Realising sets of integers as mapping degree sets

TL;DR

This paper investigates which sets of integers can be realized as the degrees of maps between closed oriented manifolds, showing limitations and possibilities for specific types of sets, including arithmetic and geometric progressions.

Contribution

It provides new results on the (non-)realisability of certain infinite and finite sets of integers as degree sets between manifolds, especially characterizing realizability for arithmetic and geometric progressions.

Findings

Infinite subsets containing zero cannot always be realized.

Finite arithmetic progressions containing zero can be realized in 3-manifolds.

Finite geometric progressions starting from 1 can be realized with manifolds.

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 $M,N$.