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stub for Blakers-Massey theorem. Need to add more references…
I went to add the pointer to the HoTT book theorem 8.10.2 to Blakers-Massey theorem and when doing so noticed that the entry had been heavily edited by R. Brown (rev 4) on 4th of October 2013.
In the course of this the abstract homotopy-theoretic statement of the result and the pointers to the modern proofs got moved all the way to below the discussion of Brown’s own work.
I have reorganized the entry now a bit to maybe find a compromise. The theorem in the original form appears as theorem 1, then below that Remark 1 points out the modern versions, Remark 2 points to Freudenthal suspension as a special case, Remark 3 is about the role of excision.
The generalizations that Brown added follow below that, highlighted by a section headline.
Yes, it is considered good nLab form to announce substantial edits to nLab articles here at the nForum, as Urs has done here. (Especially when someone decides to edit out someone else’s very sensible work – not saying this happened here, just that it sometimes happens and is rather annoying.)
Just for completeness somebody should announce here:
after waiting for a while, the other day Charles Rezk had made available his note in which he writes out the new homotopy theoretic proof of Blakers-Massey/Freudenthal by reverse-engineering from Favonia Mlatus’s formal proof in the form of Coq code, which in turn originated in translating to HoTT the new proof as found but not yet made public by Peter Lumsdaine, Eric Finster and Dan Licata.
I sure hope the original authors will find the energy to publish their original new proof, but I am also glad that the proof is visible now. I feel like highlighting the curious aspect that its, say, publication history now serves to make an additional case for HoTT all in itself: for it was most curious to see Charles Rezk – originator of higher topos theory – ask Favonia Mlatus to explain the new proof of Blakers-Massey to him, with Favonia – I think that’s fair to say – being unsure as to what might possibly be non-obvious and in need of explanation about his formal proof, while at the same time – I think that’s still fair to say – not being by far an expert on homotopy theory as Charles is. You see what I mean: with HoTT in his hands a relative newbie to homotopy theory was able to produce a proof that leading expert(s) in homotopy theory had sought in vain.
I think that’s a rather neat proof of concept.
Yes!
@Urs: Blakers-Massey’s Theorem was shown to be a consequense of Farjoun’s “cellular inequalities” already in [Chachólski, Wojciech. “A generalization of the triad theorem of Blakers-Massey.” Topology 36.6 (1997): 1381-1400] (Theorem 1.B).
Thanks for the pointer, I have added it to the entry. Might you have a pointer to those cellular inequalities (I don’t have Chachólski’s article to look at for the moment ),
The cellular inequalities appear in [Farjoun, Emmanuel. Cellular spaces, null spaces and homotopy localization. No. 1621-1622. Springer, 1996]. Chacholski’s paper is now in your mailbox.
Thanks! I have to dash off now, but I’ll try to look at it later. So you are saying this gives a proof that is entirely homotopy theoretic?
Yes, that’s what I say; and that it is the first one I know in the literature.
Does Chachólski’s argument actually prove the full strength of the Blakers–Massey Theorem? If I read Theorem 1.B of A generalization of the triad theorem of Blakers-Massey correctly, it only implies that given a homotopy pushout
$\begin{matrix} A & \longrightarrow & B \\ \downarrow & & \, \downarrow \\ C & \longrightarrow & D \end{matrix}$with $A \to B$ $p$-connected and $C \to D$ $q$-connected (in the HoTT convention), the suspension of $A \to B \times_D^{\mathrm{h}} C$ is $(p+q+1)$-connected. That seems strictly weaker than saying that $A \to B \times_D^{\mathrm{h}} C$ itself is $(p+q)$-connected (there are non-contractible spaces with contractible suspensions).
I have added a brief remark here that there is a tower of higher cubical generalization of the Blakers-Massey theorem, which jointly say equivalently that the identity functor on $\infty Grpd$ is Goodwillie-analytic.
I have indented the paragraph with abstract for Lumsdaine’s annoucement talk. I have added the reference for the Blakers-Massey in shape theory. It seems that the case of strong shape theory is still open (I have some interest in working on that case).
Zoran, what sort of result do you think might hold in strong shape? and how general a strong shape conetxt are you thinking of?
The HoTT-proof is finally out: FFLL 16.
The page points to a g+ discussion which of course is no longer supported. The wayback machine gives me a glimpse of the old page but then disappears. Do we have a way to retrieve it?
two maps out of the same domain which are $n_1$-connective and $n_2$-connective
Is ’connective’ preferred to ’connected’ in this context, or just used as a synonym?
Re 16: Urs, did you export your G+ content before it got deleted?
Re 17: Urgh, I forget – sometimes there is an off-by-one shift in the indexing. I know there is an off-by-one shift between what a classical homotopy theorist calls an $n$-connected map (or more often an $n$-equivalence) and what HoTT people call an $n$-connected map, because ours is based on the connectivity of the fiber and theirs on the connectivity of the cofiber. I forget what the indexing is for “connective”; maybe it matches the HoTT indexing and a different word was chosen to avoid conflict with the classical terminology?
Oops, no, I didn’t export it. Figured there was nothing there necessary to keep for posterity, but of course there was: this memorable exchange between Favonia and Charles Rezk. Sorry, that’s a real loss.
The discussion is still in the source of the Wayback Machine page that David Corfield linked to, it’s just not being displayed when the page is rendered by our browsers. Someone who knows more HTML than I do could alter the source to make it display correctly or clear it up so it can be read directly.
Can we copy it out of the Wayback machine into a page here?
I’m extracting the post and comments and will place it in on my GitHub Sandbox so that we have a record of it.
Edit: see here
Well done! Should we upload it to the nLab?
The file? I don’t mind.
It can be renamed to a .txt if you want. The .md just makes GitHub render it with some minimal formatting, making links active etc.
Or even better, since the format is markdown, it could be just pasted into an nLab page rather than attached as a file. If we don’t think it’s appropriate as a standalone nLab page for some reason, we could put it on the HoTT web instead.
Thanks very much by the way!
You’re welcome :-)
The urls are not proper markdown links (GH just links them automatically), and I think the names could be bumped down a notch in heading size, but I don’t mind if it’s put on the HoTT web as a page.
Yesterday I pasted it into our Sandbox, and it came out not looking well. Maybe better if we could have the source rendered by some software that does it decently (maybe just the GitHub output, or maybe there is something better) and then store this output in a pdf or the like.
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