section_path: Contributions to mathematics and physics/Second epoch (1920–1926)/Topology
section_name: Topology
paragraph_text: Topology
  
  As noted by Hermann Weyl in his obituary, Noether's contributions to topology illustrate her generosity with ideas and how her insights could transform entire fields of mathematics. In topology, mathematicians study the properties of objects that remain invariant even under deformation, properties such as their connectedness. An old joke is that " a topologist cannot distinguish a donut from a coffee mug ", since they can be continuously deformed into one another.
  
  Noether is credited with fundamental ideas that led to the development of algebraic topology from the earlier combinatorial topology, specifically, the idea of homology groups. According to Alexandrov, Noether attended lectures given by him and Heinz Hopf in the summers of 1926 and 1927, where "she continually made observations which were often deep and subtle" and he continues that,
  
  When... she first became acquainted with a systematic construction of combinatorial topology, she immediately observed that it would be worthwhile to study directly the groups of algebraic complexes and cycles of a given polyhedron and the subgroup of the cycle group consisting of cycles homologous to zero; instead of the usual definition of Betti numbers, she suggested immediately defining the Betti group as the complementary (quotient) group of the group of all cycles by the subgroup of cycles homologous to zero. This observation now seems self-evident. But in those years (1925–1928) this was a completely new point of view.
  
  Noether's suggestion that topology be studied algebraically was adopted immediately by Hopf, Alexandrov, and others, and it became a frequent topic of discussion among the mathematicians of Göttingen. Noether observed that her idea of a Betti group makes the Euler–Poincaré formula simpler to understand, and Hopf's own work on this subject "bears the imprint of these remarks of Emmy Noether". Noether mentions her own topology ideas only as an aside in a 1926 publication, where she cites it as an application of group theory.
  
  This algebraic approach to topology was also developed independently in Austria. In a 1926–1927 course given in Vienna, Leopold Vietoris defined a homology group, which was developed by Walther Mayer, into an axiomatic definition in 1928.