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

Anders Bj\"orner and Michelle L. Wachs

Abstract: 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 a 1986 paper of the second author. More explicitly, the following is done. Let $A$ be a central and essential hyperplane arrangement in $\Bbb R^d$. Let $R_1,...,R_k$ be the bounded regions of a generic hyperplane section of $A$. We show that there are induced polytopal cycles $\rho_{R_i}$ in the homology of the proper part $\oli L_A$ of the intersection lattice such that $\{\rho_{R_i}\}_{i=1,\dots,k}$ is a basis for $\tilde H_{d-2} (\oli L_A)$. This method for constructing homology bases is applied to the Coxeter arrangements of types A, B and D, and to some interpolating arrangements.

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