  A solution to the normalized Ricci flow is called non-singular if it exists
for all time with uniformly bounded sectional curvature. By using the
techniques developed by the present authors, we study the existence or
non-existence of non-singular solutions of the normalized Ricci flow on
4-manifolds with non-trivial fundamental group and the relation with the smooth
structures. For example, we prove that, for any finite cyclic group ${\mathbb
Z}_{d}$, where $d>1$, there exists a compact topological 4-manifold $X$ with
fundamental group ${\mathbb Z}_{d}$, which admits at least one smooth structure
for which non-singular solutions of the normalized Ricci flow exist, but also
admits infinitely many distinct smooth structures for which {\it no}
non-singular solution of the normalized Ricci flow exists.
  Related non-existence results on non-singular solutions are also proved.
Among others, we show that there are no non-singular $\ZZ_d-$equivariant
solutions to the normalized Ricci flow on appropriate connected sums of $\bcp
^2$s and $\cpb $s ($d>1$).
