Below is my sketch of the deformation of the top hemisphere (of a ball) into the lower hemisphere to give some intuition of what I am referring to.
What is the name for a continuous deformation of surfaces?
I want to say something like "2-homotopy" or "surface homotopy" or "homotopy of homotopies" -- although perhaps it should be a "3-homotopy" since it corresponds to a continuous map from $[0,1]^3$ or $B^3$ into the space $X$ (singular $3-$simplex), its image in non-degenerate/singular cases would be 3-dimensional, and it corresponds to $3-$morphisms in higher category theory -- of course that would make homotopies "2-homotopies", so maybe "3-path" would be a better term.
Anyway it would be nice to be able to say something like "the second homology group of the ball $B^3$ is trvial whereas that of the sphere $S^2$ is not, since there exists a 2-homotopy between the top and bottom hemispheres passing through the ball $B^3$". I.e. to extend to higher dimensions the description of the generators of 1-boundaries as closed paths enclosing homotopies.
EDIT: This page on nLab seems relevant.
