Equivariant methods in combinatorial geometry

Abstract

Partition problems are classical problems of the combinatorial geometry whose solutions often rely on the methods of the equivariant topology. The k-fan partition problems introduced in [11] and first discussed by equivariant methods in [2], [3] have forced some hard concrete combinatorial calculations in equivariant cohomology [5], [4]. These problems can be reduced, by the beautiful scheme of Bárány and Matoušek, [2], to topological problems of the existence of D2n equivariant maps V2(ℝ3) → Wn→ ∪A(α) from a Stiefel manifold of all orthonormal 2-frames in R3 to complements of appropriate arrangements. In this paper we present a set of techniques, based on the equivariant obstruction theory, which can help in answering the question of the existence of a equivariant map to a complement of an arrangement. With the help of the target extension scheme, introduced in [5], we are able to deal with problems where the existence of the map depends on more then one obstruction. The introduced techniques, with an emphasis on computation, are applied on the known results of the fan partition problems.