Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism.

*(English)*Zbl 1086.57013
Lyubich, Mikhail (ed.) et al., Graphs and patterns in mathematics and theoretical physics. Proceedings of the conference dedicated to Dennis Sullivan’s 60th birthday, June 14–21, 2001. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3666-8/hbk). Proceedings of Symposia in Pure Mathematics 73, 173-203 (2005).

The concept of finite type invariants for integral homology three-spheres (three-manifolds whose integral homology groups are the same as those of the three-sphere) was introduced by T. Ohtsuki [J. Knot Theory Ramifications 5, No. 1, 101–115 (1996; Zbl 0942.57009)]. M. Goussarov, S. Garoufalidis, M. Polyak [Geom. Topol. 5, 75–108 (2001; Zbl 1066.57015)] and K. Habiro [Geom. Topol. 4, 1–83 (2000; Zbl 0941.57015)] independently generalized it by using \(Y\)-links, framed graphs with univalent and trivalent vertices embedded in a three-manifold.

Two compact three-manifolds \(M\) and \(N\) are called equivalent if there exists a homomorphism \(\rho\colon H_{1}(M;\mathbb{Z})\to H_{1}(N;\mathbb{Z})\) inducing an isometry of the linking forms on their torsion subgroups and a homeomorphism \(\partial M\to\partial N\) consistent with \(\rho\). Let \(\mathcal{M}(M)\) be the free abelian group generated by all three-manifolds equivalent to \(M\), and \(\mathcal{F}_{n}^{Y}\mathcal{M}(M)\) be its subgroup generated by the following formal sums: \(\sum_{\Gamma'\subset\Gamma}(-1)^{| \Gamma'| }M_{\Gamma'}\), where \(\Gamma\) is a \(Y\)-link with at least \(n\) components, \(\Gamma'\) runs over all sublinks of \(\Gamma\), \(M_{\Gamma'}\) is the three-manifold obtained from \(M\) by surgery along \(\Gamma'\), and \(| \Gamma'| \) is the number of connected components of \(\Gamma'\). Put \(\mathcal{G}_{n}^{Y}\mathcal{M}(M) :=\mathcal{F}_{n}^{Y}\mathcal{M}(M)/\mathcal{F}_{n+1} ^{Y}\mathcal{M}(M)\). Note that a theorem of S. Matveev states that \(\mathcal{G}_{0}^{Y}\mathcal{M}(M)=\mathbb{Z}\), see [Math. Notes 42, No. 1/2, 651–656 (1987); translation from Mat. Zametki 42, No. 2, 268–278 (1987; Zbl 0649.57010)]. Note also that a three-manifold is equivalent to \(S^{3}\) if and only if it is an integral homology three-sphere, and that a map \(f\colon\{\text{integral homology three-spheres}\}\to\mathbb{Q}\) is a finite type invariant of degree at most \(d\) if \(f\) vanishes on \(\mathcal{F}_{d+1}^{Y}\mathcal{M}(S^{3})\). (The degree here is different from Ohtsuki’s.)

Now let us consider a closed three-manifold \(M\) and an embedded surface \(\Sigma\). Let \(M_{\phi}\) denote the three-manifold obtained from \(M\) by ‘twisting’ \(\Sigma\) by \(\phi\in\mathcal{T}(n)\), where \(\mathcal{T}(n)\) is the subgroup of the Torelli group of \(\Sigma\) generated by \(n\)-fold commutators. Then it is known [M. Goussarov, S. Garoufalidis, M. Polyak, loc.cit. and K. Habiro, loc.cit.] that \(M-M_{\phi}\in\mathcal{F}_{n}^{Y}\mathcal{M}(M)\). The main result of the paper under review is that as an element in \(\mathcal{G}_{n}^{Y}\mathcal{M}(M)\) the difference \(M-M_{\phi}\) can be described in terms of the Johnson homomorphism and Massey products modulo elements expressed by \(Y\)-links with loops.

