On Euler characteristic and fundamental groups of compact manifolds

Abstract

Let M be a compact Riemannian manifold, \(\pi :\widetilde{M}\rightarrow M\) be the universal covering and \(\omega \) be a smooth 2-form on M with \(\pi ^*\omega \) cohomologous to zero. Suppose the fundamental group \(\pi _1(M)\) satisfies certain radial quadratic (resp. linear) isoperimetric inequality, we show that there exists a smooth 1-form \(\eta \) on \(\widetilde{M}\) of linear (resp. bounded) growth such that \(\pi ^*\omega =d \eta \). As applications, we prove that on a compact Kähler manifold \((M,\omega )\) with \(\pi ^*\omega \) cohomologous to zero, if \(\pi _1(M)\) is \(\mathrm{CAT}(0)\) or automatic (resp. hyperbolic), then M is Kähler non-elliptic (resp. Kähler hyperbolic) and the Euler characteristic \((-1)^{\frac{\dim _{\mathbb R}M}{2}} \chi (M)\ge 0\) (resp. \(>0\)).