# Ralph Hartzler Fox

### Quick Info

Born
24 March 1913
Morrisville, Pennsylvania, USA
Died
23 December 1973

Summary
Ralph Fox was an American mathematician who made important contributions to differential topology and knot theory.

### Biography

Ralph Fox was born into a Quaker family. He was educated at home and did not attend school. When he was an adult he was called to testify in favour of the soundness of home education at a court case in Virginia; he certainly provided a good example that such an education can be highly effective. He was a very musical child and showed an extraordinary talent as a pianist. He began his formal education at Swarthmore College, originally founded by Quakers, in Swarthmore, Pennsylvania, but at the same time he studied piano at the Leefson-Hille Conservatory of Music in Philadelphia. In fact Swarthmore is less than 20 km from Philadelphia, so geographically this was not difficult, and Fox continued with these two studies for two years.

After taking his first degree from Swarthmore, Fox went to Johns Hopkins University where he studied for a Master's Degree in mathematics before going to Princeton where he undertook research advised by the topologist Solomon Lefschetz. He also became an expert in the board game of Go while at Princeton (he later represented the United States in the first international Go tournament in Tokyo in 1963). He graduated with a Ph.D. from Princeton in 1939 after submitting his thesis On the Lusternik Schnirelmann Category. In this work he made an extensive study of the category of subsets of a space, and the relation to homotopy and homology properties. The thesis contains theorems on homotopy properties of mappings, particularly of mappings of sets into absolute neighbourhood retracts. Relations with the "category" of a set, and with particular deformations, "expansions," are studied. He published a number of papers related to the work he had undertaken for his doctorate: On homotopy and extension of mappings (1940), On the Lusternik-Schnirelmann category (1941), Extension of homeomorphisms into Euclidean and Hilbert parallelotopes (1941), and Topological invariants of the Lusternik-Schnirelmann type (1941).

While he was a student, Fox had married Cynthia Atkinson; they had one son Robin. Fox spent the year after the award of his doctorate at the Institute for Advanced Studies at Princeton. Then he was appointed to the University of Illinois, moving to a position at Syracuse University before returning to Princeton in 1945. He remained on the Princeton University faculty for the rest of his life. The year after Fox was appointed as a lecturer at Princeton, Emil Artin left Indiana University at Bloomington and was appointed to Princeton. Artin had worked on the braid group as early as the 1920s, publishing a fundamental paper on the topic in 1926. After arriving in Princeton he began to think again about braids and he began working with Fox on certain topological ideas. They published the joint paper Some wild cells and spheres in three-dimensional space (1948). Epple writes [1]:-
On the one hand, they asked for a clear delineation of the domain of knot theory within three dimensional topology; on the other, it hinted as a relation between the complements of (maybe wild) knots or knotted arcs and the Poincaré conjecture.
Fox became the leader of a very active group of young mathematicians at Princeton studying knots, links and three dimensional topology. In 1950 the International Congress of Mathematicians took place in Cambridge, Massachusetts, and Fox was an invited speaker giving the lecture Recent development of knot theory at Princeton on the work of his Princeton group. In this lecture he pointed the way towards the new approach that they were taking:-
... classical knot theory tends, by its narrowness, to isolate the subject from the rest of topology. It is hoped that the various special theorems which make up classical knot theory will eventually turn out to be particular cases of general topological theorems. In working towards this end the following principles seem obvious: (A) The class of polygons should be replaced by a suitable topologically defined class of curves. ... (B) Euclidean 3-space should be replaced by other compact 3-manifolds. ...
In the same talk he reported on work of his Princeton group on the algebraic structure of the knot group and presentations of this group. These ideas had not been published at the time Fox gave this talk but over the next few years he published them in a number of important articles. He also mentioned in his talk to the International Congress of Mathematicians his ideas on how to study group presentations using the "free differential calculus." Five papers on this topic appeared over the following ten years: Free differential calculus. I. Derivation in the free group ring (1953); Free differential calculus. II. The isomorphism problem of groups (1954); Free differential calculus. III. Subgroups (1956); (with Roger Lyndon and Kuo-Tsai Chen) Free differential calculus. IV. The quotient groups of the lower central series (1958); and Free differential calculus. V. The Alexander matrices re-examined (1960). In the same 1950 lecture, Fox reported on work by his young student John Milnor on the total curvature of a knot. Milnor had, at this time, only just begun research for his doctorate which he was awarded in 1954.

In the summer of 1951, Fox went to the Mathematics Institute of the Universidad Nacional Autónoma de México in Mexico City where he gave a series of lectures, supported by a Guggenheim Fellowship. This was part of a year of festivities which the university organised to celebrate the Fourth Centenary of the Royal Pontifical University of Mexico. During these celebrations the university awarded honorary degrees to several leading mathematicians including Garret Birkhoff, Solomon Lefschetz and Norbert Wiener. A Fulbright grant allowed Fox to lecture at the Universities of Delft and Stockholm in 1952. In the following year he lectured to the Summer Seminar of the Canadian Mathematical Society.

