by Allyn Jackson
Everyone likes to talk to André Haefliger. His mathematical insight and vivid enthusiasm have for decades made him a beloved interlocutor among mathematicians. But talk to him about anything — music, cheese, books, people, the state of the world — and you will find the conversation buoyed along by his singular intellect as well as his joyous laugh. Likely taking part in the conversation is André’s wife Hermina (née Polakovitch), universally known by her nickname Minouche, whose intelligence, warmth, and charm increase the enjoyment.
In his 1958 PhD thesis, written under the direction of Charles Ehresmann, André Haefliger obtained ground-breaking results in foliation theory. After making significant contributions to embedding and immersion theory, he returned to foliations to again produce fundamental advances, including what is now known as the Haefliger classifying space. His work in geometric group theory set the stage for many advances in this now-burgeoning subject.
Haefliger spent two postdoctoral years as the assistant of Hassler Whitney at the Institute for Advanced Study, where his two main duties were to do mathematics and to play chamber music with Whitney. In 1961 he accepted a professorship at the University of Geneva and remained there, apart from frequent visits abroad, until his retirement in 1995.
The following article is based on a series of interviews with Haefliger, conducted by Allyn Jackson. The interviews took place in May 2018, at the Haefliger home near Nyon, Switzerland. Minouche was present during portions of the interview, as was D. Kotschick of the University of Munich.
Early life
André Haefliger was born in 1929 in Nyon, Switzerland, and
since the 1960s has lived in a small village less than a mile
away. Nyon sits on the northern shore of Lake Geneva about 15
miles east of Geneva and about 25 miles west of Lausanne. His
father worked in a bank and was good at catching arithmetic
mistakes; he once found that the date had been added to an account
balance. André’s mother, who was about 20 years younger than
her husband, also worked at the bank, although she stopped working
when her children were born. The couple already had two children,
a boy and a girl, when André and his twin brother were born on
May 22, 1929. At that time their father was 60 years old.
André’s older brother, now deceased, studied law and at the end
of his life worked in Lausanne as the head of the Land
Registrar of
the Canton of Vaud. André’s sister, who was the one who taught
André to read, loved children and became a teacher. She still
lives in Nyon near André and Minouche. They also remain close
to one of his cousins, who as a girl was adopted by André’s
mother. Now in her 90s, this cousin lives in Florence.
Haefliger: My twin brother had psychiatric problems. He had schizophrenia. It was very difficult for the whole family. Now he lives in an apartment close to us and is doing much better. This is thanks to my oldest son, Ivan, who is a medical doctor.1 He takes care of my brother and does everything for him, paying bills and so on. He is very devoted to helping people. It is in his nature!
Jackson: That’s good. So your twin brother could not do any studies?
Haefliger: He did some studies, but he had difficulties. But he is very intelligent. He used to read poetry, for example Baudelaire. He could remember many poems by heart. He also had a talent for imitating people. He would go to the front of the little church in our neighborhood and imitate the pastor! He used to play the flute and was very interested in music. He still is.
Jackson: He wasn’t interested in mathematics?
Haefliger: No, not at all.
Jackson: Did you have some early influences inspiring you to study mathematics?
Haefliger: Yes, I had a very good teacher when I was 16. He was very enthusiastic and competent. At that time I liked to go to the public library. I used to borrow the Disquisitiones Arithmeticae!
Jackson: Did you read it?
Haefliger: I tried! It was in Latin! Do you know Jean-Luc Godard?
Jackson: The filmmaker?
Haefliger: Yes. He also lived in Nyon. We were very good friends. My best friend married his sister. Godard made a documentary on René Thom.
Jackson: Did you suggest that to Godard?
Haefliger: No, I did not suggest it, but I used to see him very often in Paris, and we would talk together. Thom had great success with catastrophe theory. Godard was interested in making documentaries of people. You can listen to this interview on YouTube. It’s terrible! Terrible for Thom! Thom is trying to explain with great patience some basic catastrophe theory, and Godard is making comments. It was a big chore for Thom!
Jackson: You went to a classical secondary school, with Latin and ancient Greek.
Haefliger: Yes, in Nyon. We had very good teachers. For instance one was a specialist in Sanskrit, but he taught in Nyon because he had no position in the university. Afterward I went to the Collège Calvin in Geneva.
Jackson: Was that a religious school?
Haefliger: No. At that time there was only one Gymnasium, and it was in the Collège Calvin, a prestigious school that still exists.
When I arrived in Geneva from Nyon, the mathematics teacher told me, “Don’t push, don’t come too early into mathematics.” The reason is that there was in Geneva a genius in mathematics, who was pushed that way and got his doctorate in mathematics one year after his maturité.2 He had written a thesis that was interesting but not first-class. Then he was sent to Chicago to André Weil. It was terrible for him. André Weil was very tough!
Jackson: So your teacher gave you good advice.
Haefliger: Yes — “Don’t do like this guy!”
Jackson: Did you start studying violin very early?
Haefliger: Yes, as a child. My mother lost her father when she was 10 years old. She had begun piano, but could not continue. Her mother had no pension and therefore had to work very hard. My mother loved music, so she was insistent that all the members of our family learn an instrument.
