  Up to dimension five, we can prove that given any closed Riemannian manifold
with nonnegative scalar curvature, of which the universal covering has
vanishing homology group $H_k$ for all $k\geq 3$, either it is flat or it has
Gauss-Bonnet quantity (defined by (1.3)) no greater than $8\pi$. In the second
case, the equality for Gauss-Bonnet quantity yields that the universal covering
splits as the Riemannian product of a $2$-sphere with non-negative sectional
curvature and the Euclidean space. We also establish a dominated version of
this result and its application to homotopical $2$-systole estimate unifies the
results from [BBN10] and [Zhu20].
