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5.7 Topological field theories of Reshetikhin-Turaev type and invariants
of 3-manifolds and knots
Observation 5.7.1.
We need the following description of closed oriented three-manifolds.
• Let L ⊂ S 3 be a framed link with components L i . Choose around all components L i a
solid torus U i which is small enough not to intersect any other component of the link.
Choose a parametrization of these solid tori:
ϕ i : U i → S 1 × D 2 ,
where D 2 is a standard disc in R 2 . The parametrization should be such that the link L i
itself is mapped to the center of the discs,
ϕ i (L i ) = S 1 × {0}
and that the normal vector field of the framing is mapped to a constant vector field of
R 2 .
• Consider now a solid four-ball B 4 with boundary a three-sphere, ∂B 4 = S 3 .
Consider four-dimensional handles D 2 × D 2 which are four-dimensional manifolds with
boundary
∂(D 2 × D 2 ) = S 1 × D 2 ∪ D 2 × S 1
For any component L i , glue a handle to S 3 = ∂B 4 by identifying the boundary of the
handle with the parametrization ϕ i to the solid torus U i ⊂ S 3 .
This gives a new oriented four-dimensional manifold W L with boundary M L = ∂W L a
closed oriented three-manifold. We say that three-manifold M L has been obtained from
S 3 by surgery on the link L ⊂ S 3 .
We can view M L as S 3 \ ⋃ i
tori.
U i , with other solid tori glued in instead of the removed solid
• A theorem of Lickorish and Wallace asserts that any closed oriented three-manifold can
be obtained in this way.
• For a given modular category C, we can now define invariants of three-manifolds. Let L
be a surgery link for M. Denote by coll(L) the set of colourings of the link L by objects
in the dominating family {V i } i∈I . We then define
τ(M) = Δ σ(L) D −σ(L)−m−1
∑
λ∈coll(L)
( ∏ m
dim V λ(i)
)F (L, λ) ∈ k ,
where m is the number of components of the link L and σ(L) is another integer associated
to the link L.
Theorem 5.7.2.
The element τ(M) ∈ K is a topological invariant of M.
i=1
Proof.
The invariance on the choice of the surgery link L follows from the invariance under the
so-called Kirby moves.
✷
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