This result can be compared with a theorem by N. Habegger and G. Masbaum about Vassiliev invariants for string links [Topology 39, No.6, 1253-1289 (2000; Zbl 0964.57011)] that, roughly speaking, the Kontsevich integral of a string link can be interpreted in terms of Milnor’s \(\mu\)-invariants modulo chord diagrams with loops. (Note that such \(\mu\)-invariants are Massey products of the three-manifold obtained by \(0\)-surgery along the closure of the string link.)

For the entire collection see [Zbl 1061.00007].

Two compact three-manifolds \(M\) and \(N\) are called equivalent if there exists a homomorphism \(\rho\colon H_{1}(M;\mathbb{Z})\to H_{1}(N;\mathbb{Z})\) inducing an isometry of the linking forms on their torsion subgroups and a homeomorphism \(\partial M\to\partial N\) consistent with \(\rho\). Let \(\mathcal{M}(M)\) be the free abelian group generated by all three-manifolds equivalent to \(M\), and \(\mathcal{F}_{n}^{Y}\mathcal{M}(M)\) be its subgroup generated by the following formal sums: \(\sum_{\Gamma'\subset\Gamma}(-1)^{| \Gamma'| }M_{\Gamma'}\), where \(\Gamma\) is a \(Y\)-link with at least \(n\) components, \(\Gamma'\) runs over all sublinks of \(\Gamma\), \(M_{\Gamma'}\) is the three-manifold obtained from \(M\) by surgery along \(\Gamma'\), and \(| \Gamma'| \) is the number of connected components of \(\Gamma'\). Put \(\mathcal{G}_{n}^{Y}\mathcal{M}(M) :=\mathcal{F}_{n}^{Y}\mathcal{M}(M)/\mathcal{F}_{n+1} ^{Y}\mathcal{M}(M)\). Note that a theorem of S. Matveev states that \(\mathcal{G}_{0}^{Y}\mathcal{M}(M)=\mathbb{Z}\), see [Math. Notes 42, No. 1/2, 651–656 (1987); translation from Mat. Zametki 42, No. 2, 268–278 (1987; Zbl 0649.57010)]. Note also that a three-manifold is equivalent to \(S^{3}\) if and only if it is an integral homology three-sphere, and that a map \(f\colon\{\text{integral homology three-spheres}\}\to\mathbb{Q}\) is a finite type invariant of degree at most \(d\) if \(f\) vanishes on \(\mathcal{F}_{d+1}^{Y}\mathcal{M}(S^{3})\). (The degree here is different from Ohtsuki’s.)

Now let us consider a closed three-manifold \(M\) and an embedded surface \(\Sigma\). Let \(M_{\phi}\) denote the three-manifold obtained from \(M\) by ‘twisting’ \(\Sigma\) by \(\phi\in\mathcal{T}(n)\), where \(\mathcal{T}(n)\) is the subgroup of the Torelli group of \(\Sigma\) generated by \(n\)-fold commutators. Then it is known [M. Goussarov, S. Garoufalidis, M. Polyak, loc.cit. and K. Habiro, loc.cit.] that \(M-M_{\phi}\in\mathcal{F}_{n}^{Y}\mathcal{M}(M)\). The main result of the paper under review is that as an element in \(\mathcal{G}_{n}^{Y}\mathcal{M}(M)\) the difference \(M-M_{\phi}\) can be described in terms of the Johnson homomorphism and Massey products modulo elements expressed by \(Y\)-links with loops.

This result can be compared with a theorem by N. Habegger and G. Masbaum about Vassiliev invariants for string links [Topology 39, No.6, 1253-1289 (2000; Zbl 0964.57011)] that, roughly speaking, the Kontsevich integral of a string link can be interpreted in terms of Milnor’s \(\mu\)-invariants modulo chord diagrams with loops. (Note that such \(\mu\)-invariants are Massey products of the three-manifold obtained by \(0\)-surgery along the closure of the string link.)

For the entire collection see [Zbl 1061.00007].

Reviewer: Hitoshi Murakami (Tokyo)

##### MSC:

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

55S30 | Massey products |