# Carlos Benjamin de Lyra

### Born: 23 November 1927 in Recife, Pernambuco, Brazil

Died: 21 July 1974 in São Paulo, Brazil

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**Carlos Benjamin de Lyra**'s father, Carlos de Lyra Filho, was a journalist who was descended from a family of mill owners. Carlos de Lyra Filho was well-known as the owner of the newspaper 'Diário de Pernambuco'. Carlos Benjamin de Lyra's mother was Elizabeth Lau, his father's second wife. She was of German origin and came to Brazil with her family who were involved with the production of sugar. The family were Roman Catholics and Carlos Benjamin was one of his parents' two sons. He had a younger brother George but also had a number of older siblings from his father's first marriage. George de Lyra (1929-1999) became a well-known artist. Carlos' father died when he was eight years old and his mother remarried an American, Paul Nortz, probably in 1936. Elizabeth and Paul Nortz, along with Carlos and George, then moved to the United States. Paul Nortz was a broker on the Wall Street Stock Exchange and the family lived in the suburbs of New York.

Lyra attended Roman Catholic schools in New York. After elementary schooling, he entered the Iona Preparatory School in New Rochelle, New York, in 1939. This school, named after the Scottish island of Iona, had been founded in 1916 by the Congregation of Christian Brothers. Lyra excelled at the school and graduated 'summa cum laude' in 1945. It was while he was at this high school that Lyra decided that the subject that he wanted to continue studying was mathematics. Although this must have been, at least in part, due to his studies of the topic at school, it seems that one of the main reasons was that he came to know Richard Courant. Both Courant and Lyra lived in the New York suburbs and they travelled into the city on the same train every day. Courant had published his classic book, co-authored with Herbert Robbins,

*What is mathematics?*in 1941, shortly before getting to know Lyra on the train journeys. What better friend could young 15-year old Lyra have to enthuse him about mathematics!

After graduating from high school in 1945, Lyra had a choice. He could opt to take American citizenship, serve in the U.S. Army, and continue his studies in the United States. However, he had been born and had spent the first nine years of his life in Brazil so he felt a strong attachment to that country. Therefore, in November 1945, just before his 18

^{th}birthday, he returned to Brazil and so confirmed his Brazilian nationality. He enlisted for military duties in Brazil. At first he lived in Pernambuco, in the family home where Christiano de Lyra, one of his father's sons from his first marriage, was living. Later he moved to São Paulo where he lived with a friend of the family, the lawyer Manuel Tavares. In São Paulo in 1946 he began attending the Faculdade de Filosofia, Ciências e Letras where he met André Weil and Jean Dieudonné. André Weil had been appointed as a professor of mathematics at São Paulo in 1945 and remained there for two years. Lyra and André Weil [4]:-

Dieudonné had been appointed as professor of mathematics at São Paulo in 1946 and spent one year in that position. Lyra attended the Algebraic Topology course given by Cândido Lima da Silva Dias in 1947 [4]:-... used to take long walks on week-ends in a then sparsely inhabited quarter of town. The recollection of these walks and of the talks with Weil always remained vivid in Lyra's mind.

During his final year as an undergraduate, Lyra proved some results relating the well ordering theorem, the axiom of choice and Zorn's lemma. This became his first publicationDias himself had recently returned from the U.S. where he had studied the subject with Henri Cartan and Edwin Spanier. The deep impression left by these courses on him, plus his earlier and profound interest in point set topology, led Lyra to choose algebraic topology as his special field.

*A note on Zorn's theorem*(1949). After an outstanding undergraduate career, Lyra graduated in 1950. In 1951 he sailed to France to undertake research for his doctorate. He left Brazil having applied for a scholarship but still awaiting the decision on whether it would be granted. His confidence was rewarded, however, for he received a scholarship from the Conselho Nacional de Pesquisas. In Paris, Lyra attended Henri Cartan's seminar on fibre spaces. In the summer of 1953 he attended Witold Hurewicz's lectures on 'Homotopy' at the Collège de France.

