Multipath complexes of bidirectional polygonal digraphs

TL;DR

This work investigates the homotopy type and high-connectivity properties of multipath complexes $X(G)$ for bidirectional polygonal digraphs, extending prior results on cycle-free chessboard complexes. The authors develop a toolbox including blow-up operations, $T$-operations (suspensions), and a Mayer-Vietoris spectral sequence to compute homology, yielding explicit connectivity and homology for bidirectional polygonal digraphs $BP_n$ and related graphs. They prove that $X(BP_n)$ is $\nu_n$-connected with $\nu_n = \left\lfloor \frac{n-1}{2} \right\rfloor - 1$, and provide detailed homology groups depending on $n \pmod{4}$, showing non-shellability and leaving open whether the complexes are wedges of spheres. These results contribute to the understanding of multipath complexes and their connections to matching and chessboard complexes, with implications for related homology theories of algebras.

Abstract

In this work we study the homotopy type of multipath complexes of bidirectional path graphs and polygons, motivated by works of Vrećica and Živaljević on cycle-free chessboard complexes (that is, multipath complexes of complete digraphs). In particular, we show that bidirectional path graphs are homotopic to spheres and that, in analogy with cycle-free chessboard complexes, multipath complexes of bidirectional polygonal digraphs are highly connected. Using a Mayer-Vietoris spectral sequence, we provide a computation of the associated homology groups. We study T-operations on graphs, and show that this corresponds to taking suspensions of multipath complexes. We further discuss (non) shellability properties of such complexes, and present new open questions.

Figures (7)

• Figure 1: The coherently oriented linear graph ${\tt I}_3$ (top left), the multipath complex $X({\tt I}_3)$ (top right), and the path poset $P({\tt I}_3)$ (bottom).
• Figure 2: Transitive tournament ${\tt T}_3$ (in red on the left), its blow-up (in blue at the centre), and the associated bipartite graph ${\tt B}_3$ (in orange on the right).
• Figure 3: The bidirectional linear graph ${\tt BL}_n$.
• Figure 4: The linear graph ${\tt W}_n$ (on the left) and the graph $\widehat{{\tt BL}_n}$ (on the right).
• Figure 5: The definition of $T$-operation, which is a glueing of a copy of ${\tt W}_2$ (in blue) to a graph ${\tt G}'$.

Theorems & Definitions (33)

• Theorem : Theorem \ref{['thm:highconn']}
• Theorem : Theorem \ref{['thm:connectivity_homology']}
• Definition 1.1
• Definition 1.2
• Definition 1.3: arXiv:2401.01248
• Theorem 1.4: arXiv:2401.01248
• Definition 1.5: Omega
• Theorem 1.6: Omega
• Conjecture 1.7: Omega
• Lemma 2.1: BjorTopMeth