# Mapping degrees between spherical $3$-manifolds

## TL;DR

This paper explicitly calculates the degrees of maps between spherical 3-manifolds, linking algebraic properties of fundamental group homomorphisms to geometric degrees, thereby determining the set of possible degrees.

## Contribution

It provides explicit formulas for degrees of maps between spherical 3-manifolds based on fundamental group homomorphisms, extending previous understanding.

## Key findings

- Explicit calculation of degree sets for surjective homomorphisms
- Method to determine degrees for arbitrary homomorphisms
- Complete characterization of possible 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 $deg f\equiv \overline{deg}\psi$ $(|\pi_1(N)|)$ 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 $D(M,N)$.