Quillen’s conjecture

Imagine two switches controlling two binary settings. Flipping either twice restores the settings, and flipping both gives the same result in either order. Their four combined moves are do nothing, flip the first, flip the second, and flip both. When such small reversible collections are nested inside a larger collection of moves, can the shape of their nesting force a hidden collection preserved by all the larger moves?

A finite group records how moves compose and undo one another; a subgroup is a collection closed under those operations. Choose its nontrivial subgroups in which moves commute and every move other than doing nothing undoes itself. Form a geometric complex by joining nested collections, filling triangles for chains of three and higher-dimensional pieces for longer chains. In the two-switch example, the three two-move collections each sit inside the four-move collection, giving a three-spoked star. If the entire complex can continuously shrink to a point, must the group contain a nontrivial normal subgroup with 2, 4, 8, or another power of two elements? Normal means that doing any group move, then a chosen move, then undoing the first move still gives a move in the chosen collection.[1] [2]

Consider transformations of a finite object that undo themselves and commute with one another. Can the topology made by nesting their subgroups detect a nontrivial normal power-of-two part of any finite group? For $C_2\times C_2$, three order-two subgroups lie below the whole group, so the order complex is a contractible three-edge star. For the permutation group $S_3$, its three transposition subgroups are incomparable, giving three isolated vertices; its largest normal 2-subgroup is trivial. In general, $A_2(G)$ is the poset of nontrivial elementary abelian 2-subgroups: finite abelian groups whose nonidentity elements all have order two. A chain $H_0<\cdots<H_k$ gives a k-simplex. The target is $O_2(G)=1\Rightarrow |A_2(G)|\text{ is noncontractible}$ for every finite $G$. A nonzero reduced rational homology group implies noncontractibility, but that stronger conclusion is not assumed equivalent here.[1] [2]

The conjecture holds for solvable groups, so a group built from two commuting switches is a covered small case. Its three order-two subgroups each sit inside the whole four-element group, making a three-edge star with the whole group at its center; the star can shrink to a point. The July preprints report odd-prime results and reductions for more groups at two. They do not establish the converse for every finite group at two, especially the remaining nonsolvable cases.[1] [2]

Quillen established the converse for solvable groups, groups of Lie type in the defining characteristic and groups of p-rank at most two. The July 2026 unitary-group revision reports an odd-prime theorem, leaving the universal prime-two target separate. The component preprint gives conditional homology results and narrows possible components of a minimal counterexample. These are restricted or conditional statements about the stronger homological version; they do not prove noncontractibility for every finite group with trivial O₂.[1] [2]

Counting order-two elements loses information about how they combine. The two-switch group and the group permuting three objects both have three such elements, yet their subgroup arrangements differ: one gives a connected star, the other three isolated points. Studying nesting as a geometric shape therefore reveals structure absent from that count. These shapes also connect to group cohomology, which studies algebraic structure through topological invariants; they provide more than a catalogue of subgroups.[1] [2]

The two small groups both have three involutions, but inclusion makes one complex a star and the other disconnected. That difference illustrates information in subgroup topology that a raw element count cannot capture. Moreover, the complex of all nontrivial 2-subgroups and the elementary abelian subcomplex are homotopy equivalent, so the smaller collection can serve as topological input to group-cohomological and homology-decomposition methods. The conjecture concerns the normal subgroup information retained by this input. Keeping homotopy type separate from the stronger rational-homology conclusions prevents a convenient detecting invariant from being mistaken for the whole topological question.[1] [2]