Jackson: And your father, did he also play music?
Haefliger: Yes, he played in the Nyon orchestra. But I don’t remember him playing, he was too old at that time.
Jackson: You finished your studies at the Collège Calvin, and then you went to the University of Lausanne. Why did you not go to the University of Geneva?
Haefliger: Because the only possibility I thought I could have was to become a teacher in a secondary school. The canton of Vaud is much bigger than the canton of Geneva, so there would be more possibilities to get a job. This was before the explosion of universities, and I didn’t have the idea I could have an academic career.
Jackson: When you entered university, you had decided to study mathematics?
Haefliger: Sure. There was no question.
Jackson: But you also got a certificate from the conservatory in violin.
Haefliger: Yes, during my studies in Lausanne, I got a certificate.
Jackson: Did you consider becoming a musician?
Haefliger: No. I realized mathematics was much easier!
Jackson: Too much practicing in music?
Haefliger: Yes, and it’s hard to find a job. I never considered it. But I played with many people, including my teachers in Lausanne. Later on I played with [Henri] Cartan, [William] Browder, [Michael] Artin, [Hassler] Whitney, [Eugenio] Calabi, Dusa McDuff, as well as the wives of some of the mathematicians. Music is really a way of meeting people. Minouche also took part, as she plays piano, violin, and viola. She and I have played chamber music together ever since we first met.
Jackson: That’s wonderful to be able to share that. What did Cartan play?
Haefliger: Cartan played piano. He was a very good pianist. He had a brother who was a composer and had contact with many musicians in Paris.
Jackson: You took many courses with Georges de Rham in Lausanne. What were your initial impressions of him?
Haefliger: He was a very impressive man. He had a seminar, and we had to prepare subjects to present. He was very demanding, very precise, but still very kind. Later he became a friend. Unfortunately in his last years he completely lost his memory. We invited him regardless, very often. Once he forgot to come. One of our colleagues was here and went to pick him up. De Rham had already eaten, but he ate again with us! His memory loss was difficult, because he realized it. So it was very sad. He was a great alpinist. He loved to read Mallarmé.
Jackson: You had to do army service around this time. What did you have to do?
Haefliger: I was in aerial defense. After one month, the captain asked us to write our opinion about something. So I wrote, “I want to be in the Communist Party.” After that I was considered a terrible danger! So they put me in the kitchen! This was not long after the war. Switzerland was really afraid of communism.
Jackson: But you were not afraid of saying this?
Haefliger: No. At the end of this army service, you got a grade. My grade was “surnuméraire”[redundant]! Also during this army service, I was injured when a cannon got pushed onto my feet, so I had to go to the hospital. This was marvelous, because I did mathematics! I don’t know how I got books, but I remember reading Felix Klein and other classic books.
Jackson: Going back to your university work, how did you decide to go on to get a doctorate?
Haefliger: At first I thought I would do secondary teaching. It was Jean de Siebenthal who told me, “No, you should get an academic degree.” So I asked for a stipendium, and de Rham helped me with that. I did not want to do a doctorate with de Rham. I wanted to be more free. He recommended to me not to go to Paris. He said, “You will be lost there, it’s better for you to go to Strasbourg.” He was a very good friend of Ehresmann, who was at Strasbourg then. In fact I was also very much interested in the papers of Ehresmann. I studied them in a seminar with de Siebenthal.
Jackson: So Strasbourg was a good choice for you.
Haefliger: Yes. It was fantastic. At that time [1954], I was the only graduate student. Wu Wentsun and Georges Reeb had been students in Strasbourg, but they had left by then. It is strange that I was the only student! There were very good people there: [Jean-Louis] Koszul, [Georges] Cerf (the father of Jean Cerf), and René Thom, of course. And Ehresmann invited many, many people.
At this time I got in very good contact with Thom. Later Ehresmann was nominated as a professor in Paris, and I followed him, but I used to go very often to Strasbourg to discuss with Thom. Our difference in age was not so big. I was born in 1929, and Thom was born in 1923. Six years, it’s not so much. But when I met him, I knew almost nothing about mathematics! I was really a beginner. I didn’t know what direction to go in, how to orient myself.
Jackson: What were your impressions of Thom?
Haefliger: I admire his way of understanding mathematics. He would be listening to you and then would make remarks which were extremely helpful. And he was very kind. I can show you letters he sent me, many letters. I became like a confidant for him — I don’t know why! It was a time when he was very much interested in singularities. A concrete part of my thesis used the theory of singularities of differentiable mappings.
We were friends with him and his family, until the end. We spent hours and hours with him when he was very ill, and we sometimes stayed overnight to help the family. He had a leg amputated and was in a wheelchair. And he lost his memory. It was very hard for his wife, who had a very clear mind.3
Reeb’s thesis an inspiration
“It was René Thom who explained to me Ehresmann’s ideas about holonomy during the winter of 1954–55 (Ehresmann was often traveling). I realized pretty quickly that, in the differentiable case, the holonomy of a proper leaf \( F \) (that is, a leaf such that the leaf topology coincides with the induced topology) permitted one to reconstruct the germ of the foliation along \( F \) and to characterize it. The following year I showed the non-existence of analytic foliations of codimension 1 on compact simply connected manifolds, in particular responding, in the negative, to the question Reeb had posed concerning the existence of an analytic foliation on the 3-sphere.”
