On the simplicial volumes of fiber bundles (Q2701605)

The simplicial volume \(\|M\|\) of a manifold \(M\) is a homotopy invariant which was introduced by Gromov. It measures in some sense the complexity of the fundamental class and it tends to be non-zero for large manifolds or large fundamental groups, typically the negatively curved ones. For products of compact manifolds, Gromov proved that the simplical volume is almost multiplicative, in the sense that one has the following inequalities: NEWLINE\[NEWLINE\|M_1\times M_2\|\leq c_n\|M_1\|\|M_2\|\text{ and }\|M_1\times M_2\|\geq \|M_1\|\|M_2\|.NEWLINE\]NEWLINE The question which is addressed in this paper is to what extent these inequalities hold for non-trivial fibre bundles instead of products. The authors obtain the following:NEWLINENEWLINENEWLINETheorem: Let \(X\) be the total space of an oriented surface bundle over a surface, with base \(F\) and fibre \(B\) both of genus \(\geq 2\). Then, \(\|X\|\geq 4\chi (X)>0.\)NEWLINENEWLINENEWLINETheorem: Let \(X\) be the total space of a compact oriented fibre bundle with fibre \(F\) and base \(B\). If dim\((X)\leq 4\), then \(\|X\|\geq \|F\|\|B\|.\)NEWLINENEWLINENEWLINEAs a consequence, the authors obtain the first example of an aspherical manifold with non-zero simplicial volume, but with no metric of non-positive curvature. The paper also contains other interesting results on the subject.