Permutohedral complex and complements of diagonal subspace arrangements

The complement of an arrangement of diagonal subspaces $x_{i_1}=\cdots =x_{i_k}$ in the real space is defined by a simplicial complex $K$. In this paper, we prove that the complement of a diagonal subspace arrangement is homotopy equivalent to a subcomplex $\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{m}(K)$ 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 maps to the diagonal constructed by Li Cai for describing the product in the cohomology of a real moment-angle complex.