Online Talk

An arrangement of real or complex diagonal subspaces is defined by a simplicial complex K. We present the complex Perm(K) of faces of the permutohedron and prove that it is homotopy equivalent to the complement D(K) of a diagonal arrangement. The product in the cohomology is then described via Saneblidze and Umble’s cellular approximation of the diagonal map in the permutohedron. We establish the connection between permutohedral complexes and moment-angle complexes that links the Saneblidze--Umble diagonal and the dga-model for a real moment-angle complex constructed by Cai.

Also we discuss the K(\pi, 1) problem in case of diagonal arrangements. We consider arrangements for which the corresponding permutohedral complex admits a metric of non-positive curvature.