Equivariant obstruction cocycle as a criterion for the existence of mass partitions
by
Aleksandra Dimitrijevic Blagojevic
Faculty for Agriculture, Zemun, Serbia

A significant group of problems coming from the realm of Combinatorial Geometry can be approached fruitfully through the use of Algebraic Topology. 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. 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. Recently equivariant obstruction theory has provided answers where the previous methods failed.

We demonstrate the process of solving generalized Makeev problem of partition of a measure on the sphere S² with three planes intersecting along the common line. Using the original configuration test map scheme the problem is translated to a problem of the existence of an equivariant map from the sphere S³ to the complement of an appropriate arrangement. Application of equivariant obstruction theory on this problem provides the existence of new weighted partitions of a mass by thee planes intersecting along the common line. In this talk the problem of explicit calculation of the obstruction cocycle is particularly emphasized.

Date received: May 18, 2008