Filling Volumes and Simplicial Volume of Mapping Tori

In this thesis we introduce two numerical invariants, namely the integral and the real filling volume, defined on orientation-preserving self-homotopy equivalences f of a closed orientable manifold M. These two invariants catch the complexity of the action of the map on the space of (integral or real) singular chains and they are closely related to the (integral or real) simplicial volume of mapping tori. Indeed, if the map f is a homeomorphism of a closed manifold, the real filling volume of f coincides with the simplicial volume of the mapping torus associated to f. On the other hand, the integral filling volume of f constitutes a lower bound for the integral simplicial volume of the associated mapping torus and an upper bound for the stable integral simplicial volume of the same mapping torus.After general considerations on these invariants, we focus on low dimensional cases: we establish when the real and the integral filling volume of an orientation-preserving self-homotopy equivalence of a surface vanish or do not vanish. By employing the neat relation between the real filling volume of f and the simplicial volume of the corresponding mapping torus, we prove that the real filling volume of f is always zero when M is 3-dimensional, while the same does not hold for the integral filling volume.We finally utilize these results to prove that in dimension 3 integral simplicial volume and triangulation complexity are deeply different, where the triangulation complexity of a manifold is the minimal number of simplices required to triangulate it.