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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 14th 2011
    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 14th 2011
    • (edited Dec 14th 2011)

    I added some material to Hopf fibration, some of it in a speculative vein. In this case, I was an Anonymous Coward because I hadn’t noticed I was not logged in.

    • CommentRowNumber3.
    • CommentAuthorTim_Porter
    • CommentTimeDec 14th 2011
    • (edited Dec 14th 2011)

    I just checked that you had not spammed :-) I found some typos as well. (I fixed them.)

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 20th 2015

    I have cross-linked the material at Hopf construction and Hopf fibration a little more.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2019
    • (edited Feb 11th 2019)

    briefly added mentioning (here) of the alternative construction via quaternions that came up in discussion in another thread (here)

    (I am not happy with the structure of this entry, there is much room for making it more comprehensive and systematic, but I’ll leave it as is, for the time being)

    diff, v16, current

    • CommentRowNumber6.
    • CommentAuthorAlizter
    • CommentTimeFeb 11th 2019
    • (edited Feb 11th 2019)

    It is not strictly true that the hopf construction works for S 0S^0 since it is not path-connected. However it is very easy to give the fibration explicitly. For the other spheres the hopf construction works fine. This is true straight out of Stasheff’s book page 5. There are many places where this is incorrectly mentioned in Hopf construction and Hopf fibration.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2019

    Hm, I had another look at the Definition as stated in the entry here and plugged in S 0/2S^0 \simeq \mathbb{Z}/2 for XX. From, this I do seem to get the real Hopf fibratoipn S 12S 1S^1 \overset{\cdot 2}{\longrightarrow} S^1 all right.

    Let’s go through it in detail:

    So on the left we have four copies of the interval, labeled in /2×/2\mathbb{Z}/2 \times \mathbb{Z}/2, while on the right we have two copies, labeled by /2\mathbb{Z}/2.

    As we run once around on the left (with coordinates in [0,1]×( 2× 2)[0,1] \times (\mathbb{Z}_2 \times \mathbb{Z}_2) )

    (0,(0,0))(1,(0,0))(1,(1,0))(0,(1,0))(0,(1,1))(1,(1,1))(1,(0,1))(0,(0,1))(0,(0,0)) (0,(0,0)) \to (1,(0,0)) \sim (1,(1,0)) \to (0,(1,0)) \sim (0,(1,1)) \to (1,(1,1)) \sim (1,(0,1)) \to (0,(0,1)) \sim (0,(0,0))

    we run on the right (with coordinates in [0,1]× 2[0,1] \times \mathbb{Z}_2)

    (0,(0+0=0))(1,(0+0=0))(1,(1+0=1))(0,(1+0=1))(0,(1+1=0))(1,(1+1=0))(1,(0+1=1))(0,(0+1=1))(0,(0+0=0)) (0,(0+0 = 0)) \to (1,(0 + 0 = 0)) \sim (1,(1 + 0 = 1)) \to (0,(1 + 0 = 1)) \sim (0,(1+1 = 0)) \to (1,(1+1 = 0)) \sim (1,(0 + 1 = 1)) \to (0,(0 + 1 = 1)) \sim (0,(0 + 0 = 0))

    That’s twice around the circle on the right, for once on the left.

    Seems right to me. But let me know if I am missing something.

    • CommentRowNumber8.
    • CommentAuthorAlizter
    • CommentTimeFeb 11th 2019
    • (edited Feb 11th 2019)

    I think the problem occurs because we get a quasifibration not necesserily a fibration. I am not so sure however.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2019

    Sorry, which problem do you think occurs?

    • CommentRowNumber10.
    • CommentAuthorAlizter
    • CommentTimeFeb 11th 2019

    Actually ignore what I said above that is complete nonsense. I found that X is 0-connected can be weakened to π 0(X)\pi_0(X) is a group, so this is in fact correct. On page 4. The reason I got confused is because the multiplcation maps X×XXX\times X \to X should be weak homotopy equivalences XXX \simeq X when given one argument. It seems the book claims this can be proven on the weaker assumption that π 0(X)\pi_0(X) is a group. Which means the articles are kind of correct, but only when this condition holds.

    • CommentRowNumber11.
    • CommentAuthorAlizter
    • CommentTimeFeb 11th 2019

    In general it is not correct to say the hopf construction works for any XX but for a 00-connected XX or at least π 0(X)\pi_0(X) being a group. When these are CW-complexes it seems quasifibrations become fibrations due to Dold. I am not so familar with proving left multiplication in a H-space is a weak homotopy equivalence however, especially with the connectedness condition weakened.

