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 2 n D_{2n} -equivariant maps from spheres to complements of arrangements, motivated by the problem of finding a 4 4 -fan partition of 2 2 -spherical measures, where D 2 n D_{2n} is the dihedral group. One of the technical highlights is the determination of the D 2 n 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.