Borsuk-Ulam Theorems for Complements of Arrangements

Pavle V. M. Blagojevic; Aleksandra S. Dimitrijevic Blagojevic; John McCleary

Borsuk-Ulam Theorems for Complements of Arrangements

Algebraic Topology 2008-07-10 v2 Combinatorics

Authors: Pavle V. M. Blagojevic , Aleksandra S. Dimitrijevic Blagojevic , John McCleary

Abstract

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.

Keywords

borsuk-ulam theorem arrangements equivariant maps spectral sequence cohomology blow up

Cite

@article{arxiv.math/0612002,
  title  = {Borsuk-Ulam Theorems for Complements of Arrangements},
  author = {Pavle V. M. Blagojevic and Aleksandra S. Dimitrijevic Blagojevic and John McCleary},
  journal= {arXiv preprint arXiv:math/0612002},
  year   = {2008}
}

Comments

The authors wish to acknowledge the hospitality of MSRI whose atmosphere fosters collaboration. A great deal of gratitude goes to professor C. Schultz for sharing his insight with us

Borsuk-Ulam Theorems for Complements of Arrangements

Abstract

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