##
**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:

[1] | [AK] Akbulut, S., Kirby, R.: Branched covers of surfaces in 4-manifold. Math. Ann.252, 111–131 (1980) · Zbl 0437.57001 |

[2] | [AS] Atiyah, M.F., Singer, I.M.: The index of elliptic operators. III. Ann. Math.87, 546–604 (1968) · Zbl 0164.24301 |

[3] | [CG] Casson, A.J., Gordon, C.McA.: On slice knots in dimension three. In: Geometric Topology. Proc. Symp. Pure Math. XXXII, Providence, RI: Am. Math. Soc. 1976, pp. 39–53 |

[4] | [H] Hirzebruch, F.: The signature theorem: Reminiscences and recreation. In: Prospects in Math. Ann. Math. Studies70, Princeton, NJ: Princeton Univ. Press 1971, pp. 3–31 · Zbl 0252.58009 |

[5] | [K] Kirby, R.C.: A calculus for framed links inS 3. Invent. Math.45, 35–56 (1978) · Zbl 0377.55001 |

[6] | [KM1] Kirby, R., Melvin, P.: Evaluations of the 3-manifold invariants of Witten and Reshetikhin-Turaev forsl(2, C). In: Geometry of Low-Dimensional Manifolds. London Math. Soc. Lect. Note Ser.151, Cambridge: Cambridge Univ. Press 1990, pp. 473–545 |

[7] | [KM2] Kirby, R., Melvin, P.: The 3-manifold invariants of Witten and Reshetikhin-Turaev forsl(2, C). Invent. Math.105, 473–545 (1991) · Zbl 0745.57006 |

[8] | [KM3] Kirby, R., Melvin, P.: Dedekind sums, {\(\mu\)}-invariants and the signature cocycle. Invent. Math. (to appear) · Zbl 0809.11027 |

[9] | [MH] Milnor, J., Husemoller, D.: Symmetric Bilinear Forms. Berlin, Heidelberg, New York: Springer 1973 · Zbl 0292.10016 |

[10] | [RT] Reshetikhin, N.Yu., Turaev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math.103, 547–597 (1991) · Zbl 0725.57007 |

[11] | [W] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351–399 (1989) · Zbl 0667.57005 |

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