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Here is a quote from an article by Daniel Gorenstein on the history of the classification of finite simple groups (available here).

During that year in Harvard, Thompson began his monumental classification of the minimal simple groups. He soon realized that he didn't need to know that every subgroup of the given subgroup was solvable, but only its local subgroups, and he dubbed such groups N-groups. However, the odd order theorem was still fresh in his mind. One afternoon I ran into him in Harvard Square and noticed he had a copy of Spanier's book on algebraic topology under his arm. "What in the world are you doing with Spanier?" I asked. "Michael Atiyah has given a topological formulation of the solvability of groups of odd order and I want to see if it provides an alternate way of attacking the problem," was his reply.

What is this topological formulation of the solvability of groups of odd order?

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    $\begingroup$ I guess this would have been the time to ask: en-gb.facebook.com/events/783447311806240 $\endgroup$
    – Carlo Beenakker
    Commented Jul 6, 2017 at 19:41
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    $\begingroup$ The Times just published this piece; it was also mentioned at the end of the Groups St. Andrews meeting. thetimes.co.uk/article/… $\endgroup$
    – Arturo Magidin
    Commented Aug 12, 2017 at 18:35
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    $\begingroup$ @ArturoMagidin: Thanks! Since the article is behind a loginwall/paywall, I want to quote the most interesting part here: "The [Feit-Thompson] theorem runs to 255 pages of densely argued text. Sir Michael, without the use of a computer, has reduced it to 12 pages, simply explained. This week he sent it to 15 experts in the field and is waiting for their reaction." $\endgroup$
    – spin
    Commented Aug 13, 2017 at 9:32
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    $\begingroup$ The London Times article can be read here maths.ed.ac.uk/~aar/atiyahtimes2017.pdf $\endgroup$
    – Ursula Martin
    Commented Aug 13, 2017 at 21:40
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    $\begingroup$ Sounds like the paper should be submitted for publication if he's confident. Or better: the arXiv so more than 15 experts can look at it. $\endgroup$
    – David Roberts ♦
    Commented Aug 14, 2017 at 3:37

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In an email correspondence with Atiyah, I brought this up. The comments are meant to provide background for Thompson's remark.

"When I first heard of FT I thought there should be a simpler proof using fixed point theorems and K-theory and I propagated the idea. The problem was that fixed point theorems could only deal with fixed points of elements or cyclic groups. So I knew we needed a theory that would cover fixed points of a whole group. We could then apply it to the action of a finite group on the projective space of the reduced regular representation."

He went on further to say, "It was only recently that I realized we had to use equivariant K-theory and not its completion at the identity." Then he mentioned his completion theorem of Brauer induction (with Segal), and how Snaith had given a topological proof of Brauer induction. Although not definitive, Atiyah may have a shortened proof of the Feit-Thompson theorem, and if so it would be presented in the near future.

In case it is helpful, I see that the quote from Gorenstein's paper was around 1960. Atiyah had published the finite group version of the Atiyah-Segal completion in 1961 ("Characters and cohomology of finite groups"). So, based on this recent email correspondence, these 1961 ideas make their appearance in a formulation of FT.

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    $\begingroup$ I've no clue whether this will be important, but I once corresponded with Snaith about how his proof of Brauer induction in his "Explicit...." book is not correct. $\endgroup$
    – Echocampfire
    Commented Sep 5, 2017 at 15:40
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    $\begingroup$ I only meant to provide explanation for the quote. I do not believe an actual proof will appear by Atiyah, but your comment would be relevant if it did. $\endgroup$
    – Chris Gerig
    Commented Sep 5, 2017 at 18:18
  • $\begingroup$ Near future... or never, like the nonexistence of the complex structure on S6. $\endgroup$
    – Bernie
    Commented Jan 2, 2018 at 23:55
  • $\begingroup$ There is a recent preprint by Alain Connes which seems to have more information on Atiyah's ideas about the odd order theorem. Link: arxiv $\endgroup$
    – spin
    Commented Jan 31, 2019 at 10:38
  • $\begingroup$ Quote from abstract: "In this note we investigate the idea of Michael Atiyah of using, as a possible approach to the Theorem of Feit-Thompson on the solvability of finite groups of odd order, the iterations of the transformation which replaces a representation of a finite group G on a finite dimensional complex vector space E by the difference between the associated representation of G on the sum of exterior powers of E and the trivial representation." $\endgroup$
    – spin
    Commented Jan 31, 2019 at 10:40

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