The ring structure on the cohomology of coordinate subspace arrangements

Abstract

Every simplicial complex \(\Delta\subset2^{[n]}\) on the vertex set \([n]=\{1,\ldots,n\}\) defines a real resp. complex arrangement of coordinate subspaces in \(\mathbb R^n\) resp. \(\mathbb C^n\) via the correspondence \(\Delta\ni\sigma\mapsto{\rm span\,}\{e_i:i\in\sigma\}.\) The linear structure of the cohomology of the complement of such an arrangement is explicitly given in terms of the combinatorics of \(\Delta\) and its links by the Goresky–MacPherson formula. Here we derive, by combinatorial means, the ring structure on the integral cohomology in terms of data of \(\Delta\). We provide a non-trivial example of different cohomology rings in the real and complex case. Furthermore, we give an example of a coordinate arrangement that yields non-trivial multiplication of torsion elements.