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Witold Hurewicz (June 29, 1904 – September 6, 1956) was a Polish mathematician who worked in topology.
Hurewicz attended school in a German-controlled Poland but with World War I beginning before he had begun secondary school, major changes occurred in Poland. In August 1915 the Russian forces that had held Poland for many years withdrew. Germany and Austria-Hungary took control of most of the country and the University of Warsaw was refounded and it began operating as a Polish university. Rapidly, a strong school of mathematics grew up in the University of Warsaw, with topology one of the main topics. Although Hurewicz knew intimately the topology that was being studied in Poland he chose to go to Vienna to continue his studies.
He studied under Hans Hahn and Karl Menger in Vienna, receiving a PhD in 1926. Hurewicz was awarded a Rockefeller scholarship, which allowed him to spend the year 1927–28 in Amsterdam. He was assistant to L. E. J. Brouwer in Amsterdam from 1928 to 1936. He was given study leave for a year, which he decided to spend in the United States. He visited the Institute for Advanced Study in Princeton, New Jersey and then decided to remain in the United States and not return to his position in Amsterdam.
Hurewicz's early work was on set theory and topology. The Dictionary of Scientific Biography states: "...a remarkable result of this first period 1930 is his topological embedding of separable space metric spaces into compact spaces of the same (finite set) dimension.*"
In the field of general topology his contributions are centred on dimension theory. He wrote an important text with Henry Wallman, Dimension Theory, published in 1941. A reviewer writes that the book "...is truly a classic. It presents the theory of dimension for separable metric spaces with what seems to be an impossible mixture of depth, clarity, precision, succinctness, and comprehensiveness."
Hurewicz is best remembered for three remarkable contributions to mathematics: his discovery of the higher homotopy groups in 1935–36, his discovery of the long exact homotopy sequence for in 1941, and the Hurewicz theorem connecting homotopy and homology groups. His work led to homological algebra. It was during Hurewicz's time as Brouwer's assistant in Amsterdam that he did the work on the higher homotopy groups; "...the idea was not new, but until Hurewicz nobody had pursued it as it should have been. Investigators did not expect much new information from groups, which were obviously commutative..."
Hurewicz was also first to represent a function by an arrow, in about 1940. This rapidly displaced the old notation. It was proven fundamental for the development of category theory.
In the late 1940s, he was the doctoral advisor of Yael Dowker.
Hurewicz had a second textbook published, but this was not until 1958 after his death. Lectures on ordinary differential equations is an introduction to ordinary differential equations that again reflects the clarity of his thinking and the quality of his writing.
He died after participating in the International Symposium on Algebraic Topology at the National Autonomous University of Mexico in Mexico City. He tripped and fell off the top of a Mayan step pyramid during an outing in Uxmal, Mexico. In the Dictionary of Scientific Biography it is suggested that he was "...a paragon of absentmindedness, a failing that probably led to his death."
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