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 $\eta$?).