Borsuk-Ulam Theorems for Complements of Arrangements

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 Rm for which a coincidence of the image of this mapping with an arrangement A of linear subspaces insures a desired set of linear conditions on a configuration. BorsukUlam 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. 1 Borsuk-Ulam type results Theorems of Borsuk-Ulam type present conditions preventing the existence of certain equivariant mappings between spaces. The classical Borsuk-Ulam theorem, for example, treats mappings of the form f : S → R for which f(−x) = −f(x), that is, f is equivariant with respect to the antipodal action of Z/2 on S and the action of Z/2 on R. Such a map must meet the origin. Generalizations of the Borsuk-Ulam theorem abound and their applications include some of the more striking results in some fields (see [10]). One of the more general formulations of Borsuk-Ulam type is the theorem of Dold [7]: For an n-connected G-space X and Y a free G-space of dimension at most n, there are no G-equivariant mappings X → Y . In this paper we consider a theorem of this type for which the target space is the complement of an arrangement of linear subspaces in a Euclidean space. Such spaces have been intensely investigated in recent years and they represent natural test spaces for problems in combinatorics and geometry. The nonexistence of an equivariant mapping from a configuration space associated to a problem to a complement of an arrangement means that equivariant mappings from the configuration space to the Euclidean space containing the arrangement must meet the arrangement, that is, the image must satisfy the linear conditions defining the arrangment.