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Spectral sequences in combinatorial geometry: Cheeses, Inscribed sets, and Borsuk-Ulam type theorems
Algebraic topological methods are especially suited to determining the nonexistence of continu-
ous 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 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 Rm - 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 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 nonexistence of equivariant maps
which generalizes the Borsuk-Ulam theorem
Spectral sequences in combinatorial geometry: Cheeses, Inscribed sets, and Borsuk-Ulam type theorems
Algebraic topological methods are especially suited to determining the nonexistence of continu-
ous 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 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 Rm - 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 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 nonexistence of equivariant maps
which generalizes the Borsuk-Ulam theorem