Equivariant homotopy of posets and some applications to subgroup lattices

In this paper we consider the action of a finite group G on the geometric realization |CP| of the order complex CP of a poset P, on which a group G acts as a group of poset automorphisms. For special cases we give the G-homotopy type of |CP|. Moreover, we provide conditions which imply that the orbit space |CPvb/G is homotopy equivalent to the geometric realization of the order complex over the orbit poset PG. The poset PG is the set of orbits [x] ≔ {xg|g ϵ G} of G in P ordered by [x] ≤ [y]: → ℶg ϵ G: xg ≤ y. We apply all our results to the case P = Λ(G)0 is the lattice of subgroups H ≠ 1, G of a finite group G. For finite solvable groups G we give the G-homotopy type of Λ(G)0 and we show that |CΛ(G)0|/G and |C(Λ(G)0/G)| are homotopy equivalent. We do the same for a class of direct products of finite groups and for some examples of simple groups. Finally we show that for the Mathieu group G = M12 the orbit space |CΛ(G)0|/G and |C(Λ(G)0/G)| are not homotopy equivalent.