The notion of amenable category was introduced in [J. C. Gómez-Larrañaga et al., Algebr. Geom. Topol. 13, No. 2, 905–925 (2013; Zbl 1348.55006)] as a variant of Lusternik-Schnirelmann category. It is defined via open covers of spaces for which the inclusions of the elements of the cover induce amenable subgroups on fundamental group level. In their recent article [“Amenable category and complexity”, Preprint, arXiv:2012.00612], P. Capovilla et al. pose a question which is an analogue of Rudyak’s monotonicity question for Lusternik-Schnirelmann category: Let \(M\) and \(N\) be closed manifolds. If there exists a map \(M \to N\) of non-zero degree, does it follow that \(\mathrm{cat}_{\mathrm{Am}}(M) \leq \mathrm{cat}_{\mathrm{Am}}(N)\)?
In the present article, the author gives a positive answer to this amenable monotonicity question in the three-dimensional case. To show this, he collects various results from [J. C. Gómez-Larrañaga et al., loc. cit.] to give explicit characterizations of 3-manifolds with \(\mathrm{cat}_{\mathrm{Am}}(M)=k\), where \(k \in \{1,2,3,4\}\). Combining these characterizations with known results about prime decompositions of 3-manifolds and known results about maps of non-zero degree, the author derives the claim straight from these computations.