On the Cohen-Macaulay connectivity of supersolvable lattices and the homotopy type of posets

It is a well known fact that a supersolvable lattice is EL-shellable. Hence a supersolvable lattice (resp., its Stanley-Reisner ring) is Cohen-Macaulay. We prove that if L is a supersolvable lattice such that all intervals have non-vanishing Möbius number, then for an arbitrary element x ϵ L the poset L − {x} is also Cohen-Macaulay. Posets with this property are called 2-Cohen-Macaulay posets. In particular, in this case the type of the Stanley-Reisner ring of L is given by the absolute value of the Möbius number μ(L). On the other hand, it is a simple observation that the non-vanishing of the Möbius number on intervals is a necessary condition for a poset to be 2-Cohen-Macaulay. For the proof of the 2-Cohen-Macaulayness we will derive some remarkable results about the homotopy type of posets, in particular on posets the homotopy type of which is a wedge of spheres.