Subject: spectral language From: hrm@math.mit.edu Date: Mon, 14 Feb 2005 09:02:42 -0500 (EST) To: dmd1@lehigh.edu (Don Davis) CC: hrm@math.mit.edu (Haynes Miller) Borel told me that while he never asked Leray why he chose the word spectral, he assumed that it was by some strained analogy with the spectral theorem: the cohomology of the total space is being built up bit by bit from constituent components. Borel also said that when he was writing his thesis he did not think of a spectral sequence in terms of the "Serre picture," with the fiber up and base across, even though he was in possession of Koszul's work. In a sense Serre's coining the term "suite spectrale" represented a compromise between the Leray camp (represented by "spectrale" and the Koszul/Cartan camp (who were using the word "suite," as in "suite de Leray-Koszul"). I'm amused by Ian's question about page vs term. Borel and others were aghast at the increasing use of "page" for what was obviously a term (in a sequence). Who first used "page"? Haynes >> >> Subject: Why spectral sequence >> From: John McCleary >> Date: Fri, 11 Feb 2005 09:26:19 -0500 >> >> Dear Martin, >> You can consult my paper in the History of Topology. To summarize: >> Everything was cohomology for Leray (even though he called it >> homology). For him, the argument for his proof of the Kuenneth >> Theorem for his version of cochains led to a filtration that he >> generalized to what he called a spectral algebra. (Here is the origin >> of the term spectral and where the real question lies.) >> Koszul made a remarkable clarification of the algebra of a spectral sequence >> which he termed a sequence of homologies. Cartan (Koszul's advisor, >> and co-author with Leray) published two papers in 1947 referring to >> Leray-Koszul sequences. Borel, in his thesis under Leray, continued >> to use the term anneau spectral, but, of course, he was computing >> cohomology. >> What is clear is that a term was wanting for the case of homology. >> Not that there was such a case to consider until Serre's thesis. >> The relation between homology and homotopy groups made such >> a case interesting to consider. Serre >> coined suite spectrale to cover the case of a homology spectral >> sequence. Luckily it was NOT a coalgebre spectrale. >> >> All the best, >> John >> ____________________________________________________________ >> >> Subject: Re: three postings >> From: James Stasheff >> Date: Fri, 11 Feb 2005 10:27:32 -0500 (EST) >> >> Somewhere I was told that it is spectral >> in the sense that each page is the ghost (spectre) of the one before >> but this may be appochryful (spell checker, where art thou) >> >> Jim Stasheff jds@math.upenn.edu >> >> Home page: www.math.unc.edu/Faculty/jds >> >> On Fri, 11 Feb 2005, Don Davis wrote: >> >> >> Subject: Why "spectral" sequence? >> >> From: "Martin C. Tangora" >> >> Date: Thu, 10 Feb 2005 16:36:00 -0600 >> >> For the topology list: >> >> >> >> Why is it called a "spectral" sequence? >> >> >> >> I always assumed -- just on the face of it -- >> >> that it was because the idea is to take a differential >> >> and break it up into its components, >> >> just as a prism breaks white light into the spectrum. >> >> >> >> Does anyone know who invented the term, >> >> and whether my guess is correct? >> >> >> >> Martin C. Tangora >> >> University of Illinois at Chicago >> >> tangora@uic.edu >> ___________________________________________________________ >> >> Subject: Terrible joke >> From: Ian Leary >> Date: Fri, 11 Feb 2005 22:13:18 +0000 (GMT) >> >> When I was a research student, my father (who is not a >> mathematician) suggested an explanation. The inventor >> of the spectral sequence, before writing out the full >> proofs that it did what it was supposed to exclaimed >> "I 'spect it'll work" (you have to say it to make >> "I expect it will" sound like "spectral". >> >> By the way, why do some people use "term" for what >> should clearly be called a "page" of a spectral >> sequence? >> >> Ian Leary