Return to Answer

The main thing to visualize is the Hopf fibration of $S^2$, its suspensions, and their various compositions.

Let $f \colon S^3 \to S^2$ be the Hopf fibration.

When you suspend $f$ to get $g \colon S^4 \to S^3$, you effectively embed a 2-sphere as the equator of a 3-sphere and extend the mapping in parallel to 2-spheres of longitude. Thus away from the poles you still have circles as preimages.

You can see that $f$ and $g$ compose to give a map $h \colon S^4 \to S^2$. To get a sense of how this looks as a fibration, you can work backwards. First, the preimage of a point in $S^2$ under $f$ is a circle in $S^3$. As noted above, each pointwise preimage of this circle under the suspension $g$ is again generically a circle. When the different circles fit together cleanly, it looks like you get a torus fibration, where the tori twist and interlink within each longitudinal 3-sphere of $S^4$ analogously to the meshing of circles in $S^3$ for the Hopf fibration. If you now suspend this situation, you get a torus fibration over $S^3$ that looks like $h$ within each 2-sphere of longitude.

(I'm still not happy with this description but decided to post it in the hope it might spark some ideas.)

The main thing to visualize is the Hopf fibration of $S^2$, its suspensions, and their various compositions.

Let $f \colon S^3 \to S^2$ be the Hopf fibration.

When you suspend $f$ to get $g \colon S^4 \to S^3$, you effectively embed a 2-sphere as the equator of a 3-sphere and extend the mapping in parallel to 2-spheres of latitude. Thus away from the poles you still have circles as preimages.

You can see that $f$ and $g$ compose to give a map $h \colon S^4 \to S^2$. To get a sense of how this looks as a fibration, you can work backwards. First, the preimage of a point in $S^2$ under $f$ is a circle in $S^3$. As noted above, each pointwise preimage of this circle under the suspension $g$ is again generically a circle. When the different circles fit together cleanly, it looks like you get a torus fibration, where the tori twist and interlink within each latitudinal 3-sphere of $S^4$ analogously to the meshing of circles in $S^3$ for the Hopf fibration. If you now suspend this situation, you get a torus fibration over $S^3$ that looks like $h$ within each 2-sphere of latitude.

(I'm still not happy with this description but decided to post it in the hope it might spark some ideas.)

The main thing to visualize is the Hopf fibration of $S^2$, its suspensions, and their various compositions.

Let $f \colon S^3 \to S^2$ be the Hopf fibration.

When you suspend $f$ to get $g \colon S^4 \to S^3$, you effectively embed a 2-sphere as the equator of a 3-sphere and extend the mapping in parallel to 2-spheres of longitudes. Thus away from the poles you still have circles as preimages (the preimage of each pole is of course itself).

You can see that $f$ and $g$ compose to give a map $h \colon S^4 \to S^2$. To get a sense of how this looks as a fibration, you can work backwards. First, the preimage of a point in $S^2$ under $f$ is a circle in $S^3$. As noted above, each pointwise preimage of this circle under the suspension $g$ is again generically a circle. When the different circles fit together cleanly, it looks like you get a torus fibration, where the tori twist and interlink within each longitudinal 3-sphere of $S^4$ analogously to the meshing of circles in $S^3$ for the Hopf fibration. If you now suspend this situation, you get a torus fibration over $S^3$ that looks like $h$ within each 2-sphere of longitude.

(I'm still not happy with this description but decided to post it in the hope it might spark some ideas.)

The main thing to visualize is the Hopf fibration of $S^2$, its suspensions, and their various compositions.

Let $f \colon S^3 \to S^2$ be the Hopf fibration.

When you suspend $f$ to get $g \colon S^4 \to S^3$, you effectively embed a 2-sphere as the equator of a 3-sphere and extend the mapping in parallel to 2-spheres of longitude. Thus away from the poles you still have circles as preimages.

You can see that $f$ and $g$ compose to give a map $h \colon S^4 \to S^2$. To get a sense of how this looks as a fibration, you can work backwards. First, the preimage of a point in $S^2$ under $f$ is a circle in $S^3$. As noted above, each pointwise preimage of this circle under the suspension $g$ is again generically a circle. When the different circles fit together cleanly, it looks like you get a torus fibration, where the tori twist and interlink within each longitudinal 3-sphere of $S^4$ analogously to the meshing of circles in $S^3$ for the Hopf fibration. If you now suspend this situation, you get a torus fibration over $S^3$ that looks like $h$ within each 2-sphere of longitude.

(I'm still not happy with this description but decided to post it in the hope it might spark some ideas.)

The main thing to visualize is the Hopf fibration of $S^2$, its suspensions, and their various compositions.

Let $f \colon S^3 \to S^2$ be the Hopf fibration.

When you suspend $f$ to get $Sf \colon S^4 \to S^3$, you effectively embed a 2-sphere as the equator of a 3-sphere and extend the mapping in parallel to 2-spheres of longitudes. Thus away from the poles you still have circles as preimages (the preimage of each pole is of course itself).

You can see that $f$ and $Sf$ compose to give a map $S^4 \to S^2$. To get a sense of how this looks as a fibration, you can work backwards. First, the preimage of a point in $S^2$ under $f$ is a circle in $S^3$. As noted above, each pointwise preimage of this circle under the suspension $Sf$ is again generically a circle. When the different circles fit together cleanly, it looks like you get a torus fibration, where the tori twist and interlink within each longitudinal 3-sphere of $S^4$ analogously to the meshing of circles in $S^3$ for the Hopf fibration.

The main thing to visualize is the Hopf fibration of $S^2$, its suspensions, and their various compositions.

Let $f \colon S^3 \to S^2$ be the Hopf fibration.

When you suspend $f$ to get $g \colon S^4 \to S^3$, you effectively embed a 2-sphere as the equator of a 3-sphere and extend the mapping in parallel to 2-spheres of longitudes. Thus away from the poles you still have circles as preimages (the preimage of each pole is of course itself).

You can see that $f$ and $g$ compose to give a map $h \colon S^4 \to S^2$. To get a sense of how this looks as a fibration, you can work backwards. First, the preimage of a point in $S^2$ under $f$ is a circle in $S^3$. As noted above, each pointwise preimage of this circle under the suspension $g$ is again generically a circle. When the different circles fit together cleanly, it looks like you get a torus fibration, where the tori twist and interlink within each longitudinal 3-sphere of $S^4$ analogously to the meshing of circles in $S^3$ for the Hopf fibration. If you now suspend this situation, you get a torus fibration over $S^3$ that looks like $h$ within each 2-sphere of longitude.

(I'm still not happy with this description but decided to post it in the hope it might spark some ideas.)