La cascade d’homologie

It can be very rewarding to read old mathematical papers, in terms of accessing insights and ideas that may have been filtered out in later transformations of the results they contain. In my modest experience, this does not extend to notation and terminology, and it is much easier to appreciate the insights in question after translating them into modern language and formalism. This is an area where, maybe, progress is usually fairly steady. But still, there can be exceptions. I was recently rather struck, while reading the recently published collection of letters between Henri Cartan and André Weil, to discover that when they were exchanging many letters on algebraic topology just after 1945, they used the charming name cascade for what is now known as a “long exact sequence” (in homology or cohomology). I think it is too bad this didn’t become the standard name; one could have imagined that triangles
A\rightarrow B\rightarrow C\rightarrow A[1]\rightarrow
would be called “Escherian cascades”…

Incidentally, this book of letters is very interesting to read, in no small part because of the extensive notes and comments by Michèle Audin. It is published by the SMF in the same series where letters between Grothendieck and Serre were also published a few years ago.

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Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

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