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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.

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    $\begingroup$ 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. $\endgroup$
    – Conifold
    Nov 21 '20 at 7:51
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    $\begingroup$ 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. $\endgroup$
    – Conifold
    Nov 21 '20 at 8:15
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    $\begingroup$ @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". $\endgroup$
    – Conifold
    Nov 22 '20 at 1:03
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    $\begingroup$ 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: gdz.sub.uni-goettingen.de/id/PPN37721857X_0034 (page 361 of the scan) $\endgroup$
    – njuffa
    Nov 22 '20 at 23:21
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    $\begingroup$ @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. $\endgroup$
    – njuffa
    Nov 23 '20 at 20:54

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