On the Homology of the Real Complement of the $k$-Parabolic Subspace Arrangement

TL;DR

This paper investigates the topology of the complement of $k$-parabolic arrangements, generalizing Coxeter arrangements, by constructing a homotopy equivalent cell complex, analyzing Betti numbers, and proving torsion-free homology.

Contribution

It introduces a new cell complex $Perm_k(W)$ for $k$-parabolic arrangements, providing combinatorial Betti number interpretations and torsion-free homology results.

Findings

Constructed a homotopy equivalent cell complex for the arrangement complement.

Derived combinatorial formulas for Betti numbers.

Proved the homology groups are torsion free.

Abstract

In this paper, we study $k$-parabolic arrangements, a generalization of the $k$-equal arrangement for any finite real reflection group. When $k=2$, these arrangements correspond to the well-studied Coxeter arrangements. We construct a cell complex $Perm_k(W)$ that is homotopy equivalent to the complement. We then apply discrete Morse theory to obtain a minimal cell complex for the complement. As a result, we give combinatorial interpretations for the Betti numbers, and show that the homology groups are torsion free. We also study a generalization of the Independence Complex of a graph, and show that this generalization is shellable when the graph is a forest. This result is used in studying $Perm_k(W)$ using discrete Morse theory.