Christoforos Neofytidis (Ohio State U)

Gromov-Thurston norms for circle bundles and for fibrations over the circle

Abstract: A non-negative numerical quantity $I$ of a closed oriented manifold $M$ is said to be functorial if whenever there is a map $f\colon M \to N$ then $I(M)\geq |\deg(f)|I(N)$. A prominent example is given by Gromov's simplicial volume. We will discuss vanishing and non-vanishing results for functorial numerical invariants for two classes of manifolds: In one direction, we show that there are non-vanishing functorial numerical invariants on (non-virtually trivial) circle bundles in every dimension. The base manifolds of those circle bundles can be taken to be hyperbolic and thus we generalise in all dimensions a result of Brooks and Goldmann for (non-trivial) circle bundles over hyperbolic surfaces. In another direction, in joint work with Michelle Bucher, we show that the simplicial volume of any mapping torus vanishes only in dimensions two and four.