## Quantum invariants at the sixth root of unity.(English)Zbl 0779.57007

The authors give a formula which expresses the $$SU(2)$$-quantum invariant of a closed oriented 3-manifold at a sixth root of unity in terms of homology, Witt-invariants and signature defects of 2-fold covers. (Evaluations at the third and fourth roots of unity were given in [Invent. Math. 105, No. 3, 473-545 (1991; Zbl 0745.57006)].)
More precisely, $T_ 6(M)=\sum_{\theta \in H^ 1(M,\mathbb{Z}_ 2)} \tau_ 6 (M, \theta),$ where $$\tau_ 6(M,\theta)=\sqrt 3^{\varepsilon(\theta)+d(M_ \theta)-d(M)}i^{-w(M)+2 \theta^ 3+\text{def}_ 3(\theta) }$$ . Here $$\varepsilon(\theta)=0$$ or 1 according $$\theta$$ is zero or not, $$d(M)=rk(H^ 1(M,\mathbb{Z}_ 3))$$ and $$2\theta^ 3$$ is the image in $$\mathbb{Z}_ 4$$ of the cup cube of $$\theta$$ under multiplication by $$2:H^ 3(M,\mathbb{Z}_ 2)\to H^ 1(M,\mathbb{Z}_ 4)$$. $$w(M)$$ is the mod 3-Witt invariant of $$M$$, which is defined as follows: Let $$W$$ be a compact oriented 4-manifold bounded by $$M$$ and let $$A(W)$$ be its intersection form. Then $$w_{A(W)} \in\mathbb{Z}_ 4$$ is the mod 3 Witt class of $$A(W)$$, which is computed by diagonalizing $$A(W) \text{mod} 3$$ with diagonal entries from $$\{0,1,-1\}$$ and then taking the trace. Then $$w(M) : = \sigma_{A(W)}-w_{A(W)}\text{mod} 4$$, where $$\sigma_{A(W)}$$ is the signature. Finally let $$M_ \theta\to M$$ be the 2-fold covering determined by $$\theta$$. Let $$\widetilde W\to W$$ be a cyclic cover of oriented 4-manifolds extending $$M_ \theta\to M$$ branched along a closed surface $$F$$ in $$W$$. Then $\text{def}_ 3(\theta):=mw_{A(W)}-w_{A(\widetilde W)}-{m^ 2-1\over 2m}F\cdot F\text{mod} 4.$ It is pointed out that $$\tau_ 6(M)=\tau_ 6(M,0)$$ for each $$\mathbb{Z}_ 2$$-homology sphere and then $$\tau_ 6(M)=\sqrt 3^{d(M)}i^{-w(M)}$$, in particular $$\tau_ 6(M)=1$$ for $$\mathbb{Z}$$- homology spheres. The question whether $$\tau_ 6$$ is a homotopy invariant is related to the homotopy invariance of Witt defects resp. signature defects. Finally the range of $$\tau_ 6(M,\theta)$$ is determined, and $$\tau_ 6$$ is explicitly computed for lens spaces.
Reviewer: U.Kaiser (Siegen)

### MSC:

 57N10 Topology of general $$3$$-manifolds (MSC2010) 81T25 Quantum field theory on lattices 57M25 Knots and links in the $$3$$-sphere (MSC2010)

Zbl 0745.57006
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### References:

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