Sequences of degree-one maps between geometric 3-manifolds
Given closed, connected, orientable, \(3\)-manifolds \(M\), \(N\), we say that \(M\) dominates \(N\) if there is a nonzero degree map \(M\to N\). The author proved elsewhere that any \(3\)-manifold dominates only finitely many hyperbolic manifolds. A compact connected \(3\)-manifold \(H\) is said to be dominated by \(M\) as a hyperbolic piece if there is a Haken manifold \(N\) dominated by \(M\) such that \(H\) is homeomorphic to a hyperbolic piece in a geometric decomposition of \(N\). The main result of this paper states that given a closed, connected, orientable \(3\)-manifold \(M\) there are only finitely many pairwise nonhomeomorphic \(3\)-manifolds dominated by \(M\) as hyperbolic pieces. Combining with results of Y.~Rong et al., the author deduces the following corollary. Given a closed, connected, orientable \(3\)-manifold \(M\) there is an integer \(n=n(M)\) such that any sequence of degree one maps \(f_i:M_i\to M_{i+1}\), where \(i=0,\dots, k\) and \(M_0=M\), contains a homotopy equivalence, provided \(k>n\) and \(M_i\) is geometrizable for \(i>0\).
scientific article; zbMATH DE number 1463212