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Among the many cohomology theory's branches I asked about last time, I was curious about $d^2=0$ because I know that it is the formula that is the basis of all cohomology. So this time, I would like to ask a more detailed question than the last one. I want to know the background of singular homology, that is, the motivation and history of singular homology. I would like to receive a recommendation for a text-oriented book with little formula or an answer on the motivation and background of this singular homology.

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Regarding singular homology proper, its main development occurred in the 1930s and 1940s with Lefschetz and Eilenberg figuring prominently. For this history, Dieudonné's "A History of Algebraic and Differential Topology, 1900 - 1960" is rather authoritative (see especially pages 68-70). See also Ronnie Brown's math stack exchange answer here in which he comments briefly on singular homology and references Eilenberg's 1944 paper.

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  • $\begingroup$ Hello, thank you for your answer. I read Ronnie Brown's answer you provided as a link. But I have one question about this. Of course, I posted a question in the link's answer, but it appears that Ronnie Brown has been inactive recently. $\endgroup$ Feb 14 at 7:08
  • $\begingroup$ This is my question! It is said that $∂∂=0$ appeared in Poincaré's paper. I am curious about the motivation behind why Poincaré was able to derive $∂∂=0$ from the history of mathematics. $\endgroup$ Feb 14 at 7:13
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Poincare's original papers, Analysis situs and 5 supplements to it, contain relatively few formulas, make a good reading, and explain his motivation. If you can read French. Another good source is

H. Seifert and W. Threlfall, A textbook of topology, which exists in English translation.

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  • $\begingroup$ Hello, thanks for your answer. Does the 'Analysis situs and 2 addenda to it' you mentioned in your answer above refer to the 'second supplement to analysis situs' in this file? $\endgroup$ Feb 10 at 21:47
  • $\begingroup$ I meant all 5 supplements. $\endgroup$ Feb 11 at 3:38
  • $\begingroup$ Okay, I will see all supplements. Thanks for your reply. $\endgroup$ Feb 11 at 4:03
  • $\begingroup$ @user1274233: Thanks for the link to the English translation. $\endgroup$ Feb 11 at 4:08

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