  We find an obstruction to the existence of non-singular solutions to the
normalized Ricci flow on four-manifolds with $b^+=1$. By using this
obstruction, we study the relationship between the existence or non-existence
of non-singular solutions of the normalized Ricci flow and exotic smooth
structures on the topological 4-manifolds ${\mathbb C}{P}^2 # l
\overline{{\mathbb C}{P}^2}$, where $5 \leq l \leq 8$.
