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This is sort of a mixture of a math and history question.

First the math part: thinking about it, I do not actually know how to properly use the term "Betti cohomology". I know I should, but I don't. Betti cohomology of a variety $X$ defined over a field $k\subseteq \mathbb{C}$ refers to the singular cohomology of the associated complex space $X(\mathbb{C})$. But what coefficients, integral or rational? Or even local systems? More importantly, complex conjugation induces an involution on $X(\mathbb{C})$ and then also on the cohomology. It seems that sometimes this is part of the structure of Betti cohomology, sometimes it isn't. So this is the math part of my confusion - maybe someone can tell me how I should use the term Betti cohomology appropriately.

Next (and more seriously), assuming we have clarified how the term "Betti cohomology" is supposed to be used nowadays - how did this evolve? The german Wikipedia article claims that Poincaré coined the term "Betti numbers" for the ranks of singular homology groups because these ranks agreed with numbers Betti had defined for surfaces. So, what are the possible reasons for calling singular cohomology of the associated complex space "Betti cohomology"? Which papers were instrumental in making Betti cohomology a popular term? Can anyone shed light on the history of the terminology?

PS: I tagged the question ag.algebraic-geometry because the "Betti cohomology" seems to be prevalently used in algebraic geometry related communities. Feel free to retag if you consider this inappropriate.

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    $\begingroup$ I've only seen this usage among algebraic geometers. I assumed it was someone like Grothendieck who started this. I tend to understand it as with $\mathbb{Z}$ coefficients, unless that the author says otherwise. Finally, unless the field of definition lies in $\mathbb{R}$, there is no natural conjugation on $X(\mathbb{C})$. Of course, if you use cohomology with $\mathbb{C}$ as coefficients, then there is conjugation on that. $\endgroup$ – Donu Arapura Nov 6 '14 at 17:16
  • $\begingroup$ @DonuArapura: You are absolutely right, and this is part of my confusion that I did not explicitly realize before - so Betti cohomology can refer to singular cohomology of the associated complex space if $k\subseteq\mathbb{C}$ as well as singular cohomology of the complex space plus complex conjugation if $k\subseteq\mathbb{R}$. $\endgroup$ – Matthias Wendt Nov 6 '14 at 18:01
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    $\begingroup$ A few comments about history: The Betti numbers, as Poincaré defined (and called) them, did not exactly agree with Betti's definition. In Betti's definition, he used formal sums of simplices, but no integer coefficients. (Picard applied those Betti numbers in work about complex algebraic surfaces.) This difference raised some confusion, because Poincaré's proof of Poincaré duality did not work for Betti's original definition (as pointed out by Heegaard) so Poincaré had to write a short note in the CRAS to point out that his notion of Betti numbers was different from Betti's. $\endgroup$ – ThiKu Nov 6 '14 at 19:01
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    $\begingroup$ @ThiKu, do you have the reference to the CRAS paper? $\endgroup$ – Mariano Suárez-Álvarez Nov 6 '14 at 19:02
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    $\begingroup$ Another curiousity: Poincaré's first definition was 1 bigger as the one used later. That is, he first defined the Betti number $P_m$ such that modulo boundaries there are $P_m-1$ independent m-cycles. I.e. $P_m=b_m+1$. The same with Betti's original definition, where, as said, boundaries where only admitted sums of distinct simplices, no integer coefficients. $\endgroup$ – ThiKu Nov 6 '14 at 19:04
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Some of this stuff is explained in much detail in this paper:

In particular, after reviewing the work of Riemman and Betti, it says:

Inspired by Betti's paper, Poincaré (1854-1912) developed a more correct homology theory in his landmark 1895 paper "Analysis Situs". [...] In honor of Enrico Betti, Poincaré defined the nth Betti number of V to be $b_n+1$, where $b_n$ is the size of a maximal independent family. Today we call $b_n$ the nth Betti number, because it is the dimension of the rational vector space $H_n(V;\mathbb{Q})$.

The reference in question is:

  • Henri Poincaré, Analysis situs (1895) Journal de l'École Polytechnique

Also, the follow-up notes mentioned in the comments are:

  • Henri Poincaré, Complément à l'Analysis Situs (1899) Rendiconti del Circolo Matematico di Palermo

  • Henri Poincaré, Second complément à l'Analysis Situs (1900) Proceedings of the London Mathematical Society

There's also a moder translation of Poincaré's original paper and a total of five "supplements":

As for the evolution toward the modern use of Betti cohomology, this paragraph from Weibel's survey seems relevant:

Homological algebra in the 19th century largely consisted of a gradual effort to define the "Betti numbers" of a (piecewise linear) manifold. Beginning with Riemann's notion of genus, we see the gradual development of numerical invariants by Riemann, Betti and Poincaré: the Betti numbers and Torsion coefficients of a topological space. Indeed, the subject did not really move beyond these numerical invariants until about 1930. And it was not concerned with anything except invariants of topological spaces unit about 1945.

I still don't know where the use of Betti cohomology comes from, so this is not really an answer; hopefully someone else can help. My best guess is that it is from about the same time that the concept of Weil cohomology theory. The papers in which the classic theorems of Betti cohomology are proven never use that name, see for example:

  • Jean-Pierre Serre, Géométrie algébrique et géométrie analytique (1956)

  • Michael Artin, The étale topology of schemes (1968)

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