The Topology of k–Equal Partial Decomposition Lattices

The lattice B_n of subsets of the set {1, 2, ..., n} ordered by inclusion and the lattice \Pi_n of partitions of {1, 2, ..., n} ordered by refinement are two of the most fundamental examples in the theory of partially ordered sets (posets). A natural well-studied q–analogue of the subset lattice is the lattice B_n(q) of subspaces of the n–dimensional vector space F^n_q over the field F_q with q elements ordered by inclusion. There are many justifications for viewing this as a q–analogue. One comes from the fact that the number of maximal chains of B_n is n!, while the number of maximal chains of B_n(q) equals the q–analogue of n! which is defined by [n]_q! := [n]_q[n − 1]_q . . . [1]_q, where [n]_q := 1 + q + · · · + q^{n−1}. Another justification comes from studying the topology of a certain simplicial complex associated with the poset, called the order complex. The order complex associated with B_n is homeomorphic to a single sphere of dimension n−2, while the order complex associated with B_n(q) has the homotopy type of a wedge of q^{\binom{n}{ 2}} spheres of dimension n − 2. It is well-known that the order complex associated with \Pi_n has the homotopy type of a wedge of (n−1)! spheres of dimension n−3. Various q–analogues of the partition lattice \Pi_n have been proposed over the years, starting with the Dowling lattices introduced in a 1973 paper of Dowling. Posets studied by Welker and by Hanlon, Hersh, and Shareshian involve direct sum decompositions of vector spaces over F_q. While these posets have interesting properties analogous to those of \Pi_n, such as having the homotopy type of a wedge of spheres, none have the desirable property that the number of spheres is a q–analogue of (n−1)!. The q–analogue proposed in this thesis is the poset \Pi_n(q) of direct sum decompositions of subspaces of F^n_q whose summands all have dimension at least 2, ordered by inclusion of summands. This is actually a q–analogue of a poset that is isomorphic to \Pi_n, namely the poset of partitions of subsets of {1, 2, ..., n} in which each block has size at least 2. We show that the order complex associated with \Pi_n(q) has the homotopy type of a wedge of f(q)[n−1]_q! spheres of dimension n − 3 where f(q) is a polynomial in q that is equal to 1 when q is set equal to 1. In order to prove this result, we initiate a study of a much more general class of posets, which includes \Pi_n, \Pi_n(q), and the k–equal partition lattices introduced by Björner, Lovász, and Yao in 1992. The k–equal partition lattice \Pi^{=k}_ n is the subposet of \Pi_n consisting of partitions for which each block has size at least k or 1. In this general class, the roles of B_n and B_n(q) in the definitions of \Pi_n and \Pi_n(q) are played by an arbitrary geometric lattice L. We use shellability theory to prove that the order complex associated with a general k–equal decomposition lattice \Pi^{=k}_ L has the homotopy type of a wedge of spheres in varying dimensions when k \u3e 2 and just in dimension n − 3 when k = 2. Shellability theory also enables us to derive a complicated formula for the number of spheres in each dimension. The nontrivial step of reducing the complicated formula in the case of \Pi^{=2}_{ Bn(q)} = \Pi_n(q) to the desired f_n(q)[n − 1]_q! formula uses Stanley’s theory of exponential structures