Why did I imagine that this would be about two shapes that are merely topologically n-balls, each having part of their boundary be incident with one of the two

Why did I imagine that this would be about two shapes that are merely topologically n-balls, each having part of their boundary be incident with one of the two hemi(n-1)-spheres of the boundary of an n-ball (and otherwise not intersecting it)? (So like, in 3D, if you took some ball and two lumps of clay of different colors, and smooshed each piece of clay over half of the surface of the ball, with each of the two lumps of clay remaining topologically a 3-ball.) I don’t know that there would even be anything interesting to say about that.