Angle cocycles, Euler characteristic, and simplicial volume of smooth manifolds

Abstract

A celebrated conjecture of Gromov states that for closed aspherical manifolds, the vanishing of simplicial volume implies the vanishing of the Euler characteristic. We prove Gromov's conjecture for closed convex projective manifolds and, more generally, for closed orientable smooth manifolds admitting an immersed straightening. Generalising the straightening of simplices in properly convex domains in real projective space, we introduce the notion of immersed straightening and prove that, for every integer $d\ge 1$, there exists a constant $C_d>0$ such that every closed orientable smooth $d$-manifold $M$ admitting an immersed straightening satisfies $|\chi (M)|\le C_d||M||$.

arXiv comment37 pages. Comments are welcome!