Timeline for Does Heegaard splitting relate topological properties of a $3$-manifold to properties of subgroups of $MCG$
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Oct 21, 2019 at 20:50 | comment | added | Hans | @IanAgol Thank you very much. Both the comments and references are very helpful. | |
| Oct 18, 2019 at 9:27 | comment | added | Ian Agol | For separable subgroups of the mapping class group, check out: arxiv.org/abs/math/0505225 | |
| Oct 18, 2019 at 9:18 | comment | added | Ian Agol | 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). | |
| Oct 16, 2019 at 17:48 | comment | added | Ian Agol | 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 | |
| Oct 16, 2019 at 3:51 | comment | added | Ryan Budney | @MarkSapir: that's true for surface bundles (genus 2 or higher) over the circle, with the homeomorphism being the bundle monodromy. | |
| Oct 16, 2019 at 0:36 | comment | added | Andy Putman | @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. | |
| Oct 16, 2019 at 0:15 | review | First posts | |||
| Oct 16, 2019 at 1:06 | |||||
| Oct 16, 2019 at 0:14 | comment | added | user6976 | If the homomorphism is pseudo-Anosov, then the manifold is hyperbolic? | |
| Oct 16, 2019 at 0:10 | history | asked | Hans | CC BY-SA 4.0 |