Using Equivariant Obstruction Theory in Combinatorial Geometry

Abstract

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 Lov% ász \cite{Lovasz} through the solution of the Lovász conjecture \cite% {Babson-Kozlov}, \cite{Carsten}, 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. In this paper we are going to extract the essence of the equivariant obstruction theory in order to obtain an effective \emph{general position map%} scheme for analyzing the problem of existence of equivariant maps.