Is shape deformation a topological invariant?

In this beautiful paper, Kawabata et al.  propose a topologically-framed unification of some physical features.  
It is well-known that topology deals with the properties of space preserved under continuous deformations, such as stretching, twisting, bending and so on.  This means that two shapes of genus zero (or one, or two, and so on) are topologically invariant under homeomorphisms, i.e., they share matching topological description. 
 
Here we ask: is this tenet true?
 
Take a positive-curvature active surface, such as a spheric soap bubble.   Due to the Borsuk-Ulam theorem, the bubble's surface displays at least two antipodal points with the same description (e.g., two antipodal points with the same value of surface tension, the latter standing for a continuous function on the 2D surface of the 3D bubble).  
When a spontaneous or a mechanical stress (e.g., an internal or external force, or a torque) is applied within and onto the surface, the subsequent instability leads to the production of a deformed bubble.  The formation of this bubble's nontrivial surface shape leads to the loss of the above-mentioned antipodal points with matching description.  Therefore, once a spheric manifold's curvature is modified, an algebraic topological feature gets lost, i.e., the two antipodal points with matching description.