  We prove that a closed $n$-manifold $M$ with positive scalar curvature and
abelian fundamental group admits a finite covering $M'$ which is strongly
inessential. The latter means that a classifying map $u:M'\to K(\pi_1(M'),1)$
can be deformed to the $(n-2)$-skeleton. This is proven for all $n$-manifolds
with the exception of 4-manifolds with spin universal coverings.
