The question is triggered by the wonderful animations by Jason Hise:

https://www.youtube.com/watch?v=LLw3BaliDUQ

https://www.youtube.com/watch?v=6Ul_-ABYaYU

https://www.youtube.com/watch?v=aYVt1UiERIQ

All these animations are based on the well-known belt trick (a way to represent SU(2) as double cover of SO(3)) and suggest that a solid sphere that is glued into a soft mattress can rotate continuously, if the region of mattress close to the sphere performs certain appropriate motions around the sphere. Is this conclusion correct? Are there papers on this very counter-intuitive issue?

It is not evident that the belt trick implies that a sphere can rotate when glued in a mattress. Many discrete belts do not make a continuous and smooth mattress. So it could be that the mattress property does not follow from the belt trick - it could be that the belt trick contains discontinuities. So the question is: is the mattress trick a continuous operation on the mattress, or does it need cuts in the mattress?