  We construct an invariant for non-spin 4-manifolds by using 2-torsion
cohomology classes of moduli spaces of instantons on SO(3)-bundles. The
invariant is an SO(3)-version of Fintushel-Stern's 2-torsion instanton
invariant. We show that this SO(3)-torsion invariant is non-trivial for $2CP^2
# -CP^2$, while it is known that any known invariant of $2CP^2 # -CP^2$ coming
from the Seiberg-Witten theory is trivial since $2CP^2 # -CP^2$ has a positive
scalar curvature metric.
