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

Abstract

Let $M$ and $P$ be Seifert 3-manifolds. Is there a degree one map $f\colon M \rightarrow P\,$? The problem was completely solved by Hayat-Legrand, Wang, and Zieschang for all cases except 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 $f_{\#}\colon \pi_1 (M) \rightarrow \pi_1(P)$.