I've read in the textbooks that the non-trivial generator $\eta\_n$ of $\pi\_{n+1}(S^n)$ is the suspension of the Hopf map $S^3\to S^2$, and the generator $\chi$ of $\pi\_5(S^3)$ is given by $\eta\_3 \circ \eta\_4$. Fine. 


My question is, how I can visualize them? Is there a nice explicit way to describe these maps $\eta\_3$ and $\eta\_3\circ \eta\_4$ ? How about the generator of $\pi\_6(S^3)$ ? 


(Other questions on MO look more serious. Hopefully this question is not out of place ...)


EDIT: anyone with rudimentary understanding of basic homotopy theory would say $\eta$ and $\eta\circ\eta$ are explicit enough, but I just can't visualize the suspension. I would be happy with a nice description of $SU(2)$ bundles over $S^n$, as my first exposure to homotopy is through quantum field theory...


Further edit: Thanks everyone for answers, I'm almost inclined to accept Per's answer, but I'm not still satisfied :p



How can I visualize the nontrivial element of $\pi_4(S^3)$ and $\pi_5(S^3)$ ?
(This is a bit late, but I hope you find it interesting!)


Here's smooth representation of the generator of $\pi\_4(Sp(1))$ (and so the same homotopy group of $S^3$ and $SU(2)$). Consider $S^4 = \mathbb{HP}^1$, and $Sp(1)$ the unit sphere in $\mathbb{H}$. Then the following function $t\colon \mathbb{HP}^4 \to Sp(1)$ represents the nontrivial homotopy class $S^4 \to S^3$:
$$
t[p;q] = \frac{2p\bar{q}i\bar{p}q - |p|^4 + |q|^4}{|p|^4 + |q|^4}
$$
where [p;q] are homogeneous coordinates on $\mathbb{HP}^4$. I don't know if this has appeared previously (I would love to know!), but I presented this as part of [some slides](http://www.maths.usyd.edu.au/u/austms2013/talks/aust13Roberts.pdf) at the Australian Mathematical Society's annual conference last year (see slide 6), and originally worked it out with a pointer from Michael Murray to the Hopf fibration described using quaternions (that is, $Sp(1) \to S(Im\mathbb{H})$, the unit sphere in the pure imaginaries). That this map is the generator (i.e. is not null-homotopic) I calculated following the answer at my question [Detecting homotopy nontriviality of an element in a torsion homotopy group](https://mathoverflow.net/questions/127434/).


Note that this function followed by the inclusion $Sp(1) \hookrightarrow Sp(2)$ (as the top left entry) is the generator of $\pi\_4(Sp(2))$ (by results of Mimura and Toda). And thus we also get a representative for the generator of $\pi\_4$ of $Spin(5) = Sp(2)$.


