##
**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.

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) |

### Keywords:

\(SU(2)\)-quantum invariant; closed oriented 3-manifold; sixth root of unity; Witt-invariants; signature defects of 2-fold covers; mod 3-Witt invariant; homotopy invariance; lens spaces### Citations:

Zbl 0745.57006
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\textit{R. Kirby} et al., Commun. Math. Phys. 151, No. 3, 607--617 (1993; Zbl 0779.57007)

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### References:

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