The Cleavage Operad and String Topology of Higher Dimension

TL;DR

This paper introduces a new coloured operad based on the geometry of manifolds embedded in Euclidean space, which generalizes string topology operations to higher dimensions and relates to well-known operads like Cacti and framed little disks.

Contribution

It constructs a coloured operad acting on mapping spaces from manifolds to smooth manifolds, extending string topology to higher dimensions and connecting to E_{n+1}-operads and framed disk actions.

Findings

Operad acts on mapping spaces from embedded manifolds.

Recovers the Chas-Sullivan product for the circle.

Establishes a coloured E_{n+1}-operad structure for spheres.

Abstract

For a manifold N embedded inside euclidean space 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. For N the unit sphere, we indicate how this gives homological actions, generalizing the action of the Cacti operad and retrieving the Chas-Sullivan product by taking N to be the unit-circle in R^2. We go on to show that for S^n the unit sphere in 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 give actions of the framed little (n+1)-disks.