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An \(\text{Sp}(2g;{\mathbb Z}_{2})\)-module structure of the cokernel of the second Johnson homomorphism. (English) Zbl 1025.57024

The study of the Torelli group \(I_{g,1}\) (associated to a surface of genus \(g\) and one boundary component) led D. Johnson [Math. Ann. 249, 225-242 (1980; Zbl 0409.57009)] to the recursive definition of certain representations \[ \tau_k:{\mathcal M}(k)\to{\mathcal L}_{k+1}\otimes H \] of subgroups \({\mathcal M}(k)\) of \(I_{g,1}:{\mathcal M}(1)\) is \(I_{g,1}\), \({\mathcal M}(2)\) is the subgroup \({\mathcal K}_{g,1}\) generated by all Dehn twists along separating simple closed curves. \(H\) is the homology of the underlying surface and \({\mathcal L}= \bigoplus_k{\mathcal L}_k\) is the free Lie algebra on \(H\). \(\tau_1\) is surjective, but \(\tau_2\) is not.
Building on work by S. Morita [Topology 28, 305-323 (1989; Zbl 0684.57008)], the paper determines the \(\mathbb{Z}/2\)-rank of the cokernel of \(\tau_2\) and the action of the symplectic group on it.

MSC:

57M99 General low-dimensional topology
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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References:

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