IASM Preprint Series 2021-02: REALISING SETS OF INTEGERS AS MAPPING DEGREE SETS

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REALISING SETS OF INTEGERS AS MAPPING DEGREE SETS

 

CHRISTOFOROS NEOFYTIDIS, SHICHENG WANG, AND ZHONGZI WANG


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 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.