An Invariant of Integral Homology 3-Spheres Which Is Universal For All Finite Type Invariants

Details An Invariant of Integral Homology 3-Spheres Which Is Universal For All Finite Type Invariants An Invariant of Integral Homology 3-Spheres Which Is Universal For All Finite Type Invariants

1996-01-02

arxiv.org/abs/q-alg/9601002v2

Mathematics

Quantum Algebra

23 pages, Amslatex; Typos and minor mistakes corrected; More explanations and details added; A stronger version of operations

Scientific paper

In [LMO] a 3-manifold invariant $\Omega(M)$ is constructed using a modification of the Kontsevich integral and the Kirby calculus. The invariant $\Omega$ takes values in a graded Hopf algebra of Feynman 3-valent graphs. Here we show that for homology 3-spheres the invariant $\Omega$ is {\em universal} for all finite type invariants, i.e. $\Omega_n$ is an invariant of order $3n$ which dominates all other invariants of the same order. This shows that the set of finite type invariants of homology 3-spheres is equivalent to the Hopf algebra of Feynman 3-valent graphs. Some corollaries are discussed. A theory of groups of homology 3-spheres, similar to Gusarov's theory for knots, is presented.

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