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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].

##### MSC:
 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 55S30 Massey products
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