  Via a compactification of the cleavage operad, we describe two actions on the
mapping spaces from spheres into compact, oriented manifolds. The map from the
compactified version of the cleavage operad provides a version of intersection
theory along the underlying base manifold lifted to the mapping space. The
other action map only works for the uncompactified cleavage operad, and gives a
homotopy theoretic version of higher dimensional string topology.
