IASM Preprint Series 2021-02: REALISING SETS OF INTEGERS AS MAPPING DEGREE SETS
REALISING SETS OF INTEGERS AS MAPPING DEGREE SETS
CHRISTOFOROS NEOFYTIDIS, SHICHENG WANG, AND ZHONGZI WANG
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 Z containing zero which cannot be realised as D(M, N) for any closed oriented n-manifolds M, N.
(ii) Every ﬁnite arithmetic progression of integers containing zero 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.