Can a sphere glued into a soft 3d-mattress rotate continuously? (manifolds, SU(2) and the belt trick)

The answer is "yes":
A sphere glued into a soft 3d-mattress can rotate continuously.

Let $R_t\in SO(3)$ be the rotation by angle $t$ around the $z$-axis.

Pick a nullhomotopy $R_{t,s}$ ($s\in [0,1]$) of the map $[0,4\pi]\to SO(3):t\mapsto R_t$.
So $R_{t,0}=R_t$ and $R_{t,1}=\mathrm{id}$, for all $t\in [0,4\pi]$.

Now here's the description of the motion of the mattress:

At time $t$, the sphere of radius $1+s$ performs the rotation $R_{t,s}$.
(The mattress is immobile outside the sphere of radius 2.)

This is sometimes called Dirac's belt trick. There is a Wikipedia article under the name the plate trick. As that article says, it demonstrates the theorem that "SU(2) (which double-covers SO(3)) is simply connected."

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^{Dirac Belt Trick illustrated by George Francis. From Carlo H. Séquin, "Torus Immersions and Transformations," 2013.}

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