Spectral Sequences in Combinatorial Geometry: Cheeses, Inscribed Sets, and Borsuk-Ulam Type Theorems

Abstract

Algebraic topological methods are especially suited to determining the nonexistence 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 $\mathbb{R}^m$ in which a subspace, a discriminant, often an arrangement of linear subspaces $\mathcal{A}$, expresses a desirable condition on the configurations. Add symmetries of all these data under a group $G$ for which the mapping is equivariant. Removing the discriminant leads to the problem of the existence of an equivariant mapping from $X$ to $\mathbb{R}^m$- the discriminant. Algebraic topology may be applied to show that no such mapping exists, and hence 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 $\mathbb{R}^d$, 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 nonexistence of equivariant maps which generalizes the Borsuk-Ulam theorem.