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?
