For M a compact, orientable manifold and N „ Rn􀀀1 a submanifold, we construct the cleavage operad that acts on MN through correspondences,
analogous to the Cacti Operad acting on MS1 , formulating String Topology.
For the unit sphere, N : Sn „ Rn􀀀1 we compute the cleavage operad
to be a coloured En􀀀1-operad. We twist this structure by the orthogonal
group SOpn 􀀀 1q to obtain an operad whose actions prescribe non-unital
pn 􀀀 1q-Batalin-Vilkovisky algebras. We show that the action through correspondences transfers to a spectral action on MSn ^ SdimpMq. This action is obtained through an extension of the Cleavage Operad. Homotopically the
extension is a simplication, and it adjoins a unit to the action on MSn.
We nally give advantages of our geometric stance on generalizing String
Topology even when N S1:We improve on equivariance of group actions on
MSn, and provide apparent links between Knot Theory and String Topology.