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nLab > Latest Changes: Loops with singularities

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    • CommentRowNumber1.
    • CommentAuthorAndrew Stacey
    • CommentTimeSep 16th 2011

    I found an interesting question on MO (here) and merrily set out to answer it. The answer got a bit long, so I thought I’d put it here instead. Since I wrote it in LaTeX with the intention of converting it to a suitable format for MO, it was simplicity itself to convert it instead to something suitable for the nLab.

    The style is perhaps not quite right for the nLab, but I can polish that. As I said, the original intention was to post it there so I started writing it with that in mind. I’ll polish it up and add in more links in due course.

    The page is at: on the manifold structure of singular loops, though I’m not sure that that’s an appropriate title! At the very least, it ought to have a subtitle: “or the lack of it”.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2011

    The answer got a bit long, so I thought I’d put it here instead.

    Great, that’s the way to go!

    The style is perhaps not quite right for the nLab

    I think it’s perfect at this point. Ideally your MO discussion partners will be ecouraged to join into the editing of this entry. For that purpose the way it’s curently phrased is good. Once the dust has settled one can still move the attributions from the intro to the references.

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeSep 16th 2011

    The page name is unconventional, but I won’t move it now, since I can’t fix the cache bug from here. But I added redirects.

    • CommentRowNumber4.
    • CommentAuthorAndrew Stacey
    • CommentTimeSep 16th 2011

    What would you suggest? I know that the current one isn’t great, but I didn’t know how to make it clear without making it long.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 17th 2011
    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeSep 17th 2011

    What would you suggest?

    I suggest what I put in the redirect: manifold structure of singular loops. (The only thing violating the conventions is ‘on’, and thus also ‘the’.) David’s idea also works, so I made that a redirect too.

    • CommentRowNumber7.
    • CommentAuthorTobyBartels
    • CommentTimeSep 17th 2011

    Brain flash … what about singular loop space?

    • CommentRowNumber8.
    • CommentAuthorAndrew Stacey
    • CommentTimeSep 17th 2011
    • (edited Sep 17th 2011)

    That suits me!

    After all, one can think of a lot more to say about singular loops which is more interesting than their manifold structure (or lack of).

    • CommentRowNumber9.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 19th 2011
    In fact, what essential would a manifold structure have given?
    The discussion about self-intersections recalls Sullivan-Chase string topology
    where it is important that the loops are in a manifold (for transversality arguments)
    but from there on things are at the chain level
    • CommentRowNumber10.
    • CommentAuthorAndrew Stacey
    • CommentTimeSep 19th 2011

    The Chas-Sullivan case is a smooth manifold because in string topology one knows exactly where the coincidence should be. Actually, there’s two spaces important in string topology. One is formed by taking one circle and crossing it, the other is formed by taking two circles and making them touch. But in both cases we specify exactly where these coincidences occur and that makes them smooth manifolds.

    Once you get to chains then you’re right, being a manifold doesn’t matter. But the Cohen-Jones reformulation does need the spaces themselves to be manifolds to get the Thom maps (well, there’s ways around that but it certainly makes it much prettier if they are manifolds).

    In general, I’d say that manifolds are Very Nice Spaces and so knowing that you have one is a good thing to know just in case it ever becomes important.

1 to 10 of 10

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