On Euler characteristic and fundamental groups of compact manifolds

Abstract

Let $M$ be a compact Riemannian manifold, $π:\widetilde{M}\rightarrow M$ be the universal covering and $ω$ be a smooth $2$-form on $M$ with $π^*ω$ cohomologous to zero. Suppose the fundamental group $π_1(M)$ satisfies certain radial quadratic (resp. linear) isoperimetric inequality, we show that there exists a smooth $1$-form $η$ on $\widetilde M$ of linear (resp. bounded) growth such that $π^*ω=d η$. As applications, we prove that on a compact Kahler manifold $(M,ω)$ with $π^*ω$ cohomologous to zero, if $π_1(M)$ is $\mathrm{CAT}(0)$ or automatic (resp. hyperbolic), then $M$ is Kahler non-elliptic (resp. Kahler hyperbolic) and the Euler characteristic $(-1)^{\frac{\dim_\mathbb{R} M}{2}}χ(M)\geq 0$ (resp. $>0$).

22 pages