Non-zero degree maps and surface bundles over \(S^ 1\)
The authors study non-zero degree maps \(f:M\to N\) between 3-dimensional compact orientable irreducible \(\partial\)-irreducible manifolds. One may ask whether there exists a map of non-zero degree between given manifolds of the same dimension, and can think of the existence of such a map as defining a partial ordering on the set of homeomorphism classes of compact connected manifolds of a given dimension. As suggested by Gromov, this partial order can be defined as follows: say that \(M\) dominates \(N\), denoted by \(M\geq N\), if there is a non-zero degree allowable map from \(M\) to \(N\). Each closed hyperbolic orientable \(n\)-manifold with \(n\neq 3\) dominates only finitely many closed orientable hyperbolic \(n\)-manifolds. The authors show that this result fails in dimension 3. The authors apply the results to study a conjecture of Thurston which says that any complete finite volume hyperbolic 3-manifold is finitely covered by a surface bundle over \(S^1\).
scientific article; zbMATH DE number 955313