Permutohedral Complex and Complements of Diagonal Subspace Arrangements

T. E. Panov^abcd, V. A. Tril<sup>ca

a</sup> Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
^b Institute for Information Transmission Problems (Kharkevich Institute), Russian Academy of Sciences, Moscow, Russia
^c National Research University Higher School of Economics, Moscow, Russia
^d Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: The complement of an arrangement of diagonal subspaces $x_{i_1}=\dots =x_{i_k}$ in the real space is defined by a simplicial complex $\mathcal K$. In this paper, we prove that the complement of a diagonal subspace arrangement is homotopy equivalent to a subcomplex $\mathrm {Perm}(\mathcal K)$ of faces of the permutohedron. The product in the cohomology ring of the complement of a diagonal arrangement is then described via Saneblidze and Umble's cellular approximation of the diagonal map in the permutohedron. We consider the projection from the permutohedron to the cube and prove that the Saneblidze–Umble diagonal is mapped to the diagonal constructed by Li Cai for describing the product in the cohomology of a real moment–angle complex.

Keywords: permutohedron, complements of diagonal subspace arrangements, dga models, Saneblidze–Umble diagonal approximation.

Received: May 5, 2025
Revised: July 6, 2025
Accepted: July 9, 2025

DOI: 10.4213/tm4488

 English version: Proceedings of the Steklov Institute of Mathematics, 2025, 330, 254–269

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