On the Homology of the h,k-Equal Dowling Lattice

Abstract

We define a Dowling lattice generalization of the k-equal partition lattice and the h,k-equal signed partition lattice. We use the theory of lexicographical shellability to show that it has the homotopy type of a wedge of spheres, to describe its Betti numbers, and to give a basis for its homology in terms of labelled trees. For cyclic groups, the h,k-equal Dowling lattice arises as the lattice of intersections of a complex subspace arrangement. We use Whitney homology to compute the cohomology of the complement of this arrangement. Our results generalize and unify previous research.