Geometrically Constructed Bases for Homology of Non-Crossing Partition Lattices

TL;DR

This paper constructs a geometric basis for the homology of non-crossing partition lattices associated with finite reflection groups, linking it to existing intersection lattice bases through hyperplane arrangements.

Contribution

It introduces a novel geometric basis for homology of non-crossing partition lattices related to reflection groups, expanding the understanding of their topological structure.

Findings

Established a geometric basis for homology of non-crossing partition lattices.

Connected the new basis to the intersection lattice basis by Bj"orner and Wachs.

Utilized affine hyperplanes in the construction process.

Abstract

For any finite, real reflection group $W$, we construct a geometric basis for the homology of the corresponding non-crossing partition lattice. We relate this to the basis for the homology of the corresponding intersection lattice introduced by Bj\"{o}rner and Wachs in \cite{BW} using a general construction of a generic affine hyperplane for the central hyperplane arrangement defined by $W$.