  We describe a construction of Riemannian metrics of nonnegative sectional
curvature on a closed smooth nonorientable 4-manifold with fundamental group of
order two that realizes a homotopy class that was not previously known to
contain nonnegatively curved manifolds. The procedure yields new metrics of
nonnegative sectional curvature on any 2-sphere bundle with base space the
2-sphere or the real projective plane.
