Extensions of the Euler-Satake Characteristic of Closed 3-Orbifolds

Abstract

The project described here is a continuation of the work of several authors on orbifold invariants in low-dimensional and algebraic topology. Speci cally, my research explores applying the Euler-Satake characteristic to the 􀀀{sectors of an orbifold for a nitely-generated group 􀀀 which results in a numerical invariant of the original orbifold, the 􀀀{Euler-Satake characteristic. Most Euler characteristics have proven ine ective in giving useful information on orbifolds and, in particular, 3{orbifolds. Orbifolds can be partitioned into two categories: orientable and non-orientable. This partition is determined by the types of singularities in the orbifolds. My work previous to this project has dealt with formulating this invariant for orientable 3{orbifolds which lead to the successful determination of their point singularities when 􀀀 = F`, the free group of ` generators (see [3]). By now considering non-orientable 3{orbifolds, we have developed a formulation of the 􀀀{Euler{Satake characteristic for all closed, e ective 3{orbifolds. In light of these formulas, counterexamples exist to show that neither an in nite collection of F`{ nor Z`{Euler{Satake characteristics determine the point singularities of general closed 3{orbifolds. Furthermore, counterexamples exist which prove that even an in nite collection of both F`{ and Z`{Euler{Satake characteristics do not determine the point singularities of general closed 3{orbifolds.