Leda Lacerda was from Rio de Janeiro where she had been working with Bernhard Gross, a German physicist who had arrived in Rio de Janeiro in 1933. Around 1947 they had derived a new integral transform in rheology which they had published in the

*Journal of Applied Physics*. After graduating from the Faculdade Nacional de Filosofia in Rio de Janeiro, Leda Lacerda sailed to Europe to continue her studies at the European Institute of Physics. Carlos de Lyra and Leda Lacerda became partners and lived in the Hotel des Grands Hommes in Paris. After Leda became pregnant in August 1953 she returned to Brazil and, a short while later, Lyra followed her. Back in Brazil the two were married and their first child, Jorge Lacerda de Lyra, was born on 20 May 1954. Carlos and Leda de Lyra had two further children, Sylvia Lacerda de Lyra (born 1956), and Eduardo Lacerda de Lyra (born 1958). We now give some details of these children. Jorge studied in São Paulo and then went to Yale University where he was awarded a Ph.D. in physics. He became a professor of physics at the University of São Paulo. Sylvia was awarded a degree in Sociology. Tragically, Eduardo died in 1971 at the age of twelve in an accident on a beach.

Back in São Paulo at the end of 1953, Lyra was appointed as an assistant professor of mathematics at the Faculdade de Filosofia, Ciências e Letras. He was much influenced by Alexander Grothendieck who spent the years 1953-55 at the University of São Paulo. Lyra published

*Minimal complexes and maps*(1954) in which he proved the existence of a minimal subcomplex of the cubical singular complex of a topological space. This paper contained results he had obtained while in Paris but also some further results. He gave his first graduate course on algebraic topology and published the lecture notes in 1957. He was awarded a doctorate in 1958 for his 83-page thesis

*Sobre os espaços de mesmo tipo de homotopia que o dos poliedros*Ⓣ [4]:-

He published results from his thesis in the paperIn1956Lyra had taken part in the International Symposium on Algebraic Topology, in Mexico City, where he became interested in cohomological operations and Postnikov systems. The research which resulted in his Ph.D. was done during1957-1958, and by the end of1958he presented his thesis.

*On spaces of the same homotopy type as polyhedra*(1960). V Gugenheim writes in a review:-

Lyra received a grant from the Rockefeller Foundation to fund a two-year visit to Institute for Advanced Study at Princeton in the United States. He went there in 1960 with his wife and children and, while there, continued his research, took part in Norman Steenrod's seminar, and learnt much about differential topology from John Milnor. Of course Steenrod was one of the main reasons that Lyra wanted to go to the Institute for Advanced Study since he had published the first systematic account of fibre bundles in his bookBy considering the realisations of the singular complex of a space in the manner of J H C Whitehead and J Milnor the author proves some theorems characterising spaces having the homotopy type of locally finite or finite polyhedra ...

*The Topology of Fibre Bundles*(1951). Despite having funds for a two-year visit to Princeton, Lyra returned to Brazil after spending only one year at the Institute for Advanced Study. His reasons were two-fold. Firstly, he was receiving an inheritance from the sale of the house in Pernambuco which had been his parents' home and in which he had lived. Secondly, there was political instability in Brazil which worried him and made him frightened that he would not be allowed to return to Brazil. In fact, this worried him so much that he never left Brazil again. After returning to Brazil the family could not live in their own home having rented it out for the two-year period he was expecting to be in the United States. Therefore, using the inheritance he had received, he bought a new large three-floor property in Vila Mariana district of São Paulo, more in the style of his old Pernambuco home than the place in São Paulo he had been living in.

