Here is a quote from an article by Daniel Gorenstein on the history of the classification of finite simple groups (available [here](http://www.ams.org/samplings/math-history/hmath1-gorenstein33.pdf)).



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



What is Atiyah's topological formulation of the odd order theorem?
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.


