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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2015
    • (edited Nov 25th 2015)

    I have created an entry on the quaternionic Hopf fibration and then I have tried to spell out the argument, suggested to me by Charles Rezk on MO, that in GG-equivariant stable homotopy theory it represents a non-torsion element in

    [Σ G S 7,Σ G S 4] G [\Sigma^\infty_G S^7 , \Sigma^\infty_G S^4]_G \simeq \mathbb{Z} \oplus \cdots

    for GG a finite and non-cyclic subgroup of SO(3)SO(3), and SO(3)SO(3) acting on the quaternionic Hopf fibration via automorphisms of the quaternions.

    I have tried to make a rigorous and self-contained argument here by appeal to Greenlees-May decomposition and to tom Dieck splitting. But check.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2019
    • (edited Mar 10th 2019)

    added the following fact, which I didn’t find so easy to see:


    1. the Spin(5)-action on the 4-sphere S 4S^4 which is induced by the defining action on 5\mathbb{R}^5 under the identification S 4S( 5)S^4 \simeq S(\mathbb{R}^5);

    2. the Spin(5)-action on the 7-sphere S 7S^7 which is induced under the exceptional isomorphism Spin(5)Sp(2)=U(2,)Spin(5) \simeq Sp(2) = U(2,\mathbb{H}) by the canonical left action of U(2,)U(2,\mathbb{H}) on 2\mathbb{H}^2 via S 7S( 2)S^7 \simeq S(\mathbb{H}^2).

    Then the complex Hopf fibration S 7h S 4S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4 is equivariant with respect to these actions.

    This is almost explicit in Porteous 95, p. 263

    diff, v16, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 22nd 2019
    • (edited Mar 22nd 2019)

    added the remark

    Of the resulting action of Sp(2)×\timesSp(1) on the 7-sphere (from this Prop.), only the quotient group Sp(n).Sp(1) acts effectively.

    diff, v17, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2019

    added pointer to Table 1 in

    • Machiko Hatsuda, Shinya Tomizawa, Coset for Hopf fibration and Squashing, Class.Quant.Grav.26:225007, 2009 (arXiv:0906.1025)

    for the coset presentation

    S 3 fib(h ) S 7 h S 4 = = = Spin(4)Spin(3) Spin(5)Spin(3) Spin(5)Spin(4) \array{ S^3 &\overset{fib(h_{\mathbb{H}})}{\longrightarrow}& S^{7} &\overset{h_{\mathbb{H}}}{\longrightarrow}& S^4 \\ = && = && = \\ \frac{Spin(4)}{Spin(3)} &\longrightarrow& \frac{Spin(5)}{Spin(3)} &\longrightarrow& \frac{Spin(5)}{Spin(4)} }

    diff, v19, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2019
    • (edited Mar 31st 2019)

    added pointer to

    • Herman Gluck, Frank Warner, Wolfgang Ziller, The geometry of the Hopf fibrations, L’Enseignement Mathématique, t.32 (1986), p. 173-198

    which in its Prop. 4.1 explicitly states and proves the Spin(5)Spin(5)-equivariance of the quaternionic Hopf fibration

    (but fails to mention the coset representation that makes this manifest)

    diff, v20, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2019
    • (edited Apr 27th 2019)

    added this in the list of references:

    Noteworthy fiber products with the quaternionic Hopf fibration, notably exotic 7-spheres, are discussed in

    • Llohann D. Sperança, Explicit Constructions over the Exotic 8-sphere (pdf)

    diff, v21, current

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