— André Haefliger, “Naissance des Feuilletages, d’Ehresmann–Reeb à Novikov” (translated from the French).4
Jackson: Your thesis work was closely related to the work of Georges Reeb. Did you get to know Reeb?
Haefliger: Yes, yes.
Jackson: He was already in Grenoble when you were in Strasbourg. Did he come back to Strasbourg sometimes?
Haefliger: Yes. He was very kind and his wife also. He was truly Alsatian. He didn’t go to the École Normale, so he was really an outsider! Of course we saw him very often, in Brazil, in many places. He had a whole school around him. In the end, he was interested in nonstandard analysis. It was like a religion! He wanted to persuade people they should know about it.5
Jackson: Did you learn about foliations from Reeb?
Haefliger: Yes, by reading his thesis. I remember in 1954 I went to the International Congress in Amsterdam. I didn’t know the thesis of Reeb before that. I saw it on a table, and I opened it, and I found it fascinating. He posed several problems in his thesis.
I was also very much influenced [in my thesis] by Ehresmann’s way of thinking about mathematics. And people disliked that very much.
Jackson: Why?
Haefliger: Why? You should read the report of [Richard] Palais in Math Reviews!
Jackson: Oh yes, he wrote things like, “An extreme Bourbaki-like generalization…several hundred preliminary definitions…”
Haefliger: Yes, but he mixed up a few things! And my thesis was very long. When I arrived in Princeton, everybody was joking about this! Later Palais and I talked about this, and we had a very good relationship after that. He was sorry to have written that, because it’s clear that there was a confusion in his mind between \( H^0 \) and \( H^1 \)!
At this time I had a terrible conflict with de Rham. He wanted my thesis to be published in the Commentarii. When he heard that Reeb wanted to publish it in the Annales de l’Institut Fourier, he wrote me a terrible letter, saying that I was not thankful for the help the Swiss National Fund gave to me. This letter was so terrible that I destroyed it!
In the end, de Rham published my thesis in the Commentarii. He made a great effort to be very quick. At that time you could not get your degree before your thesis was printed. I had to send to the Sorbonne more than 100 copies of my thesis. And of course they are lost! It was the custom at that time.
Jackson: So your thesis did not appear in the Annales de l’Institut Fourier. What did Reeb think of that? Was he angry?
Haefliger: No, no — he was not that kind of man. He was very kind. I probably told him about the letter from de Rham, and he understood. But de Rham was right in a sense. He had helped me very much.
Jackson: What was Ehresmann like?
Haefliger: He was a very open and generous man. His office was open, anybody could come. He liked to work in restaurants in Paris. He had problems with his wife, who stayed in Strasbourg — she was a strong personality. So when he was in Paris he was alone. He invited me very often to very good restaurants, because he liked to eat good food, and we would discuss mathematics. We had a meal in a restaurant in Paris after my thesis defense.
When I discussed my thesis work with Ehresmann, he said, “If you could prove in every foliation on the 3-sphere there is a torus, then it is enough for your thesis.” I didn’t prove that!6 Reeb was a student of Ehresmann, and it was Ehresmann who introduced Reeb to the theory of foliations. When I was a student in Lausanne, de Siebenthal, [Georges] Vincent, and I organized a seminar on the theory of fiber bundles, very classical, and Ehresmann was mentioned there. I was attracted by his way of thinking. He was very geometrical, like Reeb.
Later Ehresmann developed a very abstract way of looking at mathematics — categories and basic structures and so on. It was not my way of looking at mathematics. So he had changed completely his way of understanding mathematics.
Jackson: Why did he change so much?
Haefliger: I don’t know.
Jackson: This occurred after he moved to Paris?
Haefliger: Somehow, yes. Maybe it was the influence of Bourbaki. He was a member of Bourbaki.
Jackson: Were you ever in Bourbaki?
Haefliger: No.
Jackson: Were you asked?
Haefliger: No. But when I was a student at Geneva, I used to go by bike to France to buy the fascicule of Bourbaki. It was not a long bike ride!
Jackson: What are your thoughts on the influence of Bourbaki?
Haefliger: It was an important enterprise. But they wanted to make something that would remain forever, and mathematics moves so quickly that the first publications were already not valuable anymore. So that is the bad thing. Bourbaki published some useful volumes, but differential topology — it was too loose. They were not able to do that, because they did not take account of the evolution. In the States, so many people were working in differential geometry, it was progressing so much, and Bourbaki was too rigid. The best volumes of Bourbaki are those dedicated to Lie groups. [Armand] Borel made a big contribution to that, and [Jacques] Tits too.
Strasbourg and Paris
Haefliger: Minouche and I got married the first of March in 1958, and we went immediately to Paris. I had a stipendium, a very small one, but it was very cheap at that time. We could not stay in the same hotel for more than two weeks. That was interesting! We used to prepare some of our meals in the hotel. It was not well accepted, but we did it!
