Integral foliated simplicial volume and S 1-actions

The simplicial volume of oriented closed connected smooth manifolds that admit a non-trivial smooth S 1 {S^{1}} -action vanishes. In the present work, we prove a version of this result for the integral foliated simplicial volume of aspherical manifolds: The integral foliated simplicial volume of aspherical oriented closed connected smooth manifolds that admit a non-trivial smooth S 1 {S^{1}} -action vanishes. Our proof uses the geometric construction of Yano’s proof for ordinary simplicial volume as well as the parametrized uniform boundary condition for S 1 {S^{1}} .

Variations on the theme of the uniform boundary condition

The uniform boundary condition (UBC) in a normed chain complex asks for a uniform linear bound on fillings of null-homologous cycles. For the [Formula: see text]-norm on the singular chain complex, Matsumoto and Morita established a characterization of the UBC in terms of bounded cohomology. In particular, spaces with amenable fundamental group satisfy the UBC in every degree. We will give an alternative proof of statements of this type, using geometric Følner arguments on the chain level instead of passing to the dual cochain complex. These geometric methods have the advantage that they also lead to integral refinements. In particular, we obtain applications in the context of integral foliated simplicial volume.