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The question Why do the homology groups capture holes in a space better than the homotopy groups? and many others here use the idea that homology counts the ``holes'' in a space. The comments on this and related questions show that many people do not like this terminology. I like it a lot and I want to know if someone deserves credit for it.

The first systematic use of the idea that I know is Atiyah's 1975 Bakerian lecture, available at

http://www.jstor.org/stable/4145047?seq=1#page_scan_tab_contents

and elsewhere. Atiyah uses it to link classical cohomology of topological spaces to \'etale cohomology and related theories.

It is easy to suppose it must go back to Poincare but it seems not to. I do not recall it in the Analysis Situs and its complements. I have checked the word "hole" does not occur in John Stillwell's translation of AS. I believe there used to be searchable pdfs of AS on line but anyway I do not find them now.

In most 20th century discussions the genus of a surface is described in terms of "handles" on the surface, not "holes" (reflecting a focus on compact surfaces). Indeed Riemann surfaces were often described topologically as "handle bodies." Compare https://en.wikipedia.org/wiki/Handlebody

Seifert and Threlfall's 1934 Textbook of Topology uses "hole" as the opposite of "handle." For them a hole is made by cutting out a bit of surface, and so it cannot exist on a boundaryless compact manifold.

Alexandroff uses it the same way, indeed his Elementary Concepts of Topology uses the word "hole" just once, and that is for a hole in a plane figure, not a handle on a compact surface. So homology does not count "holes" in his sense at all.

Is it fair to give Atiyah credit for this terminology of holes?

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Colin McLarty
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    I think this whole "hole" thing is merely a reference to Alexander duality. – Alex Degtyarev Jan 27 '15 at 14:01
  • @AlexDegtyarev Then you really should read Atiyah's essay. – Colin McLarty Jan 27 '15 at 14:02
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    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... – paul garrett Jan 27 '15 at 14:13
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    @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. – Colin McLarty Jan 27 '15 at 14:36
  • 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. – paul garrett Jan 27 '15 at 14:39
  • @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. – Colin McLarty Jan 27 '15 at 14:45
  • 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. – paul garrett Jan 27 '15 at 14:52
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    Doesn't it go back to Poincare, and the first definitions of homology? – Ryan Budney Jan 27 '15 at 17:26
  • @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. – Colin McLarty Jan 27 '15 at 18:35
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    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? :) – paul garrett Jan 27 '15 at 18:44
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    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? – paul garrett Jan 27 '15 at 19:32
  • 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... – paul garrett Jan 27 '15 at 23:21
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    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"... – Dylan Wilson Jul 03 '17 at 01:34
  • @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. – Colin McLarty Jul 03 '17 at 01:45

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