Emmy Noether helped create algebraic topology

Person: Emmy Noether
Lived: 1882–1935
Field: Mathematics
From: German

The story

Topology studies the properties of shapes that survive stretching and bending, such as connectedness or the number of holes. An old joke says a topologist cannot tell a doughnut from a coffee mug, because one can be smoothly deformed into the other. In the 1920s the subject, then called combinatorial topology, described shapes mainly with numbers, such as the Betti numbers.

Göttingen had become a magnet for mathematicians from around the world, and the Russian topologists Pavel Alexandrov and Pavel Urysohn were among the first to arrive, in 1923. Between 1926 and 1930 Alexandrov lectured regularly at the university, and he and Emmy Noether became close friends. According to Alexandrov, she attended lectures given by him and Heinz Hopf in 1926 and 1927, and continually made observations that were often deep and subtle.

When she first met a systematic treatment of combinatorial topology, Alexandrov recalled, she immediately saw that it would be better to study the groups behind the numbers. Instead of simply counting, she proposed defining a Betti group, now called a homology group, as the quotient of the group of all cycles by the subgroup of cycles homologous to zero. Alexandrov noted that the idea later seemed self-evident, but between 1925 and 1928 it was a completely new point of view.

Her suggestion was adopted immediately by Hopf, Alexandrov and others and became a frequent topic of discussion among the mathematicians of Göttingen. Noether pointed out that her approach made the Euler–Poincaré formula simpler to understand, and Hopf's own work on the subject bears the imprint of her remarks. Characteristically, she mentioned her topological ideas only as an aside in a 1926 publication, citing them as an application of group theory. The same algebraic approach was also developed independently in Vienna, where Leopold Vietoris defined a homology group in a 1926–27 course and Walther Mayer gave an axiomatic definition in 1928.

Hermann Weyl, in his obituary of Noether, noted that her contributions to topology illustrated both her generosity with ideas and how her insights could transform entire fields. The shift she encouraged, from counting features to studying algebraic structures, turned combinatorial topology into algebraic topology. Homology groups remain at the core of the subject, and the method of translating geometric questions into algebra has become one of the most powerful habits of modern mathematics.

Why it matters

Algebraic topology became one of the central fields of modern mathematics, with uses in physics, data analysis and beyond.

About Emmy Noether

Erlangen-born mathematician who linked symmetry to conservation laws and reshaped abstract algebra, despite years of unpaid work as a woman and dismissal by the Nazis as a Jew.

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Emmy Noether

1882–1935 · Mathematics

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