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In cubical homology you have to consider the group of degenerate cubics and use the group of cubes module degenerate cubes (Massey). If you not do that you get the wrong homology for one point space. In singular homology (simplices) every textbook explain you do not need to do that because you are going to get the same groups. It is easy to check that you get the correct homology groups for a point space without exclude degenerate simplices. But I would like to know a GENERAL PROOF of that. That is the equality of both groups in general order and spaces (excluding or not degenerate simplices). Because it seems that textbook writers consider that evident for I have seen the statement many times but no proof at all and I am not willing to take it by faith. I have read an old paper by Tucker available in internet by googling: Degenerate cycles bound. But I dont know how to apply that to singular homology

Thank you very much

Ismael cortazar
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I am going to answer myself. I have found an old book Hilton and Wylie Homology theory that deals with this problem in sections 8.2 8.3 and 8.4 Theorem 8.4.10 is what I was looking for. It is a pity that this book uses a rather bizarre notation so I will need more time to fully understand the proof because the need of read previous chapters to grasp the notation. So if someone knows a book, paper, etc on this content that uses standard notation I shall be thankful to him. Anyway I am not able to understand that modern books as Hatcher, Rotman, Vick, Tom Diek, etc fully bypasses this matter.

Ismael cortazar
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  • Yes, Hatcher works only with delta sets (weird) and I did not find this issue addressed for simplicial sets, the main introductory texts use simplicial complexes and other less enjoyable things. I think this is (implicitly?) proven for simplicial sets in “Simplicial Homotopy Theory” by Goerss-Jardine but I haven’t read the relevant chapter yet – FShrike May 20 '23 at 19:39
  • As already pointed out in an earlier question, one version of this for simplicial sets is here: https://math.stackexchange.com/questions/108019/the-subcomplex-of-degenerate-simplices-has-trivial-homology. Another version is at https://kerodon.net/tag/00Q9, 2.5.5.5 through 2.5.5.13. – John Palmieri May 20 '23 at 23:13
  • I've now looked at Hilton and Wylie. Their proof is based on the theory of simplicial sets (or in old language, semi-simplicial complexes; see their remark following 8.2.5). Their Theorem 8.2.6 is probably what you want. It is reproducing the previously cited work with simplicial sets. – John Palmieri May 22 '23 at 19:00