    • CommentRowNumber12.
    • CommentAuthorAlizter
    • CommentTimeFeb 11th 2019

    I have asked a question on MO hopefully we will get some correct answers rather than my vague thoughts.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2019

    XX must be an H-space. As it says in the entry.

    • CommentRowNumber14.
    • CommentAuthorAlizter
    • CommentTimeFeb 11th 2019

    @Urs I am confused what you mean. Is it in the definition of H-space that left-multiplication is a homotopy equivalence? I thought a H-space was just a unital magma in the homotopy category.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2019

    added brief mentioning (here) of the suspensions of the Hopf fibrations generating the corresponding stable homotopy groups of spheres

    diff, v17, current

    • CommentRowNumber16.
    • CommentAuthorCharles Rezk
    • CommentTimeFeb 11th 2019

    I don’t see why the Hopf construction applied to an HH-space XX must necessarily produce a quasifibration, even if we assume π 0X\pi_0X is a group or even just XX connected. (It is certainly true that the homotopy fiber of h:XXSXh\colon X\star X\to SX will be equivalent to XX under these hypotheses; I just don’t see why the actual fibers of hh should also be equivalent to XX, which is what necessarily must happen if hh is a quasifibration.)

    • CommentRowNumber17.
    • CommentAuthorDylan Wilson
    • CommentTimeFeb 11th 2019
    @Charles: According to Stasheff (page 3), Sugawara proves this thing is a quasifibration for CW-H-spaces, and Dold-Lashof prove this is a quasifibration if m(x,-) is a weak equivalence for every x. But I haven't looked at the proofs- maybe you can find either reassurance or an error there?
    • CommentRowNumber18.
    • CommentAuthorCharles Rezk
    • CommentTimeFeb 11th 2019

    @Dylan: I don’t know where Sugawara’s proof is found: there are three papers of his in the references of Stasheff’s book, and I’m not going to hunt them all down.

    Suppose XX is a grouplike HH-space, with multiplication μ\mu. Let f:X×XXf\colon X\times X\to X be any morphism in the relative mapping space Map XX/(X×X,X)\mathrm{Map}_{X\vee X/}(X\times X,X) in the path component of μ\mu. Every such ff determines a group-like HH-space structure on XX, and if we believe the theorem all fibers of all the Hopf constructions h f:XXSXh_f\colon X\star X\to SX must be weakly equivalent to XX. The map ff can be found inside h fh_f (i.e., h fh_f restricts to ff along inclusions X×XXXX\times X\subset X\star X and XS(X)X\subset S(X)), so in particular all the fibers of all the maps ff over all points of XX must be weakly equivalent to XX. This seems unlikely.

    Dold-Lashof prove that a different construction gives a quasifibration; this other construction, applied to a group-like HH-space, gives a map homotopic to the Hopf construction as usually defined.

    • CommentRowNumber19.
    • CommentAuthorCharles Rezk
    • CommentTimeFeb 11th 2019

    I’m guessing Stasheff is referring Theorem 4 (p. 119) of

    Sugawara, Masahiro. On a condition that a space is an H-space. Math. J. Okayama Univ. 6 (1957), 109–129.

    This basically shows that the fiber of h:XXSXh\colon X\star X\to SX over one of the cone points of SXSX (which is in fact homeomorphic to XX) is weakly equivalent (by the tautological map) to the homotopy fiber of hh. It does not show that every fiber of hh is weakly equivalent (via the tautological map) to the corresponding homotopy fiber, so it does not show that hh is a quasifibration.

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeFeb 15th 2019
    • (edited Feb 15th 2019)

    Just to say that I have recorded the simple example from #7 (of working out the real Hopf fibration as the special case of the Hopf construction for X=/2X = \mathbb{Z}/2) to the entry real Hopf fibration: here

    • CommentRowNumber21.
    • CommentAuthorAlizter
    • CommentTimeFeb 15th 2019

    The reason I brought it up is because in HoTT we have “0-connectedness” floating around. I was further confused by Stasheff’s book mentioning the condition. When I was learning Hopf fibrations in Algebraic topology I don’t ever remember using the hopf construction to make them. I only really found out about it when learning HoTT. The mathoverflow question hasn’t helped in the slightest, though I can’t blame it since my question wasn’t exactly well written. I am inclined to believe that it does work for S 0S^0 though I am made slightly uncomfortable about what Charles has said.

