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hereby moving the following old discussion out of the entry to here:
— begin forwarded discussion —
+– {: .query} David Roberts: For those generalised smooth spaces which give rise to a topological space (e.g. a diffeological space), is the topology known to be locally contractible, or locally nice at all?
Andrew: That’s actually a question I’d quite like to study here. All of the definitions of “generalised smooth space” (that have underlying sets) induce a topology on that underlying set. Some have it built in (Chen’s early definitions, for example, and Smith spaces and differentiable modules) but even if it is not there you can induce it from the plots or functions. They are not, in general, going to be locally contractible but there are some pathologies that are ruled out.
David R: Clearly the philosophy behind smooth spaces means we have to keep what we get, and not fuss about how ugly the spaces might be. What interests me is what the fundamental group(oid) is going to look like. Will it be a profinite group? A pro-group? A smooth group? I suppose one could start with the smooth space of loops, and form the smooth quotient space under the relation of homotopy - but what does it look like? =–
— end forwarded discussion —
Incidentally, the answer to David R.’s old question above is: The underlying D-topology of diffeological spaces gives precisely the D-topological spaces aka $\Delta$-generated topological spaces. This is discussed and referenced here.
also moving the following old discussion out of the entry:
$\,$
— begin forwarded discussion —
+– {: .query} I subtracted $20$ from the $x$-coordinates on the names in the diagram so that they would stay in the boxes on my screen, but I'm not sure if this is right; the original looks fine to me as a free-standing diagram, so I don't know why it looks wrong here.
Anyway, if anybody finds that this version is worse than the previous one, then change it back to the previous one and chalk it up to an error in my browser. —Toby
Thanks, Toby. I was just heading over to see if I could fix it myself but you beat me to it. There seem to be a few subtleties over how Instiki imports SVG and I’m learning them by trial and error (and by bugging Jacques!). The picture in the Sandbox now looks right and, thanks to you, so does this one. Text boxes seems to be the trickiest to get right when doing TikZ-to-SVG conversion. —Andrew
It it helps any, I think that the problem was that the alphabetic text (but not the dates) began where it ought to have been centred. —Toby =–
— end forwarded discussion —
Replaced
- Matthias Kreck, Stratifolds and differential algebraic topology (pdf)
which has a broken link and I cannot find the pdf anywhere (eg on the Wayback Machine), with
- Matthias Kreck, Differential Algebraic Topology: From Stratifolds to Exotic Spheres, Graduate Studies in Mathematics Volume 110, AMS (2010) (publisher page), (author pdf)
The filenames seem to me to indicate that the former broken pdf link (kreck-DA24_08_07.pdf
) is probably a preliminary version of the published book (kreck-DA.pdf
).
Also gave publication details for Diffeology.
There’s a broken include
link that I haven’t fixed. Not sure how to deal with that right now.
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