Full generality of the Euler characteristic congruence for subrack lattices of p-power racks

Establish the validity of the Euler characteristic congruence in Theorem thm:euler without any additional hypotheses on parabolic subracks: specifically, for every finite group G and prime p dividing |G|, determine whether the reduced Euler characteristic of the order complex of the proper part of the subrack lattice of the p-power rack G_p satisfies tildeχ(Δ(\overline{\mathcal{R}(G_p)})) ≡ 0 mod |G|_p if and only if the p-core O_p(G) is not equal to G_p, where G_p denotes the conjugation rack consisting of all p-power elements of G and O_p(G) is the largest normal p-subgroup of G.

Background

The paper introduces the p-power rack G_p of a finite group G (the set of all elements of p-power order with the conjugation operation) and studies the homotopy type and Euler characteristic of its subrack lattice. Theorem thm:euler proves a modular divisibility statement for the reduced Euler characteristic of the order complex of the subrack lattice under the hypothesis that for every p-subgroup J of G, the set of parabolic subracks containing J has a unique minimum element.

The authors note that they could not establish this divisibility in full generality (i.e., without the unique-minimum hypothesis). They further explain that such a general theorem would imply a sharp characterization of when the subrack complex is homotopy equivalent to a sphere, paralleling Quillen’s conjecture for subgroup complexes. This situates the open problem within a broader effort to develop rack analogues of classical results in the topology of subgroup posets.