Timeline for Fundamental group of R^2-Q^2
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 12, 2011 at 23:52 | comment | added | David Roberts♦ | ...but then one loses the concrete viewpoint of topological spaces. There are a number of reference links on the nLab page that are freely available. Otherwise email Tim Porter - he knows much much more about shape theory than I do (which is negligible, really) | |
| Jun 12, 2011 at 23:50 | comment | added | David Roberts♦ | ... If you are interested in (ordinary) covering spaces, then the above is the right way to do it. There are also approaches using topoi and localic fundamental groups (work by Marta Bunge is appropriate here), fibre functors and Galois categories (due to Grothendieck, discussed in a nice paper by Eduardo Debuc) - which is close to what I mention above, but removes the choice of resolving $X$ by the $X_{n,m}$ - it considers all covering spaces. As far as shape theory goes, it's a slippery beast. The 'right' way to do it is how Tim and others have since developed it, and that is abstractly.... | |
| Jun 12, 2011 at 23:43 | comment | added | David Roberts♦ | @Dan - it's not canonical at all (really just the first construction that came to mind) but my guess from what I know of shape theory is that this choice of 'resolution' of $X$ doesn't matter, up to some sort of weak equivalence in the so-called shape category. @Avi - actually there is a mistake in that paper in that the fundamental group is not always a topological group, only a quasi-topological group (multiplication is separately continuous in each variable). There would be such a structure floating around here, but as always, it depends on what you want to do with the fundamental group.. | |
| Jun 12, 2011 at 21:29 | vote | accept | Avi Steiner | ||
| Jun 12, 2011 at 21:27 | comment | added | Avi Steiner | Is this approach at all related to David Bliss' "A Generalized Approach to the Fundamental Group"? Also, could you point me to a (preferably online) description of shape theory that someone with only minimal experience with category theory could follow? The wikipedia article doesn't give any definitions, and the shape theory definition for topological spaces is missing from the nLabs article. @Tim Porter: It's just curiosity. | |
| Jun 12, 2011 at 17:25 | comment | added | Dan Ramras | In what sense is this construction canonical? One could produce progroups using other diagrams with limit X; are these progroups always equivalent, assuming some hypotheses on the diagram? | |
| Jun 12, 2011 at 9:37 | comment | added | Tim Porter | It is worth noting that this pro-group is not profinite so taking the limit is not a good thing to do. As David says you need to look at the pro-group as it is. No single (discrete) group will do the job for you. There are interesting old results on the fundamental group of the Hawaiian earring that discuss topologies on the group that help. I do not know if that would help here. (Why do you want to know? Is it just curiosity? I ask because the style of answer will be influenced by any intended use.) | |
| Jun 12, 2011 at 0:00 | history | answered | David Roberts♦ | CC BY-SA 3.0 |