In the spring of 1956 Fox was invited to give a series of lectures at Haverford College (an institution founded on Quaker values on a campus just outside Philadelphia) under the Philips Lecture Program.The Philips Grant consists of funds left by Haverford alumnus William Pyle Philips (who graduated in 1902) for (i) the purchase of rare books which the college would not otherwise buy and (ii) to invite distinguished scientists and statesmen to Haverford. Niels Bohr, Enrico Fermi and Erwin Schrödinger had earlier been invited as Philips Lecturers. Fox's lectures were written up as a book, with Richard Crowell, a doctoral student of Fox's who was awarded his doctorate in 1955, as co-author. The book Introduction to knot theory was published in 1963 and became a classic being reprinted in 1977. A Russian translation had been published in 1967. Lee Neuwirth writes in a review of the 1963 edition:-
This book is in many ways a very surprising work. In the first place it places topology, algebraic topology and knot theory in the position of solving a "practical problem''. ... In the second place, for a very innocent task, namely distinguishing knots, a quite respectable amount of interesting and sophisticated mathematics is introduced. ... In the third place, while there is a great emphasis on practicality, and informal discussion prior to making a definition or proof is frequent, there is no sacrifice of rigour. This book is on the side of formality for necessity's sake, rather than formality for its own sake.
H F Trotter begins his review [5] by explaining what knot theory attempts to solve:-
It is a famous theorem that any two embeddings of a circle in the plane are topologically equivalent. In contrast, it is intuitively obvious that a circle embedded in three-dimensional space can be knotted in many different ways. Mathematical curiosity at once prompts the question: "How can two knots be proved to be really different?" This natural and apparently innocuous question poses the central problem of knot theory and leads to algebraic developments of surprising variety and richness. The authors have succeeded admirably in their purpose of providing an introduction to this fascinating subject at a level accessible to graduate students and advanced undergraduates.
It was Fox's work on the braid group that formed the starting point for Joan Birman's doctoral dissertation of 1968 as her abstract shows:-
In 1962 R H Fox introduced a new definition of the braid group as the fundamental group of the space of $n$ unordered distinct points of the Euclidean plane. Fox's definition suggested a natural generalization to the concept of a braid group on an arbitrary manifold, as the fundamental group of the space of $n$ unordered, distinct points of that manifold. The present investigation begins with Fox's definition, and studies the algebraic and geometric properties of these braid groups on arbitrary manifolds.
Visits by Fox to many places around the world proved important for research collaborations but none more so than his visits to Japan (see, for example [3] and [4]). As a result of these visits, several Japanese knot theorists visited Princeton to collaborate with Fox's group there. In fact Fox was very friendly and encouraging toward all young topologists and many of them received invitations to visit Princeton. His first visit to Japan was, as we mentioned above, as an American representative to the World Go tournament in October 1963. While he was in Japan the universities of Tokyo, Osaka, Nagoya, Kyoto and Fukuoka all issued invitations to him to lecture on knot theory.

Lee Neuwirth, from whom we quoted above, was one of Fox's doctoral students, being awarded a Ph.D. in 1959. Other doctoral students of Fox with biographies in this archive include John Stallings (Ph.D. 1959) and Barry Mazur (Ph.D. 1959). Fox's health deteriorated and in 1973 he entered the University of Pennsylvania Graduate Hospital to undergo open-heart surgery. He died in the hospital following surgery. Following his death Lee Neuwirth edited Introduction to Knots, Groups, and 3-manifolds, papers dedicated to the memory of R H Fox which was published by Princeton University Press in 1975. Neuwirth gives the following fine tribute to Fox in the Introduction [2]:-
The influence of a great teacher and a superb mathematician is measured by his published works, the published works of his students, and perhaps foremost, the mathematical environment he fostered and helped to maintain. In this last regard Ralph Fox's life was particularly striking: the tradition of topology at Princeton owes much to his lively and highly imaginative presence. Ralph Fox had well defined tastes in mathematics. Although he was not generally sympathetic towards topological abstractions, when questions requiring geometric intuition or algebraic manipulations arose, it was his insights and guidance that stimulated deepened understanding and provoked the development of countless theorems.
Let us end this biography by quoting the tongue-in-cheek advice that Fox gave Joan Birman when she complained that she was having to serve on too many committees. He said:-
Speak often and not to the point, and soon they will drop you from all the committees.

### References (show)

1. M Epple, Geometric aspects in the development of knot theory, in I M James (ed.), History of topology (Elsevier, 1999), 301-358.
2. L P Neuwirth, Introduction, Knots, Groups, and 3-manifolds, papers dedicated to the memory of R H Fox (Princeton University Press, Princeton, N.J., 1975).
3. J H Przytycki, Notes to the early history of the Knot Theory in Japan. http://www.shsu.edu/~mth_jaj/math467/daniel_article.pdf
4. J H Przytycki, The interrelation of the Development of Mathematical Topology in Japan, Poland and USA, Annals of the Institute for Comparative Studies of Culture, Tokyo Woman's Christian University 63 (2002), 61-86.
5. H F Trotter, Review: Introduction to knot theory by R H Fox, Amer. Math. Monthly 71 (10) (1964), 1146.