Spectral sequences in combinatorial geometry: Cheeses, inscribed sets, and Borsuk-Ulam type theorems

Abstract

Algebraic topological methods are especially well suited for determining the non-existence of continuous mappings satisfying certain properties. In combinatorial problems it is sometimes possible to define a mapping from a space X of configurations to a Euclidean space Rm in which a subspace, a discriminant, often an arrangement of linear subspaces A, expresses a target condition on the configurations. Add symmetries of all these data under a group G for which the mapping is equivariant. If we remove the discriminant from Rm, we can pose the problem of the existence of an equivariant mapping from X to the complement of the discriminant in Rm. Algebraic topology may sometimes be applied to show that no such mapping exists, and hence the image of the original equivariant mapping must meet the discriminant. We introduce a general framework, based on a comparison of Leray-Serre spectral sequences. This comparison can be related to the theory of the Fadell-Husseini index. We apply the framework to: •solve a mass partition problem (antipodal cheeses) in Rd, •determine the existence of a class of inscribed 5-element sets on a deformed 2-sphere, •obtain two different generalizations of the theorem of Dold for the non-existence of equivariant maps which generalizes the Borsuk-Ulam theorem.