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 \CP\/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 is an element of G} of G in P ordered by [x] less than or equal to [y]: double left right arrow There Exists g is an element of G: x(g) less than or equal to y. We apply all our results to the case P = Lambda(G)(0) is the lattice of subgroups H not equal 1, G of a finite group G. For finite solvable groups G we give the G-homotopy type of Lambda(G)O and we show that \C Lambda(G)(0)\/G and \C(Lambda(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 \C Lambda(G)(0)\/G and \C(Lambda(G)(0)/G)\ are not homotopy equivalent. (c) 1995 Academic Press, Inc.