Volume and macroscopic scalar curvature

We prove the macroscopic cousins of three conjectures: (1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, (2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, (3) a conjectural bound of $$\ell ^2$$ ℓ 2 -Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of 1-balls in the universal cover.

Topological volumes of fibrations: a note on open covers

We establish a straightforward estimate for the number of open sets with fundamental group constraints needed to cover the total space of fibrations. This leads to vanishing results for simplicial volume and minimal volume entropy, e.g., for certain mapping tori.

The spectrum of simplicial volume of non-compact manifolds

We show that, in dimension at least 4, the set of locally finite simplicial volumes of oriented connected open manifolds is $$[0,\infty ]$$ [ 0 , ∞ ] . Moreover, we consider the case of tame open manifolds and some low-dimensional examples.

Integral foliated simplicial volume and circle foliations

We show that the integral foliated simplicial volume of a compact oriented smooth manifold with a regular foliation by circles vanishes.

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}} .

Surface bundles over surfaces: new inequalities between signature, simplicial volume and Euler characteristic

We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, namely the ones that arise through ramified coverings. These are among the main known examples of bundles with non-zero signature.