Jacobi identities in low-dimensional topology

Details Jacobi identities in low-dimensional topology Jacobi identities in low-dimensional topology

2004-01-30

arxiv.org/abs/math/0401427v2

"Jacobi "Jacobi "Jacobi identities in low-dimensional topology", Compositio Mathematica 143 Part 3 (2007) 780-810

Mathematics

Geometric Topology

Now closely approximates published version

Scientific paper

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. We expose the underlying topological unity between the 3- and 4-dimensional IHX-relations, deriving from a picture of the Borromean rings embedded on the boundary of an unknotted genus three handlebody in 3-space. This is most naturally related to knot and 3-manifold invariants via the theory of grope cobordisms.

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