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**Symplectic Heegaard splittings and linked abelian groups.**
*(English)*
Zbl 1170.57018

Penner, Robert (ed.) et al., Groups of diffeomorphisms in honor of Shigeyuki Morita on the occasion of his 60th birthday. Based on the international symposium on groups and diffeomorphisms 2006, Tokyo, Japan, September 11–15, 2006. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-48-8/hbk). Advanced Studies in Pure Mathematics 52, 135-220 (2008).

Let \(M_g\) be a closed oriented surface of genus \(g\) and \(\widetilde\Gamma_g\) be its mapping class group, that is, the group of isotopy classes of orientation-preserving diffeomorphisms of \(M_g\). Let \(\pi= \pi_1(M_g)\), and denote by \(\pi^{(k)}\) the \(k\)th term in the lower central series of \(\pi\), that is, \(\pi^{(1)}=\pi\) and \(\pi^{(k+1)}= [\pi,\pi^{(k)}]\). Then \(\widetilde\Gamma_g\) acts on the quotient groups \(\pi/\pi^{(k)}\), and that action yields a representation \(\rho^k:\widetilde\Gamma_g\to \Gamma^k_g\), where \(\Gamma^k_g< \operatorname{Aut}(\pi/\pi^{(k)})\). In particular, \(\rho^1\) is the trivial representation and \(\rho^2\) is the symplectic representation. In this paper, the kernels of these representations are called the Johnson-Morita filtration of \(\widetilde\Gamma_g\).

The work of the authors is motivated by the case when \(M_g\) is a Heegaard surface of a Heegaard splitting of a closed connected orientable 3-manifold \(W\) and elements of \(\widetilde\Gamma_g\) are gluing maps for the Heegaard splitting of \(W\). Then one may study \(W\) by investigating the image under the maps \(\rho^k\) of the set of all possible gluing maps that yield \(W\). This gives the possibility to study deeper invariants of \(W\), obtainable in principle from deeper quotients of the lower central series. The foundations for such deeper studies have been laid in the work of Morita who introduced the idea of studying higher representations via crossed homomorphisms, see S. Morita [Invent. Math 111, No. 1, 197–224 (1993; Zbl 0787.57008) and Duke Math. J. 70, No. 3, 699–726 (1993; Zbl 0801.57011)]. The invariants of closed 3-manifolds that can be obtained in this way are known to be closely related to finite type invariants of 3-manifolds. The simplest non-trivial example of the program mentioned above is the case \(k= 2\). Here \(\widetilde\Gamma_g\) acts on \(H_1(M_g)= \pi/[\pi,\pi]\). The information about \(W\) that is encoded in \(\rho^2(\phi)\), where \(\phi\in \text{Diff}^+(M_g)\) is the Heegaard gluing map for a Heegaard splitting of \(W\) of minimum genus, together with the images under \(\rho^2\) of the Heegaard gluing maps of all “stabilizations” of the given splitting, is what the authors have in mind when they refer to a symplectic Heegaard splitting.

The purpose of this nice fundamental paper is to review the literature on symplectic Heegaard splittings of closed 3-manifolds and the closely related literature on linked Abelian groups, with the goal of describing what is known – as completely and explicitly and efficiently as possible, in a form which will be useful for future investigations and researches.

For the entire collection see [Zbl 1154.53004].

The work of the authors is motivated by the case when \(M_g\) is a Heegaard surface of a Heegaard splitting of a closed connected orientable 3-manifold \(W\) and elements of \(\widetilde\Gamma_g\) are gluing maps for the Heegaard splitting of \(W\). Then one may study \(W\) by investigating the image under the maps \(\rho^k\) of the set of all possible gluing maps that yield \(W\). This gives the possibility to study deeper invariants of \(W\), obtainable in principle from deeper quotients of the lower central series. The foundations for such deeper studies have been laid in the work of Morita who introduced the idea of studying higher representations via crossed homomorphisms, see S. Morita [Invent. Math 111, No. 1, 197–224 (1993; Zbl 0787.57008) and Duke Math. J. 70, No. 3, 699–726 (1993; Zbl 0801.57011)]. The invariants of closed 3-manifolds that can be obtained in this way are known to be closely related to finite type invariants of 3-manifolds. The simplest non-trivial example of the program mentioned above is the case \(k= 2\). Here \(\widetilde\Gamma_g\) acts on \(H_1(M_g)= \pi/[\pi,\pi]\). The information about \(W\) that is encoded in \(\rho^2(\phi)\), where \(\phi\in \text{Diff}^+(M_g)\) is the Heegaard gluing map for a Heegaard splitting of \(W\) of minimum genus, together with the images under \(\rho^2\) of the Heegaard gluing maps of all “stabilizations” of the given splitting, is what the authors have in mind when they refer to a symplectic Heegaard splitting.

The purpose of this nice fundamental paper is to review the literature on symplectic Heegaard splittings of closed 3-manifolds and the closely related literature on linked Abelian groups, with the goal of describing what is known – as completely and explicitly and efficiently as possible, in a form which will be useful for future investigations and researches.

For the entire collection see [Zbl 1154.53004].

Reviewer: Alberto Cavicchioli (Modena)

### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |

57M99 | General low-dimensional topology |

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