Realising sets of integers as mapping degree sets

Abstract

Given two closed oriented manifolds $M,N$ of the same dimension, we denote the set of degrees of maps from $M$ to $N$ by $D(M,N)$. The set $D(M,N)$ always contains zero. We show the following (non-)realisability results: (i) There exists an infinite subset $A$ of $\mathbb Z$ containing $0$ which cannot be realised as $D(M,N)$, for any closed oriented $n$-manifolds $M,N$. (ii) Every finite arithmetic progression of integers containing $0$ can be realised as $D(M,N)$, for some closed oriented $3$-manifolds $M,N$. (iii) Together with $0$, every finite geometric progression of positive integers starting from $1$ can be realised as $D(M,N)$, for some closed oriented manifolds $M,N$.