Equivariant homotopy of posets and some applications to subgroup lattices

Abstract

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 P G . The poset P G is the set of orbits [ x] ≔ { x g | g ϵ G} of G in P ordered by [x] ≤ [y]: → ℶg ϵ G: x g ≤ 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 = M 12 the orbit space | CΛ( G) 0|/ G and | C( Λ( G) 0/ G)| are not homotopy equivalent.