**Hopf**'s work was in algebraic topology. He studied vector fields and extended Lefschetz's fixed point formula. He also studied homotopy classes and defined what is now known as the 'Hopf invariant'.

- Elizabeth Hopf was a Protestant and, in 1895, Wilhelm converted to his wife's religion.
- Heinz attended Dr Karl Mittelhaus's school from 1901 until 1904 and following this he began his studies at the König-Wilhelm Gymnasium in Breslau.
- In April 1913 Hopf entered the Silesian Friedrich Wilhelms University in Breslau to read for a degree in mathematics.
- During a fortnight's leave from military service in 1917 Hopf went to a class by Schmidt on set theory at the University of Breslau.
- After the war Hopf returned to his studies in Breslau but after about a year he left and went to the University of Heidelberg.
- At Heidelberg Hopf took courses in philosophy and psychology as well as attending courses by Perron and Stäckel.
- In 1920 Hopf went to study for his doctorate at the University of Berlin where Schmidt was now teaching.
- Hopf went to Göttingen in 1925 where he met Emmy Noether.
- Her contributions would play an important part in Hopf's developing ideas.
- During this year in Göttingen Hopf worked on his habilitation thesis which was completed by the autumn of 1926.
- Aleksandrov and Hopf spent some time in 1926 in the south of France with Neugebauer.
- This was an important year in the development of topology with Aleksandrov and Hopf in Princeton and able to collaborate with Lefschetz, Veblen and Alexander.
- During their year in Princeton, Aleksandrov and Hopf planned a joint multi-volume work on Topology the first volume of which did not appear until 1935.
- In 1930 Weyl left his chair in the ETH in Zürich to take up a chair at Göttingen and in 1931 Hopf was approached to see if he was interested in accepting this chair.
- However, before receiving the formal offer from Zürich, Hopf received the offer of a chair at Freiburg but he waited for the Zürich offer and accepted it.
- The next few years were not easy ones for Hopf.
- Hopf continued to visit his parents in Breslau up until 1939.
- Hopf was able to provide refuge in Switzerland for friends who had to flee Germany under the Nazis.
- Hopf's own position became more difficult, however, for he was still a German citizen.
- Lefschetz, realising Hopf's difficulties, invited him to Princeton but Hopf refused.
- Soon after the Oberwolfach visit, Hopf went to the United States where he spent six months and there he renewed many old friendships.
- Most of Hopf's work was in algebraic topology where he can be thought of as continuing Brouwer's work.
- Hopf extended Lefschetz's fixed point formula in work which he undertook in 1928.
- He defined what is now known as the 'Hopf invariant' in 1931.
- The ideas which he introduced in this investigation led to him defining what is today called a Hopf algebra.
- The honours which Hopf received are almost too numerous to list.
- But Heinz Hopf was not only a gifted researcher: he was also an excellent teacher and a personality of the highest integrity.

Born 19 November 1894, Gräbschen (near Breslau), Germany (now Wrocław, Poland). Died 3 June 1971, Zollikon, Switzerland.

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Origin Poland, Topology

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive