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