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 is an element of D(M), then ab is an element of D(M); a set A subset of Z so that ab is an element of A for each a, b is an element of 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 is an element of N and k >= 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 [CMV], every finite set of integers (containing 0) is the mapping degree set of some n-manifolds for all n not equal 1, 2, 4, 5.