Homology representations arising from the half cube

TL;DR

This paper constructs a CW decomposition of the half cube, analyzes its homology, and reveals connections to hyperplane arrangements, Pascal-like triangles, and Coxeter group actions, providing new insights into topological and algebraic structures.

Contribution

It introduces a CW decomposition of the half cube compatible with its polytope structure and links its homology to hyperplane arrangements and Coxeter group actions.

Findings

Homology of $C_{n,k}$ is concentrated in degree $k-1$.

$(k-1)$-st Betti number equals the $(k-2)$-nd Betti number of a hyperplane complement.

Betti numbers form coefficients of a Pascal-like sequence (A119258).

Abstract

We construct a CW decomposition $C_n$ of the $n$-dimensional half cube in a manner compatible with its structure as a polytope. For each $3 \leq k \leq n$, the complex $C_n$ has a subcomplex $C_{n, k}$, which coincides with the clique complex of the half cube graph if $k = 4$. The homology of $C_{n, k}$ is concentrated in degree $k-1$ and furthermore, the $(k-1)$-st Betti number of $C_{n, k}$ is equal to the $(k-2)$-nd Betti number of the complement of the $k$-equal real hyperplane arrangement. These Betti numbers, which also appear in theoretical computer science, numerical analysis and engineering, are the coefficients of a certain Pascal-like triangle (Sloane's sequence A119258). The Coxeter groups of type $D_n$ act naturally on the complexes $C_{n, k}$, and thus on the associated homology groups.