**Béla Kerékjártó** is sometimes called Béla von Kerékjártó or, in French, Béla de Kerékjártó. He studied at the University of Budapest, being awarded his doctorate in 1920. He habilitated at the University of Szeged in 1922 becoming a privatdozent, 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 titled *Mathematische Betrachtungen zur Kosmologie* (Mathematical considerations in Cosmology). The first of these courses was enlarged into a book *Vorlesungen über Topologie* (Lectures on Topology) which appeared in the series *Grundlehren der Mathematischen Wissenschaften* in 1923. 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* [4]:-

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, centering 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 [1]:-

... 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*. He spent the years 1923-24 and 1924-25 as a visiting lecturer at Princeton where he gave courses on topology and on continuous groups. 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.

In 1925, Kerékjártó was appointed to full 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. It had opened in Szeged in October 1921 after the Treaty of Trianon had given Kolozsvár (now Cluj) to Romania so the Hungarian University moved from Kolozsvár to Szeged. Before Kerékjártó's appointment, there were four members of staff in mathematics: Frigyes Riesz, Alfréd Haar, Rudolf Ortvay, who held the Chair of Mathematical Physics, and Tibor Radó, Haar's assistant and the only assistant in the Department. Examples of paper 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 [2]:-

... 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 at7.30, wash with cold water and go into the dining room for breakfast. ... At9I 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, a distinguished geometer, founder of the theory of curves with maximal index, formerly professor of the Teacher's Training College in Szeged. 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. While in Budapest, he seems to have had a difference of opinion with his colleague Lipót Fejér who once remarked:-

What Kerékjártósays is only topologically equivalent to the truth.

An overview of Kerékjártó's contributions is given in [5]. 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 a review:-

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:-

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.

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. He was also an editor of *Acta Universitatis szegediansis, Acta scientiarum mathematicarum * from 1933 until his death.

The authors of [5] 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.

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

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