    Now I know it isn’t exactly a good motivation to think this way because of HoTT, but this still poses the question of why it is needed in HoTT. In the HoTT proof that the hopf construction gives a fibration, an essential part of which is showing left and right multiplication in the H-space is an equivalence. For this one “needs” 0-connectedness.

    Overall I couldn’t care less if the hopf construction works for S 0S^0, since the real hopf fibration is so easy to give explicitly in HoTT and AT. The only thing confusing me is the hopf construction working for S 0S^0. And on this I am unfortunately still confused.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeFeb 15th 2019

    I have spelled it out twice now, in some detail here. Both times you come back just re-iterating the question.

    • CommentRowNumber23.
    • CommentAuthorAlizter
    • CommentTimeFeb 15th 2019
    • (edited Feb 15th 2019)

    @Urs and both time I agree, what I am confused about is why this doesn’t work in HoTT. I am not at all disputing what you have written both times about S 0S^0. The nlab article on Hopf construction, attributes Theorem 3.1 to Sugawara. Unless I am mistaken, Charles has just said that Sugawara does not show this is a quasifibration, which is what Stasheff’s book and the article claim. Is this not a problem?

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeFeb 15th 2019

    I may never have understood what your question actually is. You seemed to claim some problem for the case S 0S^0, where I showed you that one gets the double cover map all right, which is as fibration as it gets.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeFeb 15th 2019
    • (edited Feb 15th 2019)

    I have made a brief edit to record Charles’ observations from #18 and #19 here. No time for more.

    • CommentRowNumber26.
    • CommentAuthorAlizter
    • CommentTimeFeb 15th 2019

    @Urs I have no problem with the case for S 0S^0, perhaps my question is better phrased as “when is the hopf construction a fibration”? The problem I have for S 0S^0 is that the article reads like the hopf construction gives you that it is a fibration, fortunately, as you said twice it is easy to see that this is actually a fibration for S 0S^0. I just want to clear up the conditions needed for the hopf construction to give a fibration.

    • CommentRowNumber27.
    • CommentAuthorDylan Wilson
    • CommentTimeFeb 15th 2019
    @Alizter: If what you really care about is HoTT, doesn't this not matter so much? You can use the Dold-Lashof result instead, which (as Charles explains) gives you the Hopf construction up to equivalence (which is invisible to HoTT) and identifies the homotopy fibers (indistinguishable from fibers in HoTT!) as what you want them to be.
    • CommentRowNumber28.
    • CommentAuthorCharles Rezk
    • CommentTimeFeb 15th 2019
    • (edited Feb 15th 2019)

    Actually, Sugawara proves everything you need: he shows that if XX is a *grouplike HH-space” (i.e., a unital multiplication such that left multiplication by any element is an equivalence, and likewise for right multiplication), then the homotopy fiber of h:XXS(X)h\colon X\star X\to S(X) over a particular point of S(X)S(X) is equivalent to XX. Since S(X)S(X) must be connected whenever XX is non-empty, this means that the homotopy fiber over any point of S(X)S(X) is equivalent to XX.

    The HoTT proof is some sort of instantiation of the following observation: we have a map from the diagram

    Xπ 1X×Xπ 2X X \xleftarrow{\pi_1} X\times X \xrightarrow{\pi_2} X

    to the diagram

    *X* * \leftarrow X \rightarrow *

    where the map in the middle column μ:X×XX\mu\colon X\times X\to X is multiplication. The “grouplike” condition lets you prove that both commutative squares you get are homotopy pullbacks. Then “descent” applies to identify the fibers of the induced map from the homotopy colimit of the first diagram to the homotopy colimit of the second diagram. (Actually, the multiplication doesn’t even need to be unital.)

    • CommentRowNumber29.
    • CommentAuthorAlizter
    • CommentTimeFeb 17th 2019

    This makes much more sense now. Thank you @Urs, @Charles and @Dylan.

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeMar 18th 2019

    added the following pointers:


    Discussion of supersymmetric Hopf fibrations:

    diff, v19, current

    • CommentRowNumber31.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 29th 2019

    Added something on Spin(3)Spin(3)-equivariance.

    diff, v21, current

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