A group of academics, in which Lyra played a central part, worked hard through the 1960s to establish an Institute of Mathematics and Statistics. They were successful and it was created as part of the University of São Paulo in January 1970. Lyra worked at the new Institute of Mathematics and Statistics from 1970 to 1972, then was appointed there as a Senior Professor. In 1968 he was given the title of "livre-docente" at the Faculdade de Filosofia, Ciências e Letras and, in 1974, he became Associate Professor of the Instituto de Matemática e Estatistica of the University of São Paulo. The committee's report on Lyra when he applied for this promotion to Associate Professor is as follows:-

Another important contribution that Lyra made was in running his weekly seminar [4]:-On his scientific work: the seriousness and coherence of the candidate in a recognisedly difficult subject - Algebraic Topology - must be mentioned; special mention must be given to his Ph.D. thesis and his thesis for the title of 'livre docente'. These works got international notice and served as a starting point for new research by the candidate, some of which he is concluding now. On his influence in the development of mathematics in Brazil: special emphasis must be given to the dedication of the candidate to all problems which arise in the development of mathematics in Brazil. This dedication has had remarkable results not only within the Institute of which he is a member but also in the whole of the University and of the country, through his activities in the Colóquios de Matemática, in the Sociedade Brasileira de Matemática, IMPA, etc. On his teaching activity: the Committee gives special emphasis to the dedication of the candidate to his work on advising graduate students, keeping up the interest on Algebraic Topology, a subject in which he is the most active lecturer in Brazil. This has never affected his activity as lecturer on the undergraduate level, as is made evident both by the variety of the courses he has taught as by his work on planning curricula and programme reforms. Taking all this into account, the Committee unanimously recommends Professor Carlos Benjamin de Lyra for Professor of the Department of Mathematics of this Institute.

It is remarkable how many leading mathematicians visited São Paulo and we have already mentioned some of these above. Lyra made special efforts to attract many others and we mention, in particular, Peter Hilton who visited São Paulo in 1968, 1969, 1972, 1973 and 1974.One other aspect of Lyra's activity on the subject of algebraic topology deserves special mention. From1956on he directed a weekly seminar on the subject covering the most important topics of the field as for instance: simple homotopy type, topological localization, homology of fibre spaces, cohomological operations, applications of cohomological operations, Postnikov systems, localization and applications. His role in supervising more than a dozen advanced students and a number of beginners was also remarkable.

In addition to the contributions mentioned, we should also explain Lyra's contributions to teaching and course development in São Paulo (we have slightly modified this quote from [4]):-

The male side of Lyra's family tended to die young. In particular his father and one of his brothers had died young. Lyra had little confidence that he would live out his three score and ten, and he would tell his friends that he expected to die soon. In 1974, when he was 46 years old, he felt unwell and spent four days in hospital. Doctors diagnosed a brain tumour and he was allowed to return home on condition that he returned to the hospital in three days time for an operation. He spent these three days working on his articleBesides all this activity he had an important role in the teaching of mathematics: from1963with Professor Elza Furtado Gomide(1925-2013)he worked in the reorganisation of courses on Calculus, and in the planning of honours courses in Advanced Calculus. He also taught courses in the History of Mathematics, and in Calculus, Analysis, Algebra, Topology, as well as advanced courses in Algebraic Topology, Group Theory, etc.

*SHM-maps of CW-groups*. After Lyra's death, this paper was revised by Peter Hilton who explains the circumstances in a note appended to the paper which was published in 1976:-

Professor de Lyra submitted an original version of this paper in January1974. It was returned to him, along with substantial suggestions for revision by the referee, on March15,1974. Lyra very much welcomed the referee's suggestions for improvement, and was able to embark on a revised version in the beginning of June,1974. Unfortunately he was stricken with cerebral haemorrhage and died, aged46, on July13,1974, leaving behind rough drafts of early sections of the paper. Peter Hilton was in Brazil at the time of Lyra's death and undertook to write the revised version in accordance with the referee's suggestions and Lyra's notes. It is this version which we now publish. Peter Hilton has asked the editors to explain the circumstances ...

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

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