Timeline for 3-manifold proof of Grushko's theorem
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Jun 23, 2021 at 3:17 | comment | added | Ian Agol | I would do this with # S^1x D^2 instead of S^2 since a handlebody is irreducible. Transversality is purely local, so you can do it for the midpoint of an interval connecting K(π,1)’s instead of using a strict wedge. Then do compressions and boundary compressions to get down to the preimage being a collection of embedded disks. | |
| Jun 22, 2021 at 16:28 | comment | added | user101010 | @MoisheKohan Sorry, I think I am being really thick with this, but then don't I just get the preimage to be subsimplical complex? For what I outlined above I was always thinking of it as a submanifold. Scott and Wall indeed give a fun topological proof of Gruschko's theorem using a very similar setup, but it doesn't involve this same 3-dimensional manifold-topology flair. | |
| Jun 22, 2021 at 16:21 | comment | added | user101010 | It just occurred to me (while reading a bit further in Stallings' paper) that he probably had in mind first killing off the higher homotopy groups of $K_1$ and $K_2$ first, and then proceeding as I said. This ensures that the aforementioned map $f$ actually exists, since the wedge of Eilenberg-MacLane spaces is again an Eilenberg-MacLane space. | |
| Jun 21, 2021 at 23:45 | comment | added | Moishe Kohan | Hint: Instead of join, connect $K_1, K_2$ by an interval and then take the preimage of a generic point if that interval under a simplicial map of your 3-manifold. This should be in the paper of Scott and Wall. | |
| Jun 21, 2021 at 13:48 | history | asked | user101010 | CC BY-SA 4.0 |