Computer Calculation of the Degree of Maps into the Poincare Homology Sphere

Abstract

Let M and P be Seifert 3-manifolds. Is there a degree one map f : M → P? The problem was completely solved by Hayat Legrand, Wang, and Zieschang for all casesexcept when P is the Poincaré homology sphere. We investigate the remaining case by elaborating and implementing a computer algorithm that calculates the degree. As a result, we get an explicit experimental expression for the degree through numerical invariants of the induced homomorphism f# : π₁(M) → π₁(P).