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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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    $\begingroup$ I think this whole "hole" thing is merely a reference to Alexander duality. $\endgroup$ Jan 27, 2015 at 14:01
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    $\begingroup$ 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... $\endgroup$ Jan 27, 2015 at 14:13
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    $\begingroup$ @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. $\endgroup$ Jan 27, 2015 at 14:36
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    $\begingroup$ Doesn't it go back to Poincare, and the first definitions of homology? $\endgroup$ Jan 27, 2015 at 17:26
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    $\begingroup$ 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? :) $\endgroup$ Jan 27, 2015 at 18:44

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