ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

6. Topology

1991 MS Classification: 57M25 Le, Thang, SUNY Buffalo, USA On Quantum and Finite Type Invariants of Homology 3-Spheres In 1995, we (joint work with J. Murakami and T. Ohtsuki) constructed an invariant Ω of 3-manifolds with values in the space of 3-valent graphs. The construction utilizes the Kontsevich integral which is a universal finite type invariant of links. Later the author showed that Ω is a universal finite type invariant of homology 3-spheres. For 𝔤 = sl_{2}, Ohtsuki extracted a power series invariant τ^{𝔤} of homology 3-spheres from quantum 𝔤-invariants (the Witten-Reshetikhin-Turaev invariants) at roots of unity. This formal power series can be regarded as the perturbative expansion of the Witten path integral. We generalize this result, constructing τ^{𝔤} for other simple Lie algebras, and explain the relation between the universal invariant Ω and τ^{𝔤}. We will also present some properties of the Kontsevich integral and Ω, including symmetry and integrality properties.