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 $degf\equiv \overline{deg}\psi$ $(|{\pi }_1(N)|)$ where $0\le \overline{deg}\psi <|{\pi }_1(N)|$ and $\overline{deg}\psi$ only depends on the induced homomorphism $\psi =f_{\pi }$ on the fundamental group. In this paper, we calculate explicitly the set $\{\overline{deg}\psi \}$ when $\psi$ is surjective and then we show how to determine $\overline{deg}(\psi )$ for arbitrary homomorphisms. This leads to the determination of the set $D(M,N)$.

24 pages