Jackson: You had gone from Strasbourg to Paris. What differences in the mathematical climate did you see?
Haefliger: Oh, it was completely different. In Paris there was the Cartan seminar. This was very important, with a lot of people. At that time there was the École Normale Supérieure and the Institut Henri Poincaré. The École Polytechnique was also in the center of Paris, but there were no other universities. Orsay didn’t exist then. So people would meet each other very often.
[Jean-Pierre] Serre and [Roger] Godement were in Paris, and [Jean] Leray. The director of the thesis of Borel was Leray. So Borel had a very good knowledge of sheaf theory, and he explained the theory of Leray to people in Zurich before they could understand it!7 In fact Serre said that Borel explained all that stuff to him.
Jackson: Did you speak in the Cartan seminar?
Haefliger: Yes, I spoke about singularities. [Claude] Chevalley and André Weil were in the audience. It was hard for me.
Jackson: Why is that?
Haefliger: They wanted to understand the details. I gave maybe three talks, but I could not give all the details.
Thom gave seminars on singularities, and Serre could not understand this.
Jackson: Why could Serre not understand Thom?
Haefliger: Because Thom was not precise enough for him, especially his talks — and later on, his publications!
Jackson: They were very different as mathematicians.
Haefliger: Yes, very different ways of understanding mathematics.
Jackson: And Cartan? Could he understand both Thom and Serre?
Haefliger: Oh, yes. He had great esteem for Thom.
Jackson: Did Serre have esteem for Thom even though he did not understand him?
Haefliger: At first, yes. I can show you letters he wrote to Thom. They are very friendly.
Jackson: And later?
Haefliger: No, not at all. Thom wrote letters to Serre, and Serre destroyed most of them. Serre, and also Borel, did not realize the impact of the non-mathematical works of Thom in many fields. They were not interested. But Borel was very interested in things outside mathematics, more than Serre. Borel was fascinated by Indian music and jazz. He wrote many articles about the history of mathematics. He had a more profound view of the world. And Borel was very critical of what Bourbaki had done.
Jackson: You got to know Borel very well.
Haefliger: Yes. Minouche met him before me. She studied mathematics in Geneva and was in Borel’s algebra course. When he was at Princeton he sometimes spent his summers in La Conversion. The mother of his wife Gaby had a house there, which still exists, with a beautiful view of the lake. Sometimes we would cross Lake Geneva by sailboat and stop in Lutry, near La Conversion. We would meet there and have a very good time!
As I said, I first went to Strasbourg on the recommendation of de Rham. There were a lot of visitors, for instance, Robert Hermann, an American who was interested in physics and mathematics. Like an American, he sat in a chair with feet on the table. Koszul did not like that!
I also remember meeting [Reinhold] Remmert and going with him to the Strasbourg cathedral, to the top of the tower. He got me interested in complex analysis. My second thesis was on this subject.
Jackson: What is the second thesis?
Haefliger: At that time you wrote a second thesis on a topic not too close to the subject of your thesis, to be sure that you learned something in another domain. Pierre Lelong was my examiner for the second thesis. It was not published because there was nothing so interesting in it. But Serre and Borel had second theses that were published and were important papers!
Reeb stability
Around the time of Haefliger’s thesis, he and Reeb
discovered a way to generalize the stability theorem for
foliations that Reeb had proved in his own thesis. They never
published their paper, but an undated draft is deposited with
Haefliger’s papers at the University of Geneva. Much later their
result was superseded in a paper of
William Thurston
[e7].
Then Reeb,
together with
Paul Schweitzer,
used nonstandard analysis to
produce a one-page proof of Thurston’s result
[e9].
An addendum to the
Reeb–Schweitzer paper, written by
Walter Schachermayer,
uses two pages to present a translation into standard
analysis
[e10].
Haefliger: Reeb and I were supposed to write a paper together, a very good paper, but as I went to the States, I did not have time to take care of it. The topic was a stability result that Thurston developed later on. I changed subjects when I arrived in Princeton in the fall of 1959. I didn’t do anything about foliations. It was not fashionable!
In Princeton I met Smale and discussed a lot with him. At this time he was doing immersions, handlebodies, classification of manifolds, things like that. He was supposed to stay in Princeton another term or so, but he left early to go to Brazil and got interested in dynamical systems.
Jackson: In Princeton you got interested in embeddings and immersions and so on. Did Whitney have anything to do with that?
Haefliger: No. He was interested in many fascinating subjects, like real algebraic manifolds. Whitney wrote marvelous papers in many different subjects. It’s really impressive.
Before working on immersions and embeddings, I wrote a joint paper with Borel [2], which is in fact my first publication after my arrival at Princeton. I will describe the genesis of that paper.
I gave a talk in the framework of the seminar organized by André Weil, called Current Literature, at Princeton University. The topic of my talk was a report on the work of Thom on singularities of differentiable mappings that I gave with [Antoni] Kosinski in the Cartan seminar in 1957 [1], as mentioned before. A few days afterward I met Borel at the traditional tea at the Institute [for Advanced Study] and mentioned to him Thom’s claim that a real algebraic analytic set carries a fundamental class modulo 2, a fact admitted without proof. Borel realized immediately that the so-called Borel–Moore cohomology was the natural tool to prove it. Borel left Princeton in December 1959 to begin a long trip to India, so most of our collaboration was made by postal mail exchanges.
