In the proof of Lickorish-Wallace theorem, we use Heegaard splitting of a closed, orientable and connected $3$-manifold and obtain a surface diffeomorphism which glues the two handle-body. I wonder whether this setup a bridge between the topological/geometrical property of the $3$-manifold on the one side and the group-theoretical property of mapping class group on the other side. For example, we can get a homology $3$-sphere by maps from Torelli subgroup. In this case, vanishing of homology of the resulting manifold is related to the action of the Torelli subgroup on the homology of the surface. If we naively prescribe a class of subgroup, like a separable subgroup of a mapping class group on a given genus-$g$ surface, and consider the class of $3$-manifolds produced from the gluing, can we say anything about the geometrical or topological properties of the $3$-manifold? Conversely, if we have a class of $3$-manifold with some topological or geometrical properties, what can we say about the class of surface diffeomorphism used in the Lickorish-Wallace theorem? This is of course complicated by the issue that a given $3$-manifold may have many different Heegaard splitting. Any reference in this direction welcomes. Thank you.
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$\begingroup$ If the homomorphism is pseudo-Anosov, then the manifold is hyperbolic? $\endgroup$– user6976Commented Oct 16, 2019 at 0:14
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$\begingroup$ @MarkSapir: Sadly, that's not true. The issue is that there are many pseudo-Anosov mapping classes that extend over the handlebody. A Heegaard splitting using one of these will yield the 3-sphere. $\endgroup$– Andy PutmanCommented Oct 16, 2019 at 0:36
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$\begingroup$ @MarkSapir: that's true for surface bundles (genus 2 or higher) over the circle, with the homeomorphism being the bundle monodromy. $\endgroup$– Ryan BudneyCommented Oct 16, 2019 at 3:51
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1$\begingroup$ The gluing map is only well defined up to double cosets by the handlebody groups (subgroups extending over a chosen handlebody). So there’s not a close relation between the gluing maps and the topology. Nevertheless, see: arxiv.org/abs/1312.2293 $\endgroup$– Ian AgolCommented Oct 16, 2019 at 17:48
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$\begingroup$ The paper sciencedirect.com/science/article/pii/0040938389900116 gives some relation between the Casson invariant and the mapping class group of Heegaard splittings. In particular, there is a certain subgroup of the mapping class group for which modifying a Heegaard splitting by an element of this group does not change the Casson invariant (see Proposition 3.5 of this paper for a more precise statement). $\endgroup$– Ian AgolCommented Oct 18, 2019 at 9:18
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