerékjártó, Acta Universitatis szegediansis, Acta scientiarum mathematicarum 11 (1946-48), 5-7.
  21. The history of the Institute, Bolyai Institute, University of Szeged (2021).
    https://www.math.u-szeged.hu/mathweb/index.php/en/component/content/article/24-erdekessegek/52-a-bolyai-intezet-toertenete
  22. J A Todd, Review: Les Fondements de la Géométrie. Tome deux. Géométrie Projective, by Béla Kerékjártò, The Mathematical Gazette 52 (379) (1968), 88.

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Written by J J O'Connor and E F Robertson
Last Update March 2026