Minouche: While we were in Princeton we went to Baltimore for a conference at the institute of [Solomon] Lefschetz.
Haefliger: When I saw him for the first time, Lefschetz said, “Where do you come from?” I said, “Geneva.” And he said, “I hate Geneva!”
Jackson: How did it come about that Lefschetz invited you?
Haefliger: I was looking for offers, and he gave me an offer
from his institute, though later he wrote to tell me it would not be
possible. During the visit, I was on duty to explain to him the work
of some Russians, or to find a mistake in it.
In 1958, Solomon Lefschetz came out of retirement to establish the mathematics division of the Research Institute for Advanced Studies, founded by the Glen L. Martin Company of Baltimore, which is now Martin Marietta. He later moved the group to Brown University, and it continues to this day as the Lefschetz Center for Dynamical Systems. In 1960, Lefschetz invited Haefliger to visit RIAS and gave him a paper by two Russian mathematicians.8 The paper claimed a solution to the second part of Hilbert’s 16th Problem, which concerns the number and location of limit cycles for a planar polynomial vector field. Haefliger found the mistake in the paper, as did Yu. Ilyashenko and S. P. Novikov around the same time in Russia (see [e16]). In the early 1980s, when Shi Song Ling visited Haefliger in Geneva, he found an explicit counterexample [e12].
Professorship in Geneva
By 1961, at the end of his time in Princeton, Haefliger had
several offers from institutions in the United States, including
the University of California at Berkeley and Stanford University.
Georges de Rham worked quickly to entice Haefliger back to
Switzerland with an offer from the University of Geneva, which
Haefliger accepted. During 1961–62, the Cartan seminar in Paris
was devoted to the work of Smale, in particular his paper “On the
structure of manifolds”
[e3]
and was organized jointly with the de Rham
seminar in Geneva and Lausanne. Haefliger, who had lived next
door to Smale in the IAS housing and had had many discussions with
him, coordinated the seminar activities through many phone calls
and letters with Cartan and also presented several of the
lectures.
Haefliger: When I started at Geneva, professors had no offices. When de Rham and I were to phone the Minister of Education in Geneva, we had to go to a public phone because there was no phone available at the university! But the physicists already had phones. They were well established, because of the proximity of CERN.
One of the first mathematics professors we hired was [Alfred] Frölicher. We also hired Michel Kervaire and Raghavan Narasimhan. We were very ambitious. We wanted to have Borel, we wanted to have [Pierre] Cartier, [Jacques] Tits. We were very pretentious!
In 1970, when Kervaire came to Geneva, de Rham was beginning to retire. De Rham was the editor of Commentarii, and we had a discussion with Kervaire about the journal. De Rham said, “Maybe it’s Haefliger who has to take care of Commentarii, and Kervaire L’Enseignement Mathématique.” So I became an editor of Commentarii. It was fantastic! I could have the work of Thurston published and other interesting papers. I was an editor for a long time.
Jackson: Were you the editor in chief?
Haefliger: No, never, because if you are the editor in chief, you have to be an administrator — you receive a lot of papers, you must refuse some of them, you must find referees, and so on. It’s a huge job. [Johann Jakob] Burckhardt at the University of Zurich did this for a long time [1950–1982].
Jackson: What is the history of the journal L’Enseignement Mathématique?
Haefliger: It was founded a long time ago [1899], by Henri Fehr from Geneva and Charles-Ange Laisant from Paris. It was in collaboration with the French Mathematical Society and the International Commission on Mathematical Instruction. All departments of mathematics in France would receive L’Enseignement Mathématique and would contribute to it. It does not publish papers in mathematics education, but the papers should be understandable to graduate students. They should not be specialized.
One thing that is very important for our library is that
L’Enseignement Mathématique would advertise new books that
came out. We got the books free, in exchange for a subscription to
the journal. So the mathematical library at Geneva got essentially
all books because of that.
In 1966, Borel, Haefliger, and de Rham organized a symposium
to be held in the Swiss Romande, just before the International
Congress of Mathematicians was to take place in Moscow that year
(at this time de Rham was president of the International
Mathematical Union). Among the participants in the Swiss
symposium were
Walter Baily,
William Browder,
Morris Hirsch,
Nicolas Kuiper,
John Milnor,
Anthony Phillips,
Stephen Smale,
Dennis Sullivan,
René Thom,
Jack Wagoner,
and
C. T. C. Wall.
Phillips grew up in Paris, and as it happened the editor of the
local newspaper Tribune de Genève was an old family
friend. Intrigued by the high-level international conference
taking place in the city, the editor asked Phillips to write an
article about it. Phillips complied, but before submitting the
piece, he read it out loud to another symposium participant, the
blind topologist
Bernard Morin
— who then did a complete rewrite.
