The complement of an arrangement of diagonal subspaces (x_{i_1}=...=x_{i_k}) in the real space is defined by a simplicial complex K. The complement of a diagonal subspace arrangement is shown to be homotopy equivalent to a subcomplex Perm(K) of faces of 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. This reveals and clarifies the known additive descriptions of cohomology via resolutions of the Stanley-Reisner ring. The projection from the permutohedron to the cube maps the Saneblidze-Umble diagonal to the diagonal constructed by Li Cai for the description of the product in the cohomology of a real moment-angle complex (D_1,S^0)^K.

This is a joint work with Vsevolod Tril.