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. He attended the Gymnasium until 1913 and it was at this school that his talent for mathematics first became clear to his teachers. In his other subjects, however, his results were less good and it is probable that he devoted too much time to sport, he was particularly fond of swimming and tennis, and not enough to his academic subjects. He left the Gymnasium with the mathematics report stating:-
He has shown an extraordinary gift in this topic, especially in the algebraic direction.In April 1913 Hopf entered the Silesian Friedrich Wilhelms University in Breslau to read for a degree in mathematics. There he was taught by Kneser, Schmidt, and Rudolf Sturm. He also attended lectures by Dehn and Steinitz who taught at the polytechnic in Breslau. However, his studies were interrupted by the outbreak of World War I in 1914. He immediately enlisted and for the duration of the war he fought on the Western front as a lieutenant. 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. From that time on he knew that he wanted to undertake research in mathematics. He wrote in  about the influence Schmidt's lectures had on him :-
I was fascinated; this fascination - of the power of the method of the mapping degree - has never left me since, but has influenced major parts of my work. And when I look for the cause of this effect, I see particularly two things: firstly, Schmidt's vividness and enthusiasm in his lecture, and secondly my own increased receptiveness during a fortnight off after many years of military service.After the war Hopf returned to his studies in Breslau but after about a year he left and went to the University of Heidelberg. By this time Schmidt had left Breslau and it appears that Hopf wanted to go to Heidelberg to be with his sister who had begun her studies there in the previous year. 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. He attended several courses by Schur in Berlin and he received his doctorate in 1925 with a thesis, supervised by Schmidt, studying the topology of manifolds. Among other results, he classified simply connected Riemannian 3-manifolds of constant curvature in this thesis. It was an impressive piece of work which received the following praise from Schmidt in his report (see for example ):-
The boldness of the questions deserves as much admiration as the surprising results of the solutions. But the most beautiful thing in the thesis is the method of proving, which is, particularly rarely found in works in that area, abstract and comprehensible in every step, and which, due to the abstractness, shows equally clearly the richness of the concrete geometric imagination.Bieberbach and Schmidt examined him in mathematics, while Planck examined him in physics.
Hopf went to Göttingen in 1925 where he met Emmy Noether. Her contributions would play an important part in Hopf's developing ideas. Perhaps even more significant was the fact that Aleksandrov was also spending time in Göttingen and Hopf wrote in :-
My most important experience in Göttingen was to meet Pavel Aleksandrov. The meeting soon became friendship; not only topology, not only mathematics was discussed; it was a fortunate and also a very happy time, not restricted to Göttingen but continued on many joint journeys.During this year in Göttingen Hopf worked on his habilitation thesis which was completed by the autumn of 1926. The thesis contains a different proof of the fact just shown by Lefschetz that for any closed manifold the sum of the indices of a generic vector field is a topological invariant, namely the Euler characteristic. Aleksandrov and Hopf spent some time in 1926 in the south of France with Neugebauer. Then the two spent the academic year 1927-28 at Princeton in the United States. 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. This was the only one of the three intended volumes to appear since World War II prevented further collaboration on the remaining two volumes.
Hopf married Anja von Mickwitz in October 1928. He was offered an assistant professorship by Princeton in December 1929 but he rejected the offer. 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. In part the offer had been prompted by a very positive recommendation which Schur had sent to Zürich:-
Hopf is an excellent lecturer, a mathematician of strong temperament and strong influence, a leading example in his discipline ... I cannot wish you a better colleague in respect to his manners, his education and his sympathetic nature.Hopf replied to the approach of the ETH in Zürich indicating that he would accept a formal offer:-
A call to Switzerland, to the beautiful city of Zürich, could indeed tempt and honour me, particularly to such a famous chair. I therefore declare that I am in principle willing to accept such an offer.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. He took up his duties in Zürich in April 1931. The next few years were not easy ones for Hopf. After the Nazis came to power in Germany in 1933, Hopf's father, being Jewish, came under increasing pressure. Hopf continued to visit his parents in Breslau up until 1939. Seeing the difficulties that his father faced Hopf arranged for his parents to receive immigration papers for Switzerland. However, his father fell ill and could not travel.
Hopf was able to provide refuge in Switzerland for friends who had to flee Germany under the Nazis. In particular Schur came for a while before finally going to Palestine in 1939. 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. Then in 1943 he was told to move back to Germany or he would lose his German citizenship. Faced with this he had little choice but to quickly apply for Swiss citizenship, which was soon granted.
With the end of World War II Hopf was able to help his German friends again. He did much more than this, however, for he put much energy into trying to re-establish a mathematical community in Germany. His visit to the research centre in Oberwolfach in August 1946 was part of his efforts. Soon after the Oberwolfach visit, Hopf went to the United States where he spent six months and there he renewed many old friendships. He was offered professorships by many of the most prestigious of the American universities but, after careful consideration, he decided to remain loyal to Zürich.
Over the next few years he enjoyed invitations to lecture at leading international conferences, and he visited many places including Paris, Brussels, Rome and Oxford. He spent the academic year 1955-56 with his wife in the United States.
Most of Hopf's work was in algebraic topology where he can be thought of as continuing Brouwer's work. He studied homotopy classes and vector fields producing a formula about the integral curvature.
Hopf extended Lefschetz's fixed point formula in work which he undertook in 1928. It is in this 1928 paper that he first explicitly used homology groups. His work on the homology of manifolds, undertaken in Princeton in 1927-28, led to his definition of the intersection ring by defining a product on cycles by their intersection. This idea was later seen to be connected to cohomology.
He defined what is now known as the 'Hopf invariant' in 1931. This was done in his work on maps between spheres of different dimensions which cannot be distinguished homologically so required the introduction of a new invariant. In 1939 he examined the homology of a compact Lie group. This was to attack questions posed to him by Élie Cartan. The ideas which he introduced in this investigation led to him defining what is today called a Hopf algebra.
In the early 1940s Hopf published :-
The paper Fundamentalgruppe und zweite Bettische Gruppe Ⓣ [which] is legitimately regarded to be the beginning of homological algebra. It opened the way for the definition for the homology and cohomology of a group. This step was made independently at different places shortly after the paper became known ...The honours which Hopf received are almost too numerous to list. He was President of the International Mathematical Union from 1955 until 1958. He received honorary doctorates from many universities including Princeton, Freiburg, Manchester, Sorbonne, Brussels, and Lausanne. He was awarded many prizes including the Gauss-Weber medal and the Lobachevsky award. He was elected to honorary membership of many learned societies throughout the world.
Freudenthal gives this description of Hopf in :-
Hopf was a short, vigorous man with cheerful, pleasant features. His voice was well modulated, and his speech slow and strongly articulated. His lecture style was clear and fascinating; in personal conversation he conveyed stimulating ideas.Frei and Stammbach in  pay this tribute to Hopf:-
Without doubt Heinz Hopf was one of the most distinguished mathematicians of the twentieth century. His work is closely linked with the emergence of algebraic topology; it is most decisively thanks to his early works that this area established itself as a new and important branch of mathematics. his work has influenced profoundly the evolution not only of topology but of a large part of mathematics. But Heinz Hopf was not only a gifted researcher: he was also an excellent teacher and a personality of the highest integrity. at the same time, he effervesced with charm and subtle humour.
Article by: J J O'Connor and E F Robertson
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