The article appeared in the Tribune de Genève on
August 2, 1966. It discussed the hope of the symposium organizers
to create an Institut Romande de Mathématique — a mathematics
research institute in the Swiss Romande, similar to ventures like
the Institute for Advanced Study in Princeton (where Borel was on
the faculty) and the Forschungsinstitut für Mathematik at the
ETH in Zurich (founded by
Beno Eckmann
in 1964).
Haefliger: This institute in the Suisse Romande was the dream of Borel.
Jackson: What became of it?
Haefliger: Nothing. [laughter]
You know, the name of Borel is never evoked in Lausanne. He is really a foreigner for them. As a boy he was a student in a public secondary school in Geneva and later on a student for one year in a private school close to Nyon. You cannot enter university without a certificate de maturité, and he did that essentially by himself. The examination is very hard, and the grades of the year before don’t count at all. He was a university student at the ETH. Later, in the 1980s, he joined the ETH faculty, and then there was huge tension in Zurich between Borel and Eckmann and Moser.
Minouche: And we were friends with all of them!
Haefliger: One person from the University of Zurich, before Borel would enter a lecture room, would write on the blackboard, “Borel is not welcome.” It was terrible! Borel had a strong character. I remember when he died, there were people on the internet saying, “It is time that the earth is getting rid of him!”
Minouche: He was not a diplomat! We were very good friends with him, but maybe we have not such strong characters!
Jackson: Was he really interested in coming back to Switzerland?
Haefliger: Oh, yes. He liked Switzerland a lot.
Connections with Asia
Starting early in his career, Haefliger had contacts with
mathematicians from Asia. While a student in Strasbourg, for
example, he met
Shoshichi Kobayashi
and
Yozo Matsushima.
Later on, his contacts with
Itiro Tamura
and
Takashi Tsuboi
helped to
build a thriving school for foliations in Japan. In 1979, at the
invitation of Wu Wentsun, André and Minouche went for the first
time to China, and on the same trip made a visit in Japan.
Minouche: We arrived in China on the first of October 1979, when there was a big meeting for the 30th anniversary of Mao Zedong’s leadership of China and the Cultural Revolution. Wu had to be at this meeting, so he invited us to it. There was an incredible feast. The whole of China was represented, with people in costume.
Haefliger: There were people who had survived the Long March with Mao. It was very impressive. The meeting was held in the Great Hall of the People in Tiananmen Square. When we arrived in Beijing, Wu told me, “I have sad news to tell you. Ehresmann died.” I didn’t know that. He had also been a student of Ehresmann.
Jackson: Was there a mathematical meeting in China that you went to?
Haefliger: No, but I gave a course.
Minouche: We were not allowed to go out by ourselves. We had a guide. So one day the guide came to pick me up. We went to the university, but André had not yet finished his lecture. The guide entered in the lecture room and said, “It’s time to leave!”
Jackson: So your lecture was over! Why were you not allowed to go out by yourselves?
Haefliger: It was a difficult time in China. It was just after the Cultural Revolution. Things were not yet stable.
Jackson: But Wu Wentsun was still able to invite you to China.
Haefliger: Yes, because he was very respected by the government. During the Cultural Revolution, he was asked to do research on mathematics in China. He wrote about this time when he got the Shaw Prize.9
Our last trip to China was close to the time when Wu received a big prize from the Chinese government. A specialist in agriculture received one of the prizes, and Wu Wentsun got the other prize.10
In China we also met Li Bang He. He was a student of Wu Wentsun and later stayed two years in Geneva, with another Chinese, Shi Song Ling.
Minouche: Li was already 40 when we met him. His studies had been interrupted because of the Cultural Revolution. To thank us, Li later invited us for one month in China, and Tsuboi invited us in Japan for two or three weeks. Because of the time they spent in Switzerland, they could develop themselves.
Jackson: On your first trip to China in 1979, you also visited Japan, and that was when you first met Tsuboi and Tamura.
Haefliger: Yes.
Jackson: When did the Japanese start working in foliations seriously?
Haefliger: I would say Tamura was the first, around 1970.
Jackson: Did Tamura ever come to Europe?
Haefliger: I don’t think so. There were many Japanese visiting Strasbourg when I was a student. But they were not doing foliations. The subject was completely unknown to most people then.
Jackson: Right after your first trip to Japan, an active exchange in foliations started.
Haefliger: Yes. You know, at that time, the Japanese didn’t travel so much, unless they already were big professors. Japanese students didn’t travel. To say, “I want to go to Geneva to study” — it was out of the question. But Tsuoboi spent two years here while he was a graduate student. I got a stipendium for him from the Swiss National Fund. Then he got his PhD [in 1983] under Tamura.
Of course Tsuboi is a marvelous person. He traveled all over Europe and was invited by many universities, mainly in France. He learned French in Fribourg, at a school for foreign students. The school had a very strict and efficient way of teaching the French language. Later Vaughan Jones and his wife learned French at the same place.
Jackson: How long was this language course?
Haefliger: About three months.
Jackson: That’s not a long time.
Haefliger: Yes, but they worked very hard!
From classifying space to group theory
“André Haefliger is a geometer. In the worlds of form that mathematicians sound in order to discover the deep structure within, he was able to perceive forms of which no one before him was aware. He thus showed the many ways in which the simplest surfaces wrap themselves up inside spaces of higher dimension. The technical terms adopted by specialists often carried his name: these are the knotted Haefliger spheres, or the Haefliger classifying space in the theory of foliations. This classifying space formalizes one of his most fertile ideas, which allows one to understand in what sense it is possible to classify all the ways in which multiple objects of small dimension fold themselves inside structures of large dimension — like the layers in puff pastry or in a block of mica.”
— from the citation for the Prix de la ville de Genève, awarded to Haefliger in 2003 (translated from the French).
Kotschick: Dennis Sullivan once said that there are two schools of differential topology in the United States. There is the Princeton school, where they do naked manifolds without any additional structure.
Haefliger: Not singular spaces.
Kotschick: Right, just naked manifolds. Then there is the Berkeley school, where they do manifolds with some additional structure — a dynamical system, or a foliation. Of course, most people are not completely in one school or the other. But you started with manifolds with extra structure, with a foliation, and then you went to naked manifolds after that.
Haefliger: Yes, because it was a subject that interested all the people!
Kotschick: That was when you met Smale, and he was also doing naked manifolds. But afterwards he started adding on dynamical systems. You did embeddings and immersion theory for a while, but then you returned to foliations.
Haefliger: I returned to foliations in 1970, following discussions with Tony Phillips.
Kotschick: Who had also been active in immersion theory!
Haefliger: But the main influence was the theorem of Bott about the annihilation of characteristic classes — the Bott vanishing theorem.
Kotschick: When did your introduction of the classifying space happen?
Haefliger: It was in 1969, when I gave a talk in Mont-Aigoual, which is a conference center close to Montpellier. I remember Gilbert Hector — he was a student of Reeb — said, “It’s foliation theory, but you don’t see the leaves! Where are the leaves?” I had introduced the notion of \( \Gamma \)-structure already in my thesis. But I had not yet had the idea of classifying space for that.
Kotschick: With the classifying space, you introduced a lot more algebraic topology to the study of foliations. Before that, it was more geometric.
Haefliger: Yes, because, how do you compute this huge classifying space? When I met Gelfand for the first time, he was collaborating with Fuks, who is a giant. And Gelfand said to me, “But you are so small!” — he had the idea of such a big classifying space!
Kotschick: It seems the Bott vanishing theorem and the discovery of Godbillon and Vey were two independent events. But they immediately fed into the development of characteristic classes for foliations. Then there was a burst of activity.
Haefliger: Godbillon once listed all papers published on foliations in the 1960s and 1970s — it’s incredible.11
Kotschick: During this burst of activity, there was an announcement by you and [Raoul] Bott [5], but it seems the full paper was never published. What happened there?
Haefliger: Bott was working with many other people, like Graeme Segal. And I am lazy to write up things! There was also a mistake in the announcement, which maybe was removed just before publication.
We spent days and days talking with Bott in his house, with his family, with his daughters. He had very good contact with Minouche, because he was Hungarian and Minouche’s father had lived in Budapest. Bott worked a lot with [Michael] Atiyah, who was very quick to write papers. Many ideas came from Bott, but Bott by himself was not so quick to write things.
Kotschick: How did you first get interested in group theory?
Haefliger: It was fashionable! There was this book of Serre, Arbres, Amalgames, et \( \mathrm{SL}_2 \) [e8]. Serre looks at an oriented graph, but one can also look at just a graph. Then the language somehow is simpler. This is the basic idea of the book Bridson and I wrote on CAT(0) [6].
Kotschick: How did the collaboration with Bridson happen?
Haefliger: Bridson was a PhD student of Karen Vogtmann, at Cornell. I got a grant from the Swiss National Fund to bring Martin to Geneva for a couple of years. We spent the summer in the mountains with him. We rented a chalet.
Minouche: One summer, and they thought they would finish the book in the autumn. It lasted for years!
Haefliger: Ten years! From 1990 until 1999. The book is quite long. The structure of the book became clear only five years after we began, because there was a lot of competition. There were several books published on CAT(0) space, for example the one by Ballmann, Gromov, and Schroeder [e13].
So we decided to write a book for beginners, so that we explain everything and give every definition, to get a different flavor from those books that were more advanced.
Jackson: Your book with Bridson has become a standard reference for this area.
Haefliger: It’s completely out of fashion now, but it is good for beginners.
Kotschick: Many of the people in that area, like some of the ones you mentioned, Ballmann, Schroeder, and so on — came from Riemannian geometry and differential geometry. Your interests and background had been completely different.
Haefliger: No — I wrote many papers on pseudogroups of isometries. But the subject of Riemannian foliations is huge now.
A pair of pants, a pair of brilliant students
In 1976, Paul Schweitzer organized a conference at his home
institution, the Pontifical University in Rio de Janeiro. The
conference attracted many of the leading lights in geometry and
topology, including Haefliger and Raoul Bott. In autobiographical
remarks in his Collected Papers, Bott writes that
many participants at this conference wound up as robbery victims.
During a beach outing, a robber stole Haefliger’s pants, which he
had bought at a store in Rio just a short while before going to
the beach.
