Integral foliated simplicial volume and $S^1$-actions

Abstract

The simplicial volume of oriented closed connected smooth manifolds that admit a non-trivial smooth $S^1$-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 $S^1$-action vanishes. Our proof uses the geometric construction of Yano's proof for ordinary simplicial volume as well as the parametrised uniform boundary condition for $S^1$.

18 pages; v2: typo in abstract, improved introduction, added missing notation; v3: added applications, generalisation to compact manifolds possibly with boundary; v4: updated the introduction and references