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I was surprised to learn that the note published by Emmy Noether in 1925 that suggested that Homology was better thought through as actual groups rather than numerically as Betti numbers consisted of fourteen lines. This is mentioned in Weibels History of Homological Algebra.

What was the content of this note?

edit

I had originally said two lines but checking back with Weibel I noted that I had made a mistake and it should have been fourteen.

Mozibur Ullah
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    In Eine Verallgemeinerung der Euler-Poincaréschen Formel Hopf writes (my translation):"In the course of a lecture I gave in Gottingen in the summer of 1928, I found out, under the influence of Freulein Emmy Noether, that my original proof of this generalization of the Euler-Poincare formula can be made easier and more transparent by using group-theoretic concepts." He does not mention a note. Neither do Dieudonne or MacLane when writing about this episode. – Conifold Nov 21 '20 at 07:51
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    In 1964 Hopf added a 2-line footnote:"The above note was probably the first publication in which Emmy Noether's group theoretical view of the topology comes into play" (he was mistaken about that), see Hirzebruch. Hirzebruch suggests that Noether told Hopf that Betti numbers and torsion coefficients should be replaced by homology groups orally, which seems more natural. Noether did not publish her ideas about algebraisation of topology, but she shared them with Gottingen colleagues even before 1928. – Conifold Nov 21 '20 at 08:15
  • The Science Direct article on Betti numbers, in the section History of Homological Algebra writes: "This changed in 1925, when Emmy Noether (1882–1935) pointed out in her 14-line report [145], and in her lectures in Göttingen, that homology was an Abelian group, rather than just Betti numbers and torsion coefficients, and perceptions changed forever." The citation "[145]" is for E. Noether, Ableitung der Elementarteilertheorie aus der Gruppentheorie dated 27, Jan 1925. The quote is from IM James History of Topology, 1999. – nwr Nov 22 '20 at 00:09
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    @Nick The only reference to topology in Ableitung is:"in the applications of group theory, e.g. Betti and torsion numbers in topology, therefore, a return to the elementary divisor theory is not necessary". In the book (p.996) James also mentions Aleksandrov recalling a conversation about homology groups from 1925, and says that they were "adopted at once" by Hopf and Vietoris. But Hopf's first draft had no groups, and Vietoris denied Noether's influence. So either Aleksandrov misremembered the date or James was too eager to tie Noether to "changing perceptions forever". – Conifold Nov 22 '20 at 01:03
  • @Conifold I should have made the effort to locate a English translation of Ableitung. Reading 14 lines is not a huge effort, even for me. Given that you have highlighted at least two inaccuracies in James History, it seems that James has erred. I also note that other sources such as wikipedia date Ableitung as 1926 rather than the early 1925 date given by James. I'm starting to doubt Ableitung has 14 lines? – nwr Nov 22 '20 at 02:43
  • @Nick I am not sure. Ableitung may not be available online, Hirzebruch says it is not included in Noether's Collected Papers. The quote I translated is given by him (p.5). James might have given the submission date rather than the publication date. Hopf was in Gottingen also in the summer of 1926. Alexandrov and Hopf do credit Noether for changing their perceptions on homology in their Topology text (1935). The credit was apparently backdated to the sentence quoted by Hirzebruch by James and Weibel. Vietoris maintained that he came to the same realization in 1926 independently of Noether. – Conifold Nov 22 '20 at 21:57
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    A scan of: E. Noether, "Ableitung der Elementarteilertheorie aus der Gruppentheorie" dated January 27, 1925, In Jahresberichte der Deutschen Mathematiker-Vereinigung, Vol. 34, 1926, part 2, p. 104 can be found at the Goettingen Digitalization Center: https://gdz.sub.uni-goettingen.de/id/PPN37721857X_0034 (page 361 of the scan) – njuffa Nov 22 '20 at 23:21
  • @Nick Fourteen lines is correct (for nitpickers: using only one third of the first line, but extending one word into a fifteenth line). Four sentences altogether, with some half-sentences separated by semicolons. – njuffa Nov 23 '20 at 00:15
  • @njuffa Thanks for that. I don't own a copy of James' text and had to rely on a google books search for the citation details. It's good to know that the date of 25 Jan 1925 is indeed correct. – nwr Nov 23 '20 at 20:45
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    @Nick: Note that Noether's brief notice is dated January 27, 1925 (presumably the date it was received by the editors of Jahresberichte?). Unfortunately my mathematical knowledge is too limited to provide an English translation of the notice or even a summary. Otherwise I would have supplied a proper answer by now. – njuffa Nov 23 '20 at 20:54

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