  The period map for a smooth closed 4-manifold assigns to a Riemannian metric
the space of self-dual harmonic 2-forms. This map is from the space of metrics
to the Grassmannian of maximal positive subspaces in the second cohomology,
where positivity is defined by cup product. We show that the period map has
dense image for every 4-manifold, and that it is surjective if $b^+=1$. Similar
results hold for manifolds of dimension a multiple of four. The proofs involve
families of metrics constructed by stretching along various hypersurfaces.
