ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

6. Topology

1991 MS Classification: 57M Ohtsuki, Tomotada, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8552, Japan A filtration of the set of integral homology 3-spheres

In 1989, Witten [3] proposed his famous formula of topological invariants of 3-manifolds, based on Chern-Simons gauge theory. The formula is given by using a path integral over all G connections on a 3-manifold, where G is a fixed Lie group. Following combinatorial properties of the invariants predicted by Witten’s formula, the invariants, what we call the quantum G invariant, have been rigorously reconstructed by many researchers, say by using surgery presentations of 3-manifolds. Since we have many Lie groups, we have obtained many quantum invariants of 3-manifolds in this decade. To control these many invariants we consider the following two approaches.

• Unify them into an invariant.

• Characterize them with a common property.

By the first approach, we expect the existence of the universal invariant among quantum invariants, though the universal quantum invariant of 3manifolds is not found yet. Instead of universal quantum invariants, we have the universal invariant Ω among perturbative invariants of 3-manifolds, where the perturbative G invariant is obtained from the quantum G invariant by “asymptotic expansion”. By the second approach, we obtain the notion of finite type such that the d-th coefficient of a perturbative invariant is of finite type of degree d. For detailed arguments see [2]. We prepare some notations. A web diagram on the empty set ϕ is a trivalent graph such that a cyclic order of three edges around each trivalent vertex of the graph is fixed. We denote by A(ϕ; C) (resp. A(ϕ; Z)) the quotient vector space over C (resp. Z) subject to the AS and IHX relations. We define the degree of a web diagram to be half the number of trivalent vertices of the graph. We denote by A(ϕ; C)^{(d)} and A(ϕ; Z)^{(d)} their subspaces of degree d. We denote by Â(ϕ; C) the completion of A(ϕ; C) with respect to the degree; values of the above universal invariant Ω belong to this space Â(ϕ; C). We have a map A(ϕ; C) → C obtained by substituting a Lie algebra to web diagrams; we call the map weight system. For precise definitions of these notations see for example [2]. For simplicity we restrict our attention to integral homology 3-spheres (ZHS’s) in the following.

Theorem 1 (see [2] for precise statement) (1)(Ohtsuki for G = SO(3)) The invariant Ω is universal among perturbative invariants, i. e. , the perturbative G invariant recovers from the invariant Ω by the weight system derived from substitution of the Lie algebra of G. (2)(Le) The invariant Ω is universal among all finite type invariants i. e. , the map

{ZHS’s} ⟶^{Ω} Â(ϕ; C) ⟶^{proj} A(ϕ; C)^{(≤d)} ⟶^{W} C

is of finite type of degree d for any linear map W, and conversely any finite type invariant of degree d is obtained as the above composed map for some linear map W.

How fine do the invariants in Theorem 1 distinguish ZHS’s? (Since it is conjectured that the perturbative G invariant dominates the quantum G invariant, this question might be equivalent to the question: how fine do quantum invariants distinguish ZHS’s?) We define the d-th equivalence relation ∼_{d} among ZHS’s as follows. Two ZHS’s M and M′ are d-th equivalent, denoted by M ∼_{d} M′, if v(M) = v(M′) for any finite type invariant v of degree ≤ d. We have the ascending series of the sets of the equivalence classes;

{ZHS’s}/ ∼_{1} ⟵ {ZHS’s}/ ∼_{2} ⟵ {ZHS’s}/ ∼_{3} ⟵ · · · .

Such equivalence relations have been originally studied by Habiro by using his “claspers”; to be precise his definition of the equivalence relations is slightly stronger than ours. Habiro showed that generators of {ZHS’s}/ ∼_{d+1} are given by web diagrams of degree d in some sense. As a corollary of results of Habiro [1] we have

Theorem 2 (a corollary of results in [1]) The set {ZHS’s}/ ∼_{d+1} becomes a commutative group which is isomorphic to A(ϕ; Z)(≤d)_{conn}, where A(ϕ; Z)_{conn} denotes the subspace of A(ϕ; Z) spanned by connected web diagrams. By taking the direct limit of these isomorphisms we have the following homomorphism

\{ZHS’s\} ⟶ lim_{\overleftarrow{d}}\{ZHS’s\}/ ∼_{d} ≅ Â(ϕ; Z)_{conn}.  (1)

It is known that there exists the invariant ω(M) ∈ A(ϕ; C)_{conn} satisfying Ω(M) = exp ω(M). The map (1) gives a reconstruction of the invariant ω, which dominates all invariants in Theorem 1. If we could identify {ZHS’s} with a subset of Â(ϕ; Z)_{conn} by the map (1), all invariants in Theorem 1 would be understood as functions of weight systems on the space Â(ϕ; Z)_{conn}.

References

[1] Habiro, K. , Clasper Theory and its Applications, in preparation.

[2] Ohtsuki, T. , Combinatorial quantum method in 3-dimensional topology, lecture note at Oiwake seminar, 1996, available at http://www.is.titech.ac.jp/labs/ohtsukilab.

[3] Witten, E. , Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 360–379.