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- Title
Jacobi identities in low-dimensional topology.
- Authors
James Conant; Rob Schneiderman; Peter Teichner
- Abstract
The Jacobi identity is the key relation in the definition of a Lie algebra. In the last decade, it has 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 in terms of Whitney towers. This paper contains the first proof of the four-dimensional version of the Jacobi identity. We also expose the underlying topological unity between the three- and four-dimensional IHX relations, deriving from a beautiful picture of the Borromean rings embedded on the boundary of an unknotted genus 3 handlebody in 3-space. This picture is most naturally related to knot and 3-manifold invariants via the theory of grope cobordisms.
- Subjects
JACOBI identity; LOW-dimensional topology; LIE algebras; INVARIANTS (Mathematics); HANDLEBODIES; COBORDISM theory
- Publication
Compositio Mathematica, 2007, Vol 143, Issue 3, p780
- ISSN
0010-437X
- Publication type
Article
- DOI
10.1112/S0010437X06002697