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 Kahler manifold $(M,\omega)$ with $\pi^*\omega$ cohomologous to zero, if $\pi_1(M)$ is $\mathrm{CAT}(0)$ or automatic (resp. hyperbolic), then $M$ is Kahler non-elliptic (resp. Kahler hyperbolic) and the Euler characteristic $(-1)^{\frac{\dim_\mathbb{R} M}{2}}\chi(M)\geq 0$ (resp. $>0$).