The cleavage operad and string topology of higher dimension

For a manifold N embedded inside euclidean space ℝⁿ⁺¹, we produce a coloured operad that acts on the space of maps from N to M, where M is a compact, oriented, smooth manifold. Our main example of interest is N, the unit sphere, and we indicate how this gives homological actions, generalizing the action of the spineless cacti operad and retrieving the Chas-Sullivan product by taking N to be the unit circle in ℝ². We go on to show that for Sn, the unit sphere in ℝⁿ⁺¹, the operad constructed is a coloured E_n+1-operad. This E_n+1-structure is finally twisted by SO(n + 1) to homologically agree with actions of the operad of framed little (n + 1)-disks.