Aleksandra S. Dimitrijevic Blagojevic

arXiv2008

Borsuk-Ulam Theorems for Complements of Arrangements

In combinatorial problems it is sometimes possible to define a $G$-equivariant mapping from a space $X$ of configurations of a system to a Euclidean space $\mathbb{R}^m$ for which a coincidence of the image of this mapping with an arrangement $\mathcal{A}$ of linear subspaces insures a desired set of linear conditions on a configuration. Borsuk-Ulam type theorems give conditions under which no $G$-equivariant mapping of $X$ to the complement of the arrangement exist. In this paper, precise conditions are presented which lead to such theorems through a spectral sequence argument. We introduce a blow up of an arrangement whose complement has particularly nice cohomology making such arguments possible. Examples are presented that show that these conditions are best possible.

math.AT↗

arXiv2007

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 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.

math.AT↗