Using Equivariant Obstruction Theory in Combinatorial Geometry

A significant group of problems coming from the realm of Combinatorial Geometry can only be approached through the use of Algebraic Topology. From the first such application to Kneser’s problem in 1978 by Lovász [13] through the solution of the Lovász conjecture [1], [8], many methods from Algebraic Topology have been used. Specifically, it appears that the understanding of equivariant theories is of the most importance. The solution of many problems depends on the existence of an elegantly constructed equivariant map. For example, the following problems were approached by discussing the existence of appropriate equivariant maps.A variety of results from algebraic topology were applied in solving these problems. The methods used ranged from well known theorems like Borsuk Ulam and Dold theorem to the integer / ideal valued index theories. Recently equivariant obstruction theory has provided answers where the previous methods failed. For example, in papers [19], [15] and [5] obstruction theory was used to prove the existence of different mass partitions. In this paper we are going to extract the essence of the equivariant obstruction theory in order to obtain an effective general position map scheme for analyzing the problem of existence of equivariant maps. The general position map algorithm will be used to obtain the following results: (A) the ”half a page” proof of the Lovász conjecture due to Babson and Kozlov [1](one of two key ingredients is the Carsten’s map [8]), (B) a generalization of the result of V. Makeev [14] about the sphere S measure partition by 3-planes (Section 2.1), and (C) the new (a, b, a), class of 3-fan 2-measures partitions (Section 3.1). These three results, sorted by complexity, share a spirit of analyzing equivariant maps from spheres to complements of arrangements of subspaces. 1. Equivariant obstruction theory The basic question of any obstruction theory is to produce an invariant associated with a specific construction in such a way that the nature of the invariant points out whether the construction can or can not be performed. The (equivariant) 1991 Mathematics Subject Classification. Primary 55S35, 52C35; Secondary 68U05.