Permutohedral complex and complements of diagonal subspace arrangements

TL;DR

The paper shows that the complement of a diagonal subspace arrangement $D_{\mathbb{R}}(\mathcal{K})$ is homotopy equivalent to a cellular subcomplex $\mathrm{Perm}(\mathcal{K})$ inside the permutohedron, and provides a concrete algebraic model for its cohomology via the $(1,\dots,1)$-component of the reduced bar construction of the exterior Stanley-Reisner algebra $\Lambda[\mathcal{K}]$. It introduces the Saneblidze-Umble diagonal $\Delta_{SU}$ to compute the cohomology product on $\mathrm{Perm}(\mathcal{K})$, and proves that under the projection $\rho: \mathrm{Perm}^{m-1} \to I^{m-1}$ this diagonal maps to Li Cai's diagonal $\Delta_{LC}$ on the real moment-angle complex, thereby aligning the permutohedral and cube pictures. The construction yields an explicit cellular diagonal approximation and demonstrates a tight connection between combinatorial permutohedra and real diagonal arrangements via a derived real moment-angle complex $\mathcal{R}_{\mathcal{L}(\mathcal{K})}$. Together these results provide computable, combinatorial models for the cohomology rings of diagonal arrangement complements and their multiplicative structures. The work broadens connections between toric topology, subspace arrangements, and moment-angle theory, with concrete descriptions of both homotopy types and ring structures.

Abstract

We prove that the complement of a diagonal subspace arrangement is homotopy equivalent to a cellular subcomplex $\mathrm{Perm}(K)$ in the permutohedron. The product in the cohomology ring of a diagonal arrangement complement is described via the cellular approximation of the diagonal map in the permutohedron constructed by Saneblidze and Umble. We consider the projection from the permutohedron to the cube and prove that the Saneblidze--Umble diagonal maps to the diagonal constructed by Cai for the description of the product in the cohomology of a real moment-angle complex.

Figures (5)

• Figure 1: The retraction $r_F$
• Figure 2: Permutohedral complex $\mathop{\mathrm{Perm}}\nolimits(\mathcal{K})$
• Figure 3:
• Figure 4:
• Figure 5:

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1: BP00, Proposition 5.2.2
• Proposition 2.2: BP00, Proposition 5.3.2
• Theorem 3.1: BP00, Theorem 5.2.5
• Theorem 3.2: LiCai
• Theorem 4.1
• proof : Proof of Theorem \ref{['Perm-face']}
• Proposition 4.2
• proof