an 2 00 4 GEOMETRICALLY CONSTRUCTED BASES FOR HOMOLOGY OF PARTITION LATTICES OF TYPES

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 [19], 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 R d. Let R 1 ,. .. , R k be the bounded regions of a generic hy-perplane section of A. We show that there are induced polytopal cycles ρ Ri in the homology of the proper part L A of the intersection lattice such that {ρ Ri } i=1,...,k is a basis for H d−2 (L A). 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.