Computational topology of equivariant maps from spheres to complements of arrangements

Abstract

The problem of the existence of an equivariant map is a classical topological problem ubiquitous in topology and its applications. Many problems in discrete geometry and combinatorics have been reduced to such a question and many of them resolved by the use of equivariant obstruction theory. A variety of concrete techniques for evaluating equivariant obstruction classes are introduced, discussed and illustrated by explicit calculations. The emphasis is on D 2n-equivariant maps from spheres to complements of arrangements, motivated by the problem of finding a 4-fan partition of 2-spherical measures, where D 2n is the dihedral group. One of the technical highlights is the determination of the D 2n-module structure of the homology of the complement of the appropriate subspace arrangement, based on the geometric interpretation for the generators of the homology groups of arrangements.