Geometrically Constructed Bases for Homology of Partition Lattices of Types A, B and D

We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A, B and D. This extends and explains the “splitting basis” for the homology of the partition lattice given in [20], thus answering a question asked by R. Stanley. More explicitly, the following general technique is presented and utilized. Let A be a central and essential hyperplane arrangement in Rd. Let R1, . . . , Rk be the bounded regions of a generic hyperplane section of A. We show that there are induced polytopal cycles ρRi in the homology of the proper part LA of the intersection lattice such that {ρRi}i=1,...,k is a basis for H̃d−2(LA). This geometric method for constructing combinatorial homology bases is applied to the Coxeter arrangements of types A, B and D, and to some interpolating arrangements.