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Morel has defined the motivic Hopf map $\eta$ (in the motivic stable homotopy category $SH(k)$). I suspect that the following facts are valid for it and its topological "cousin"; please correct me if they are false and give me some (nice) references if they are true.

1) For the topological Hopf map we have $\eta^4=0$.

2) The action ot the topological $\eta$ on the values of oriented cohomology theories is zero.

3) If $k$ is the field of complex numbers then the "topological realization" of motivic $\eta$ is the topological Hopf morphism in $SH$ (also denoted by $\eta$?).

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  • $\begingroup$ Have you looked at any of the work of Dugger and Isaksen? $\endgroup$
    – Sean Tilson
    Commented Jun 26, 2017 at 12:58
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    $\begingroup$ Here is a first thought: 1) This is true; I don't actually know who this is first due to, but it must be in Toda's book "Composition Methods in Homotopy Groups of Spheres". 2) Do you mean complex oriented cohomology theories? For such an $E$, the Hurewicz map $\pi_*S \to \pi_*E$ factors through the torsion free ring $\pi_*MU$, so this should follow. $\endgroup$
    – Drew Heard
    Commented Jun 26, 2017 at 13:15
  • $\begingroup$ 3) Realization takes $S^{1,1}$ to $S^{1}$ and $S^{0,0}$ to $S^0$, so you just have to show the realization is the non-zero element of $\pi_1(S^0)$. But the cofiber of the motivic Hopf map is $\Sigma^{-2,-1} \Sigma^\infty \mathbb{P}^2$, which realizes to $\Sigma^{-2} \mathbb{C}P^2$, I think. $\endgroup$
    – Drew Heard
    Commented Jun 26, 2017 at 13:15
  • $\begingroup$ I don't know this terminology well: I probably mean complex oriented cohomology theories. Thank you for the comments! And I have surely read some texts of Dugger and Isaksen; yet I am not sure that they say much about "simple things". $\endgroup$
    – Mikhail Bondarko
    Commented Jun 26, 2017 at 13:53
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    $\begingroup$ 3) is true unstably: take as model for motivic η the map $\mathbb{A}^2 \setminus \{0\} \to \mathbb{P}^1$ which is $S^{3,2} \to S^{2,1}$ and realizes to the topological η $\mathbb{C}^2 \setminus \{0\} \to \mathbb{CP}^1$ which is $S^3 \to S^2$. $\endgroup$
    – Konrad Voelkel
    Commented Jun 29, 2017 at 14:03

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