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In 2006, Gaunce Lewis died at the age of 56. He'd done important work setting up equivariant stable homotopy theory, and I think it's fair to say his work was far ahead of its time. In recent years, thanks in part to the solution of the Kervaire Invariant One Problem by Hill, Hopkins, and Ravenel, interest in equivariant stable homotopy has blossomed.

Peter May wrote a wonderful memorial tribute to Lewis (and also Mark Steinberger). In a footnote on page 2 he writes

Unfortunately, much influential work of his in this direction remains unpublished.

The context is a sentence about Lewis's work on Mackey functors and "standard results, like projective implies flat for modules over a ring, can actually fail in such more general contexts."

I searched, but could not find any preprints of Gaunce Lewis online. His memorial page says he published 15 papers and the equivariant stable homotopy theory book (with co-authors). On arxiv, there are just two preprints, both with co-authors and both already published now. Hence my questions:

What was Lewis working on in the last years of his life? Are his ideas written down anywhere? Which ones have been worked out, and which ones are still open?

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    $\begingroup$ By coincidence, through a mutual non-mathematical friend, I was fortunate enough to meet Lewis just after starting graduate school, shortly before he died. He asked what I had been thinking about, and I told him I'd been working on universal coefficient and Kunneth theorems in some nonclassical settings for homology theory, and he told me he'd been working on much the same thing. In retrospect I imagine he was likely thinking about equivariant universal coefficient and Kunneth theorems. Unfortunately I do not know any specifics of what he was working on, or whether he wrote up that work. $\endgroup$
    – A.S.
    Jun 26 at 17:42

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Lewis wrote, but never published, a very influential paper setting foundations for the multiplicative theory for Mackey functors. The paper is called "The Theory of Green functors" and Doug Ravenel's paper archive has a scanned copy. The date listed there is 1980, and there are numerous references to it in the literature. From the introduction:

Thus this project became, for the analogs of rings, a rough draft equivalent of an undergraduate text on the basics of ring theory.

I strongly suspect that this is one of the influential works that Prof. May had in mind. These predate Tambara's work on Green functors that also possess a multiplicative transfer (now "Tambara functors").


On a more personal note, I once had correspondence with Prof. Lewis asking some questions - mostly trying to get some effective calculational methods in the homological algebra of Mackey functors. He wrote back a very kind response. I will attach some relevant snippets from his email:

The only thing I can suggest is an old idea of mine that I have have intending to explore for years, but have never gotten around to looking at. If there is any context in which this idea might be useful, it is probably the one you are asking about [...]

Anyway, assign nonnegative integers to the subgroups of a finite group by assigning 0 to the trivial subgroup, 1 to each of the cyclic subgroups of prime order, and so forth so that the number assigned to each subgroup is the length of the longest chain of subgroups starting with that subgroup and ending with the trivial group. Filter any Mackey functor M by looking at the kernel of restriction to the collection of subgroups one notch below the whole group, the kernel of the restriction to all subgroups 2 notches below the top, and so forth. This gives a natural filtration on the whole category of Mackey functors [...] this is a very simple idea that lots of people may have tried already and found useless. On the other hand, it may be one that no one has tried because everyone assumed someone else had looked at it or because no one could get a useful description [...]

Regardless of the specifics, I think the problem Lewis addresses here is one that is still very present in equivariant stable homotopy theory. Namely, when working with Mackey and Green functors, one often wants to induct on subgroups and try to filter calculations by simpler ones. In equivariant stable homotopy theory we often do so by using "isotropy separation" techniques, allowing us to attack one subgroup at a time (see, for example, Greenlees-May's paper "Some remarks on the structure of Mackey functors"). However, to my knowledge nobody has successfully figured out how to package this isotropy-separation information in a way that makes concept and calculation more approachable in equivariant homological algebra (eg. using such filtrations to produce spectral sequences with effective $E_2$-terms).

If it strikes you as worth pursuing, feel free. I'm not likely to think more about it anytime soon.

Unfortunately, this is from February 2006, and he passed away about three months later; I regret not having the chance to discuss more.

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    $\begingroup$ Fantastic answer! Thanks so much. I suspect we might get another answer to this question as well, as a few MathOverflow regulars knew Lewis very well. $\endgroup$ Jun 22 at 15:14
  • $\begingroup$ Well, it's been several days and we haven't heard from Peter May, so I will accept this answer, and might just ask Peter next time I see him, if he had anything else in mind when he wrote that footnote. $\endgroup$ Jun 27 at 11:37

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