Mapping degrees between spherical $3$-manifolds

Abstract

Let $D(M,N)$ be the set of integers that can be realized as the degree of a map between two closed connected orientable manifolds $M$ and $N$ of the same dimension. For closed $3$-manifolds with $S^3$-geometry $M$ and $N$, every such degree $deg f\equiv \overline{deg}ψ$ $(|π_1(N)|)$ where $0\le \overline{deg}ψ<|π_1(N)|$ and $\overline{deg}ψ$ only depends on the induced homomorphism $ψ=f_π$ on the fundamental group. In this paper, we calculate explicitly the set $\{\overline{deg}ψ\}$ when $ψ$ is surjective and then we show how to determine $\overline{deg}(ψ)$ for arbitrary homomorphisms. This leads to the determination of the set $D(M,N)$.