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nLab > Latest Changes: Thom collapse map

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2011
    • (edited Jul 19th 2013)

    basic definition at Thom collaps map

    • CommentRowNumber2.
    • CommentAuthorAndrew Stacey
    • CommentTimeMay 31st 2011

    Spell-checker not working today? I just corrected “collaps” to “collapse”. As this included the title, I left in the redirects.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2011
    • (edited May 31st 2011)

    Oops. Thanks.

    By the way, later today I might be tempted to start some entry on Godin’s “propagating flows”, which she is attributing to you. Maybe you would enjoy joining me in nnLabifying some of this.

    • CommentRowNumber4.
    • CommentAuthorAndrew Stacey
    • CommentTimeMay 31st 2011

    Absolutely! If it’s what I think it is, then it is (based on) the proof that coincidence submanifolds have tubular neighbourhoods. It’s part of my “differential topology on loop spaces” work which I (still!) intend to nLabify (and generalise to more general mapping spaces).

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2011

    If it’s what I think it is,

    In lemma 5 of Godin, Higher string topology operations it says:

    These ideas are largely based on [Stacey, math/0510097]

    I am second reader of a Master thesis that generalizes Godin’s construction of these “propagating flows”.

    • CommentRowNumber6.
    • CommentAuthorAndrew Stacey
    • CommentTimeMay 31st 2011

    Yes, I remember an email conversation with her about that. One reason she decided to “include them for completeness” is that she found a small (and easily fixable) mistake in that paper which I fully intended to correct … but never got round to. I’ll have to watch out for that as we “nLabify” it.

    What’s the generalisation?

    • CommentRowNumber7.
    • CommentAuthorAndrew Stacey
    • CommentTimeMay 31st 2011

    And as we nLabify it, we can draw decent TQFT diagrams thanks to my new TQFT LaTeX package!

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2011
    • (edited May 31st 2011)

    What’s the generalisation?

    He solves Godin’s conjecture that her construction goes through in the presence of D-branes. For that he has to refine some technical constructions a bit (more refine them than generalize them, maybe)

    For the moment I’ll send you a copy of the thesis by email.

    we can draw decent TQFT diagrams thanks to my new TQFT LaTeX package!

    You have a “TQFT LaTeX package”? Sounds good.

    • CommentRowNumber9.
    • CommentAuthorAndrew Stacey
    • CommentTimeMay 31st 2011

    You have a “TQFT LaTeX package”? Sounds good.

    Yes, it’s responsible for the “A” in the following picture:

    Showing off TQFT

    I’m also quite please with how the braid was done. Both the braid and the TQFT will soon be on CTAN, and if anyone’s interested in testing preliminary versions then they are available from launchpad; just download the .dtx file and run pdflatex on it. That generates both the style file and the documentation. Warning: they require PGF2.10.

    (Just previewed this post. That image is a little big! But it does make it easier to see the individual components, I guess.)

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2011

    Cute. Why is the snake on the far right mirror-reflected? Just so that it doesn’t look like a question mark? ;-)

    • CommentRowNumber11.
    • CommentAuthorAndrew Stacey
    • CommentTimeMay 31st 2011

    No, otherwise it doesn’t look like an “S”! Though I guess for someone who’s a bit confused with their spelling, then “MATHZ” probably looks about right.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2011

    Oh, I see. I was thinking it was supposed to read just MATH. I am tempted to ask when you say “maths” instead of “math”, but maybe I won’t. ;-)

    • CommentRowNumber13.
    • CommentAuthorAndrew Stacey
    • CommentTimeMay 31st 2011

    It’s a US/UK thing. We say “maths”, they say “math”. The “Bluffers’ Guide to Mathematics” says that it is definitely maths and that

    “Math” ith a Roman Catholic thervice

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2011

    I thee.

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeJun 1st 2011

    I added a remark to Thom collapse map about the relationship to n-duality.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJun 1st 2011
    • (edited Jun 1st 2011)

    I added a remark to Thom collapse map about the relationship to n-duality.

    Thanks. I know this concept from an article by Kate Ponto mostly, have created a stub n-duality. I guess it’s also in May-Sigurdsson, but I can’t seem to find the relevant page right now.

    Do you know more generally: what’s a good general abstract characterization of Thom collapse? Most of the literature on Thom collapse and fiber integration is all about the construction, little about its abstract properties.

    • CommentRowNumber17.
    • CommentAuthorzskoda
    • CommentTimeJun 1st 2011
    • (edited Jun 1st 2011)

    This is more of a reminder of context. I guess the discussion of it abstract properties could be related to abstract treatment of Spanier-Whitehead duality (I do not understand precisely how though).

    • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeJun 1st 2011

    The only abstract characterization that I know is that one: that it shows the Thom spectrum to be the dual of X in the stable homotopy category. But I’ve only thought about it from the point of view of duality and traces, so there could certainly be more to say in different generality.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJun 2nd 2011

    I suspect that I am going to be very happy with the perspective on Thom spectra as given in

    • CommentRowNumber20.
    • CommentAuthorjim_stasheff
    • CommentTimeJun 2nd 2011
    How can I shrink that marvelous graphic so I can see it all at once
    instead of having to move over and up-down?
    • CommentRowNumber21.
    • CommentAuthorAndrew Stacey
    • CommentTimeJun 2nd 2011

    Jim, I presume you mean my picture in #9. If so, hopefully one of these will be a better size for you.

    Fancy Maths

    Fancy Maths

    • CommentRowNumber22.
    • CommentAuthorjim_stasheff
    • CommentTimeJun 3rd 2011
    yes, much easier to appreciate FULLY
    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2013

    added to the Definition section at a new subsection Abstract definition in terms of duality.

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeFeb 8th 2016

    I have expanded a little more and brushed-up a little at Pontryagin-Thom collapse map – Component definition in topological spaces. In particular I made the connection to Thom’s theorem Ω •⟶≃π •(MO)\Omega_\bullet \stackrel{\simeq}{\longrightarrow} \pi_\bullet(M O) more explicit.

    • CommentRowNumber25.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 8th 2016
    • (edited Feb 8th 2016)

    now #21 looks like trolling, but at the time I guess it made sense.

    • CommentRowNumber26.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 10th 2016

    I edited at Thom space to emphasize that the quotient D(V)/S(V)D(V)/S(V) is supposed to be a quotient of the total spaces of the bundles in TopTop, not a bundle quotient in Top/VTop/V. (The wording there didn’t make that completely clear.)

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeFeb 10th 2016

    Thanks.

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