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basic definition at Thom collaps map
Spell-checker not working today? I just corrected “collaps” to “collapse”. As this included the title, I left in the redirects.
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 $n$Labifying some of this.
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).
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”.
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?
And as we nLabify it, we can draw decent TQFT diagrams thanks to my new TQFT LaTeX package!
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.
You have a “TQFT LaTeX package”? Sounds good.
Yes, it’s responsible for the “A” in the following picture:
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.)
Cute. Why is the snake on the far right mirror-reflected? Just so that it doesn’t look like a question mark? ;-)
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.
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. ;-)
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
I thee.
I added a remark to Thom collapse map about the relationship to n-duality.
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.
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).
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.
I suspect that I am going to be very happy with the perspective on Thom spectra as given in
Jim, I presume you mean my picture in #9. If so, hopefully one of these will be a better size for you.
added to the Definition section at a new subsection Abstract definition in terms of duality.
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 $\Omega_\bullet \stackrel{\simeq}{\longrightarrow} \pi_\bullet(M O)$ more explicit.
now #21 looks like trolling, but at the time I guess it made sense.
I edited at Thom space to emphasize that the quotient $D(V)/S(V)$ is supposed to be a quotient of the total spaces of the bundles in $Top$, not a bundle quotient in $Top/V$. (The wording there didn’t make that completely clear.)
Thanks.
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