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Lagrangian mapping class groups from a group homological point of view

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arxiv 0910.5262 v2 pith:FL5DWRSZ submitted 2009-10-27 math.GT math.AT

classification math.GTmath.AT
keywords classgroupmappinghomologygroupslagrangiansurfacehomological
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We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-type invariants of 3-manifolds. In this paper, we discuss these groups from a group (co)homological point of view. The results include the determination of their abelianizations, lower bounds of the second homology and remarks on the (co)homology of higher degrees. As a by-product of this investigation, we determine the second homology of the mapping class group of a surface of genus 3.

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  1. Johnson homomorphisms and the second rational cohomology of handlebody Torelli groups

    math.GT 2025-09 conditional novelty 6.0 of 10

    For the handlebody Torelli groups HI and HBI, the kernels of the Johnson-detected cup product maps in second rational cohomology are computed explicitly as sums of irreducible SL_g(Q) modules.

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