Henri Cartan

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Lore & Background

Henri Cartan was born on 8 July 1904. According to his own words, he was interested in mathematics at a very young age, without being influenced by his family. He moved to Paris with his family after his father's appointment at Sorbonne in 1909 and attended secondary school at Lycée Hoche in Versailles. In 1923 he started studying mathematics at École Normale Supérieure, receiving an agrégation in 1926 and a doctorate in 1928. His PhD thesis, entitled Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires et leurs applications, was supervised by Paul Montel. Cartan taught at Lycée Malherbe in Caen from 1928 to 1929, at the University of Lille from 1929 to 1931 and at the University of Strasbourg from 1931 to 1939. After the German invasion of France the university staff was moved to Clermont Ferrand. He recommended the mathematicians Marie-Hélène Schwartz and Laurent Schwartz move to work at the University of Strasbourg in Clermont Ferrand for both their research and for their safety. In 1940 he returned to Paris to work at Université de Paris and École Normale Supérieure. From 1969 until his retirement in 1975 he was professor at Paris-Sud University. Cartan died on 13 August 2008 at the age of 104. His funeral took place the following Wednesday on 20 August in Die, Drome.

Reader's Guide

Henri Cartan's significance lies in his foundational contributions to algebraic topology and homological algebra. He was a founding member of the Bourbaki group in 1934 and one of its most active participants. After 1945 he started his own seminar in Paris, which deeply influenced Jean-Pierre Serre, Armand Borel, Alexander Grothendieck and Frank Adams, amongst others of the leading lights of the younger generation. The number of his official students was small, but includes Joséphine Guidy Wandja (the first African woman to gain a PhD in mathematics), Adrien Douady, Roger Godement, Max Karoubi, Jean-Louis Koszul, Jean-Pierre Serre and René Thom. Cartan's first research interests, until the 40's, were in the theory of functions of several complex variables, which later gave rise to the theory of complex varieties and analytic geometry. Motivated by the solution to the Cousin problems, he worked on sheaf cohomology and coherent sheaves and proved two powerful results, Cartan's theorems A and B. Since the 50's he became more interested in algebraic topology. Among his major contributions, he worked on cohomology operations and homology of the Eilenberg–MacLane spaces, he introduced the notion of Steenrod algebra, and, together with Jean-Pierre Serre, developed the method of 'killing homotopy groups'. His 1956 book with Samuel Eilenberg on homological algebra was an important text, treating the subject with a moderate level of abstraction with the help of category theory. They introduced fundamental concepts, including those of projective module, weak dimension, and what is now called the Cartan–Eilenberg resolution. Among his other contributions, in general topology he introduced the notions of filter and ultrafilter and in potential theory he developed the fine topology and proved Cartan's lemma. The Cartan model for equivariant cohomology is also named after him.

Did You Know?

• Henri Cartan was the son of mathematician Élie Cartan, nephew of mathematician Anna Cartan, and oldest brother of composer Jean Cartan, physicist Louis Cartan and mathematician Hélène Cartan.
• He was a founding member of the Bourbaki group in 1934 and one of its most active participants.
• Cartan used his influence to help obtain the release of dissident mathematicians Leonid Plyushch, Jose Luis Massera, and Sion Assidon.
• He was awarded the Wolf Prize in 1980 and the CNRS Gold Medal in 1976.
• Cartan lived to age 104, dying on 13 August 2008.