Homotopy of Non - Modular Partitionsand the Whitehouse

We present a class of subposets of the partition lattice n with the following property: The order complex is homotopy equivalent to the order complex of n?1 ; and the Sn-module structure of the homology coincides with a recently discovered lifting of the S n?1-action on the homology of n?1 : This is the Whitehouse representation on Robinson's space of fully-grown trees, and has also appeared in work of Getzler and Kapranov, Mathieu, and Hanlon and Stanley. One example is the subposet P n?1 n of the lattice of set partitions n; obtained by removing all elements with a unique nontrivial block. More generally, for 2 k n ? 1; let Q k n denote the subposet of the partition lattice n obtained by removing all elements with a unique nontrivial block of size equal to k; and let P k n = \ k i=2 Q i n : We show that P k n is Cohen-Macaulay with free integral homology, and that P k n and Q k n are homotopy equivalent, with Betti number n!=k ? (n ? 1)!: The posets Q k n are neither shellable nor Cohen-Macaulay. We give a simple formula for the homology as an Sn-module, in terms of the homology modules of the partition lattices k and n: We also present a short proof of the well-known result that rank-selection in a poset preserves the Cohen-Macaulay property. 0. Introduction In this paper we consider subposets of the partition lattice n obtained by removing various modular elements. Recall that n is the lattice of set partitions of an n-element set, ordered by reenement. We say a block of a partition is nontrivial if it consists of more than one element. The modular elements of n are precisely those partitions with a unique nontrivial block (for this and other basic deenitions see St3]). For a bounded poset P we denote by ^ P the proper part of P; i.e., the poset P with the greatest element ^ 1 and the least element ^ 0 removed. We write (P) for the order complex of P; the simplices of (P) are the chains of ^ P: By the ith (reduced) homology ~ H i (P) of P we mean the ith (reduced) simplicial homology of its order complex (P): All homology in this paper is taken with integer coeecients except for representation theoretic discussions, in which …