Haefliger: But the pants were so cheap! They cost something like 10 Swiss francs.
At this meeting I spoke about the work of my student [Augustin] Banyaga. He came from Rwanda, which was a Belgian colony. His father had a banana plantation and never learned to read. It was a family of many children. A teacher noticed that Banyaga was very talented, so he could go to school. He got a scholarship to come to Switzerland [in 1967] to study mining engineering, because there were a lot of mines in Rwanda. He knew French very well. He had to take a course in mathematics, and then for him it was clear that he would like to do mathematics.
Jackson: Was he in a math course you were teaching?
Haefliger: Yes. He attended many lectures of mine, and I talked a lot with him. He used to go to the Troisième Cycle in Lausanne.12 I remember explaining many things to him about Thurston’s work. He wrote a diplôme [an original work written at the end of undergraduate studies], a very interesting work, inspired by a paper of Jürgen Moser. I sent it to Lichnerowicz for the Comptes Rendus, and it was published there [e6]. Afterward Banyaga wrote a PhD thesis [e11] inspired by the work of Thurston.
Jackson: When Banyaga came to Geneva, was it very difficult for him to adjust?
Haefliger: No, because the university had a program for students from foreign countries. They would get lodging and a French course if they needed it — but he already knew French very well. They also organized some sightseeing. Banyaga was like a rayon de soleil! He came to visit us many, many times.
Jackson: When he first came to Geneva, was he married?
Haefliger: No, but he was engaged. When the war began in Rwanda between Tutsi and Hutu, he was quickly able to get his fiancée to come to Geneva.
Minouche: They married and had their first child here. At one point he was asked to be a minister in the government in Rwanda, but he refused.
Haefliger: He did not want that, because if you become a minister, you have to do politics, and you can’t do mathematics anymore. He started a PhD program in Rwanda, but it died after a while.
Minouche: At some point he was afraid to go back to Rwanda and not to be able to get out.
Haefliger: Several years ago, there was a conference at the Institut Henri Poincaré, for Marc Chaperon. Banyaga gave a talk at the meeting, and he asked for a minute of silence, because this was at the time of an earthquake and tsunami in Haiti.
Jackson: We are talking about one of your outstanding students, Banyaga. Of course you had another outstanding student, Vaughan Jones, who also came from very far away, from New Zealand. Can you tell the story of how Vaughan Jones ended up in Geneva?
Haefliger: He wanted to work with a physicist, and it happened that the physicist died when Vaughan arrived. Vaughan is also like a rayon de soleil, and people loved him.
Minouche: As this professor had died, Vaughan went one day and asked André if he can give him a job.
Haefliger: And I said, “You can begin today!” He had already followed my lectures.
Jackson: Was his talent apparent when he came to you?
Haefliger: Of course! I think I feel immediately who has talent.
Jackson: His work was very far from what you were doing.
Haefliger: Yes. Pierre de la Harpe was really the only one in Geneva who was close to his subject. In the French parts of Switzerland, many people knew that subject, so Vaughan was invited to give talks. I was formally his thesis advisor, and Alain Connes was on his thesis committee. It was clear that, for a genius like Vaughan, one had to ask for an expert of the same level.
He got his PhD in 1979, so he was here for 4 or 5 years. When he got the Fields Medal at the ICM in Kyoto in 1990, he wanted very much that I come to Kyoto. But I thought they would present me as his thesis director, and I had nothing to do with it!
Minouche: Vaughan always claimed he learned a lot from André, but André always says he didn’t give him anything.
Haefliger: He knew a lot of mathematics, more than me!
Spreading the word about Gromov’s work
In 1969, Mikhail Gromov upended the world of differential topology with his doctoral thesis, written under V.A. Rokhlin at Leningrad University. His work contained far-reaching approximation theorems generalizing the known embedding and immersion theorems that had been developed throughout the 1950s and 1960s.
The first Westerner to hear about this was probably Anthony Phillips: “I was in the USSR in Spring 1969, on the inter-academy exchange program. My mentor Sergei Novikov sent me to Leningrad to meet Gromov, who had just completed his thesis. Misha gave me a copy to take back to the hotel and I immediately realized what he had accomplished (since it extended my own work on submersions, I knew exactly where to look).”13
Phillips proceeded to Warwick, where he participated in a summer-long symposium and spoke about Gromov’s work. In October 1969 Haefliger wrote to Phillips: “I hear that you spoke at Warwick about a very general invention of Gromov that axiomatizes theorems of this type. Can you give me any leads on this question?” In another letter two months later, Haefliger thanks Phillips for supplying a copy of Gromov’s thesis. Having also worked in immersion theory for the previous ten years, Haefliger immediately grasped the import of Gromov’s work and in 1970 lectured about it at the École Polytechnique in Paris14 and at the University of Paris Orsay.15
Another sixteen years would pass before Gromov’s book Partial Differential Relations appeared in the Ergebnisse series of Springer Verlag. The theory laid out in this book, and in works of Yakov Eliashberg and others, popularized the concepts emanating from Gromov’s thesis, now known as the “h-principle”.
