Geometrically constructed bases for homology of partition lattices of types A, B and D

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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 [Wa96], 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 ρᵣᵢᵢ₌₁,...,ₖ is a basis for Hd₋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.