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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