Jacobi identities in low-dimensional topology

The Jacobi identity is the key relation in the definition of a Lie algebra. In the last decade, it also appeared at the heart of the theory of finite type invariants of knots, links and 3-manifolds (and is there called the IHX-relation). In addition, this relation was recently found to arise naturally in a theory of embedding obstructions for 2-spheres in 4-manifolds [20]. We expose the underlying topological unity between the 3and 4-dimensional IHX-relations, deriving from a picture, Figure 3, of the Borromean rings embedded on the boundary of an unknotted genus 3 handlebody in 3-space. This is most naturally related to knot and 3-manifold invariants via the theory of grope cobordisms [4].