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Unfortunately, much influential work of his in this direction remains unpublished.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?
Best Answer
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:
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:
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).
Unfortunately, this is from February 2006, and he passed away about three months later; I regret not having the chance to discuss more.