Borsuk-Ulam Theorems for Complements of Arrangements

Abstract

In combinatorial problems it is sometimes possible to define a $G$-equivariant mapping from a space $X$ of configurations of a system to a Euclidean space ${\mathbb{R}}^m$ for which a coincidence of the image of this mapping with an arrangement $\mathcal{A}$ of linear subspaces insures a desired set of linear conditions on a configuration. Borsuk-Ulam type theorems give conditions under which no $G$-equivariant mapping of $X$ to the complement of the arrangement exist. In this paper, precise conditions are presented which lead to such theorems through a spectral sequence argument. We introduce a blow up of an arrangement whose complement has particularly nice cohomology making such arguments possible. Examples are presented that show that these conditions are best possible.

The authors wish to acknowledge the hospitality of MSRI whose atmosphere fosters collaboration. A great deal of gratitude goes to professor C. Schultz for sharing his insight with us