Quillen's conjecture in rational homology

The explanation

Take the group S₃ of all reorderings of three objects, and the prime 2. An elementary abelian 2-subgroup is a subgroup where every element squares to the identity and any two commute. S₃ has three such subgroups besides the trivial one. Each is a single swap of two objects together with the identity. None contains another. So Quillen's shape for S₃ is three separate points. Three points cannot shrink to one point. Reduced homology counts this: it has rank 3 − 1 = 2.

Now take A₄, the even reorderings of four objects. It has three subgroups of order 2 and one of order 4 that contains all three. So the shape is a cone over the three points, which shrinks to a point. The difference is that A₄ has a normal 2-subgroup, the one of order 4. A normal subgroup is one that conjugation by any element maps to itself. A nontrivial normal p-subgroup always makes the shape contractibleSource 1.

Quillen conjectured the converse in 1978. He proved it for solvable groups and some other familiesSource 2. The almost-simple case came nextSource 3. Aschbacher and Smith then reduced the problem for primes above 5 to special cases of unitary groups. Díaz Ramos later deduced the rational-homology form for every finite group at odd primesSource 4. At the prime two, successive steps narrowed the remaining familiesSource 5. The obstacle was the last families of classical groups at the prime two.

This paper claims the conjecture for all finite groups and all primesSource 6. It claims a stronger form. If the group has no nontrivial normal p-subgroup, the shape has nonzero rational homologySource 7. Rational homology counts holes with rational-number coefficients. So the shape has a real hole, not just a failure to shrink. With the converse, a fixed field of coefficients detects exactly when such a normal subgroup existsSource 8.

To build the needed holes, the paper uses frames. A frame splits a vector space into linesSource 9. Flipping signs on each line gives an elementary abelian subgroup. The paper links frames that differ only inside a plane, so that boundary terms cancelSource 10. This produces cycles, which are the holes it needs.

How sure is this? There is no Lean check. A computer proof checker has not looked at any part of it. The proof depends on several earlier reductions and on case-by-case work for families of groups. No specialist review of the paper has been published.

What it doesn't show. This does not give a new proof of the odd-prime case from scratch; it uses earlier reductions and results for most cases.