# Heinrich Tietze

Heinrich Tietze (right), together with Fritz Hartogs

Heinrich Franz Friedrich Tietze (born August 31, 1880 in Schleinz near Neunkirchen , † February 17, 1964 in Munich ) was an Austro-German mathematician who was particularly active in the field of topology .

## Life

Heinrich Tietzes father was Emil Tietze (1845-1931), who was director of the Imperial and Royal Geological Institute . His mother was Rosa von Hauer, daughter of the geologist Franz Ritter von Hauer .

Tietze studied from 1898 at the Technical University of Vienna . During this time he made a close friendship with Paul Ehrenfest , Hans Hahn and Gustav Herglotz , with whom he was known as the "inseparable four".

Herglotz suggested that Tietze spend a year in Munich. He followed this suggestion in 1902 to continue his studies in Munich. After his return to Vienna he began a doctoral degree, supervised by Gustav von Escherich , and finished this in 1904 with the acquisition of a doctorate (doctoral thesis: functional equations whose solutions do not satisfy any algebraic differential equation ). Then, in 1905, he listened to Wirtinger's lectures on algebraic functions and their integrals and, under the influence of these lectures, developed the sustained interest that made topology his main research area.

In 1908 he submitted his habilitation thesis on topological invariants in Vienna and in 1910 became associate professor in Brno , Moravia . In 1913 he became a full professor, but the outbreak of the First World War in 1914 interrupted his professional career.

Tietze was called up for military service in the Austrian armed forces and was a soldier until the end of the war, after which he returned to Brno. In the following year, 1919, he accepted a call to the University of Erlangen and six years later to Munich, where he stayed until the end of his life.

In 1925 he was President of the German Mathematicians Association . In 1929 he was elected a full member of the mathematics and natural sciences department of the Bavarian Academy of Sciences . He was also a member of the Austrian Academy of Sciences (1959).

He retired from teaching at the university in 1950, but carried on mathematical research almost until his death in 1964. He reached the age of 83.

## plant

Tietze mainly worked on topology, but also z. B. on elementary geometry ( constructions with compasses and ruler ), number theory, ordinary and generalized continued fractions . Tietze made important contributions in the early years of topology.

In connection with the map coloring problem , he proved a theorem about neighboring areas on non-orientable areas . He also gave a simple proof (which can also be found in his popular science book on mathematics) why the problem analogous to the four-color theorem does not exist in three or more dimensions. To do this, he constructed bodies in three-dimensional space that all touch each other: Place bars in a row 1 next to each other and further bars, rotated by ninety degrees, in a row 2 above. Then define the bodies as being composed of one bar each from row 1 and row 2. ${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle n}$

He also worked on the Jordan curve theorem and showed that each on a closed set of dimensional space limited , continuous function on the whole space as a continuous function continue leaves ( Continued set of Tietze ). Tietze also investigated nodes using combinatorial group theory methods . In convex geometry he proved Tietze's theorem . ${\ displaystyle n-}$

In 1908 he examined the fundamental group and homology groups that Henri Poincaré introduced into topology in 1895 in order to classify topological spaces. Tietze represented the fundamental group in terms of generators and relations and proved (with his Tietze transformations between the representations of the fundamental group) its topological invariance. In this context he formulated the isomorphism problem for groups (namely, whether there is an algorithm to decide whether two groups described by a finite number of generators and relations are isomorphic). Poincaré tried to prove the topological invariance of homology groups by showing that they remained invariant with refined triangulations of space. This posed the problem of showing that such triangulations are unambiguous except for subdivisions, which Poincaré had implicitly assumed. Tietze pointed out that proof was needed, and the problem has made its way into the history of topology as the main guess (the name comes from Hellmuth Kneser ). It was only introduced in the 1960s by John Milnor , Dennis Sullivan , Robion Kirby and others. a. proven or (in higher dimensions) disproved.

The introduced by Tietze lens premises managed James Waddell Alexander in 1919 to disprove a conjecture of Poincaré as they examples of non-homeomorphic provide rooms with the same fundamental group.

He is also known for his book Solved and Unsolved Problems from Old and Modern Times (Munich 1949).