On the realisation problem for mapping degree sets

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 [NWW] and, on the other hand, every finite set of integers (containing \(0\)) is the mapping degree set of some \(3\)-manifolds [CMV]. We show the following: (i) Not every multiplicative set \(A\) containing \(0,1\) is a self-mapping degree set. (ii) For each \(n\in\mathbb N\) and \(k\geq3\), 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 [CMV], every finite set of integers (containing \(0\)) is the mapping degree set of some \(n\)-manifolds for all \(n\neq 1,2,4,5\).