**Heinrich Tietze** was an Austrian mathematician, known for the Tietze extension theorem. He also developed the Tietze transformations for group presentations, and was the first to pose the group isomorphism problem.

- Tietze was a student at the Technische Hochschule in Vienna, starting his studies there in 1898.
- It was his friend Herglotz who suggested that Tietze spend a year in Munich, and indeed he went there in 1902 to continue his studies.
- Returning to Vienna, Tietze was supervised during his doctoral studies by Gustav von Escherich and he was awarded his doctorate in 1904.
- He lectured on algebraic functions and their integrals in his first year back in Vienna, and Tietze attended these lectures and because of them formed an instant liking for topological notions which would from that time on be his main research topic.
- Tietze submitted his habilitation thesis to Vienna in 1908 and this was on a topological topic considering topological invariants.
- At the start of the war Tietze was called up to serve in the Austrian army.
- After six years in Erlangen, Tietze accepted a chair at the University of Munich.
- Of course this means that Tietze spent the difficult years of Nazi control of Germany at Munich and this had many unfortunate consequences.
- Caratheodory was a colleague of Tietze's at Munich until he retired in 1938.
- Tietze and his colleagues drew up a short list of three candidates to replace Caratheodory.
- Tietze contributed to the foundations of general topology and developed important work on subdivisions of cell complexes.
- Fundamental groups were introduced by Poincaré in 1895 and, in 1908, Tietze recognised that from the abelianised fundamental group of a space all the earlier invariants could be calculated.
- In that 1908 paper, Tietze produced a finite presentation for the fundamental group and invented the now well-known Tietze transformations to show that fundamental groups are topological invariants.
- The now famous Tietze transformations change one presentation of a finitely presented group to another presentation without changing the group which is defined by the presentation.
- It is possible to go from any given finite presentation of a group to any other using Tietze transformations.
- In the same 1908 paper Tietze gives the first reference to the isomorphism problem for groups, namely: if two groups are defined by finite presentations, is there an algorithm to decide whether they are isomorphic or not?
- Of course Tietze gives the problem in the context of fundamental groups of topological spaces.
- It is probably fair to say that von Dyck initiated the study of combinatorial group theory but then Tietze made the first major step forward.
- Among the topics in topology which Tietze worked on were knot theory, Jordan curves and continuous mappings of areas.
- Tietze also wrote on map colouring and wrote a well known book Famous Problems of Mathematics.
- Topics outside topology which Tietze worked on included ruler and compass constructions, continued fractions, partitions, the distribution of prime numbers, and differential geometry.
- Tietze received many honours for his contributions.

Born 31 August 1880, Schleinz (E of Neunkirchen), Austria. Died 17 February 1964, Munich, Germany.

View full biography at MacTutor

Algebra, Group Theory, Origin Austria, Topology

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