The cleavage operad and string topology of higher dimension

Abstract:

For a manifold $N$ embedded inside euclidean space $\mathbb {R}^{n+1}$, 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 $\mathbb {R}^2$. We go on to show that for $S^n$, the unit sphere in $\mathbb {R}^{n+1}$, 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.