Magnitude homology of real hyperplane arrangements

Abstract

We initiate the study of magnitude theory of real hyperplane arrangements. Magnitude is a cardinality-like invariant of metric spaces or enriched categories measuring the effective size. Its categorification, the magnitude homology, is a more powerful invariant. For a real hyperplane arrangement, or more generally, an oriented matroid, the tope graph encapsulates considerable amount of information. Since tope graphs are equipped with the shortest path metric, we feed them to the magnitude and magnitude homology machinery to derive new invariants of real hyperplane arrangements. We prove some structural results of the magnitude of arrangements, including reciprocity, palindromic numerator and denominator. For magnitude homology of arrangements, we give combinatorial descriptions in small length and prove that tope graphs are diagonal if and only if the arrangement is Boolean. We present a face decomposition of magnitude homology, using which we obtain a combinatorial formula of diagonal magnitude Betti numbers. Many open problems are posted for future study. In particular, we conjecture that magnitude and magnitude homology of arrangements are determined by the intersection lattice.

Figures (2)

• Figure 1: A geometric visualization of the non-commutative Tits product in the $A_2$ arrangement. Let $F$ be a ray (blue) and $G$ be a chamber (red). (a) The product $FG$ represents taking a small step from $F$ towards $G$, which immediately lands in the adjacent chamber. (b) The product $GF$ represents taking a small step from $G$ towards $F$. Because $G$ is an open chamber, a sufficiently small step remains entirely within $G$, so $GF = G$.
• Figure 2: The tope graph of the $A_2$ arrangement embedded as a $C_6$ hexagon (blue/bold) inside the Boolean $Q_3$ cube. Suppose $b^{\lambda}=(+\xrightarrow{X_1}-\xrightarrow{X_2}+)\otimes (+\xrightarrow{Y_1}-)\otimes (-\xrightarrow{Z_1} +)$, where $\lambda=(2,1,1)$. A 4-step shuffle walk $Z_1X_1X_2Y_1$ (orange/dotted) in $\nabla(b^\lambda)$ is shown hitting the forbidden vertex $G^+$. Also notice that this walk cannot appear in other $\nabla(b^{\mu})$.

Theorems & Definitions (89)

• Theorem 1.1: Theorem 4.2.14 of Bjorner1999
• Definition 2.1
• Proposition 2.2: Proposition A.1 of Aguiar2017
• proof
• Proposition 2.3: Lemma 3.2 of Leinster2019
• proof
• Example 2.4
• Proposition 2.5: Lemmas 3.5 and 3.6 of Leinster2019
• Theorem 2.6: Theorem 4.9 of Leinster2019
• Proposition 2.7: Proposition 3.9 of Leinster2019