Proceedings of the American Mathematical Society

Abstract:

The set of degrees of maps $D(M,N)$, where $M,N$ are closed oriented $n$-manifolds, always contains 0 and the set of degrees of self-maps $D(M)$ always contains $0$ and $1$. Also, if $a,b\in D(M)$, then $ab\in D(M)$; a set $A\subseteq \mathbb {Z}$ so that $ab\in A$ for each $a,b\in A$ is called multiplicative. On the one hand, not every infinite set of integers (containing $0$) is a mapping degree set (Neofytidis, Wang, and Wang [Bull. Lond. Math. Soc. 55 (2023), pp. 1700–1717]) and, on the other hand, every finite set of integers (containing $0$) is the mapping degree set of some $3$-manifolds (Costoya, Muñoz and Viruel [Finite sets containing zero are mapping degree sets, arXiv:2301.13719]). We show the following:

1. Not every multiplicative set $A$ containing $0,1$ is a self-mapping degree set.
2. For each $n\in \mathbb {N}$ and $k\geq 3$, every $D(M,N)$ for $n$-manifolds $M$ and $N$ is $D(P,Q)$ for some $(n+k)$-manifolds $P$ and $Q$.

As a consequence of (ii) and Costoya, Muñoz and Viruel, every finite set of integers (containing $0$) is the mapping degree set of some $n$-manifolds for all $n\neq 1,2,4,5$.