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{\bf Anders Bj\"orner and Michelle L. Wachs}
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{\bf 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 by M.~L.~Wachs, thus answering
a question asked by R.~Stanley.
More explicitly, the following general technique is presented and
utilized. Let ${\cal A}$ be a central and essential hyperplane
arrangement in ${\Bbb{R}}^d$. Let $R_1,\dots,R_k$ be the bounded
regions of a generic hyperplane section of ${\cal A}$. We show that
there are induced polytopal cycles $\rho_{R_i}$ in the homology of the
proper part $\overline{L}_{\cal A}$ of the intersection lattice such
that $\{\rho_{R_i}\}_{i=1,\dots,k}$ is a basis for $\widetilde{H}_{d-2}
(\overline{L}_{\cal A})$. 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.
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