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