Timeline for Who first talked about "holes" in homology?
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18 events
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| Jul 3, 2017 at 1:45 | comment | added | Colin McLarty | @DylanWilson I doubt they had the same picture. I have not read Betti, but Riemann and Poincare seem to me to focus much more on what is in the space, and not what is missing from the space. I suspect holes came more into it with the development of duality theorems such as Alexander duality, on the (co-)homology of complements to manifolds in spheres. | |
| Jul 3, 2017 at 1:34 | comment | added | Dylan Wilson | Is it really fair to focus on the actual use of the word "hole"? Poincare, and Betti and Riemann before him, very geometrically defined betti numbers in terms of unions of submanifolds that couldn't be filled in, and they were all doing this in different languages. For sure they all had the same picture in their head as Atiyah in his essay, despite not literally using the word "hole"... | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jan 27, 2015 at 23:21 | comment | added | paul garrett | Please pardon yet-one-more anecdotal-personal comment: as a kid, I found the cutting-up of "handle bodies" completely unpersuasive. Rather, the idea that a ("closed") thing that was not a "boundary" of something corresponded to a missing "something", that is, a "hole". The bar (etc) constructions in group (co)homology struck me as making simplicial complexes whose "physical" (co)homology reflected whatever-the-heck-it-was about groups that was desired. It was only later (esp. Nick Katz' lectures on Weil II c. 1974) that is became clear (to me) that "hom-things" did much more... | |
| Jan 27, 2015 at 19:32 | comment | added | paul garrett | One more dim recollection, possibly irrelevant to questions about more-professional history of the terminology: speaking of "what would a 2-D hole be?", I think by the 1960s (or earlier) the sort of "Flatland" sci-fi explicitly suggested to kids (including me) to think that a sphere would look to 4-D entities like a loop looks to us... etc... justifying thinking of the inside of a sphere as a "hole", perhaps. Maybe the sci-fi writers made it up? | |
| Jan 27, 2015 at 18:44 | comment | added | paul garrett | Colin, I don't necessarily disagree at all, but, nevertheless, somehow in the late 1960s, I did manage to convince myself that a 2-sphere has a 2-dimensional hole inside it, and that, indeed, a compact, connected, oriented surface has 2-D hole in it. By now, I don't know whether this was rationalization or suggested by some external source... What would a higher-dimensional hole be, after all? :) | |
| Jan 27, 2015 at 18:35 | comment | added | Colin McLarty | @paulgarrett I really appreciate this input, but I see that as Alexandroff uses the term "hole," there cannot be any holes in a closed surface (i.e. in a compact boundary-less surface). Homology certainly does not count holes in his sense. | |
| Jan 27, 2015 at 18:31 | history | edited | Colin McLarty | CC BY-SA 3.0 |
Notice that even Alexandroff does not say that torus, for example, has "holes".
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| Jan 27, 2015 at 17:47 | history | edited | Colin McLarty | CC BY-SA 3.0 |
Added discussion of where the term is not used.
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| Jan 27, 2015 at 17:26 | comment | added | Ryan Budney | Doesn't it go back to Poincare, and the first definitions of homology? | |
| Jan 27, 2015 at 14:52 | comment | added | paul garrett | Yes, certainly people talked that way by the mid-1970s there, and/but I was trying to think of earlier precedents. Certainly my perceptions in the late 1960s were naive, but I do have a vivid recollection of the "holes" business by that point. It may have been a forced interpretation in terms intelligible to a naive kid, given the flimsy excuse of that off-hand comment in Alexandroff. | |
| Jan 27, 2015 at 14:45 | comment | added | Colin McLarty | @paulgarrett Also, we see what is going on around us. With a 1977 Princeton PhD you might well have heard Atiyah himself use the term. You certainly knew people who did. | |
| Jan 27, 2015 at 14:39 | comment | added | paul garrett | My dim recollection of my own reaction was that, once mentioned, the notion of "holes" counted by homology didn't need to be repeated! I still cannot recall where I got the idea that the $n$-th Betti number counted the $n$-dimensional holes... and the "paradox" of having torsion, etc. It would have been late 1960s, whatever the source. But I guess one sees what one wants to see. | |
| Jan 27, 2015 at 14:36 | comment | added | Colin McLarty | @paulgarrett Nice call. That is a stunning book. But I think he only uses the term "hole" in passing and does not offer it as a general explanation of the genus of a surface or of homology groups. | |
| Jan 27, 2015 at 14:13 | comment | added | paul garrett | An inexpert comment: I recall getting the impression that homology counted "holes" in the late 1960s from Alexandroff's little book (essentially on combinatorial alg top over coefficients in a field with 2 elements, to avoid wrangling over signs, I suppose). Other sources (now forgotten by me) gave a similar impression late-1960s. Also, I dimly recall a comment that it was Emmy Noether who recommended that homology groups be groups, as opposed to "mere" Betti numbers... | |
| Jan 27, 2015 at 14:02 | comment | added | Colin McLarty | @AlexDegtyarev Then you really should read Atiyah's essay. | |
| Jan 27, 2015 at 14:01 | comment | added | Alex Degtyarev | I think this whole "hole" thing is merely a reference to Alexander duality. | |
| Jan 27, 2015 at 13:56 | history | asked | Colin McLarty | CC BY-SA 3.0 |