Non-Haken 3-manifolds are not large with respect to mappings of non-zero degree
A compact \(3\)-manifold \(M\) dominates (resp. \(d\)-dominates) a \(3\)-manifold \(N\) if there is a map \(f:M\to N\) of non-zero degree (resp. degree \(d\)). The authors consider the question of which \(3\)-manifolds dominate at most finitely many \(3\)-manifolds and the related question of which \(3\)-manifold groups map epimorphically to at most finitely many \(3\)-manifold groups. The main results are Theorem 1.2: An (irreducible) non-Haken \(3\)-manifold \(d\)-dominates only finitely many geometric \(3\)-manifolds, for each positive \(d\) . Theorem 1.3: For a non-Haken hyperbolic \(3\)-manifold \(M\) there are only finitely many finite volume hyperbolic \(3\)-manifolds \(N\) for which the fundamental group of \(M\) maps onto the fundamental group of \(N\) and \(M\) dominates at most finitely many hyperbolic manifolds. An analogous result is obtained for cusped hyperbolic \(3\)-manifolds that do not contain closed essential surfaces. As examples the authors show: (1) The Weeks manifold \(M_W\), which is conjectured to have the smallest volume of a hyperbolic \(3\)-manifold, does not dominate any other closed hyperbolic \(3\)-manifold or a Seifert fiber space with infinite fundamental group; and the only geometric manifolds 1-dominated by \(M_W\) are the lens spaces \(L\)(5,1) and \(L\)(5,2). (2) The result \(M\) of (1,2) Dehn surgery on the figure eight knot complement does not dominate any other geometric \(3\)-manifold.
maps of non-zero degree between 3-manifolds
scientific article; zbMATH DE number 1295155