Deformation fields would struggle to fundamentally change the topological type, particularly where the transformation would need to “tear” the manifold, such as

Deformation fields would struggle to fundamentally change the topological type, particularly where the transformation would need to “tear” the manifold, such as turning a sphere into a donut or dividing a cell into two child cells. HyperNeRF’s exploit and extend Level Set Methods, which are rooted in Morse Theory. https://en.wikipedia.org/wiki/Level-set_method https://en.wikipedia.org/wiki/Level-set_method https://en.wikipedia.org/wiki/Morse_theory https://en.wikipedia.org/wiki/Morse_theory