Contort rectangle to torus minus splice=homomorphism?

[aheight]

I understand the transformation in general is not homomorphic but what about the transformation minus the splices, that is, contort it all the way up to and not including splicing the edges? Isn't that a homomorphism? Can't we define a bijective function (rotation matrices) to map the two spaces? Seems also to be diffeomorphism as well or no?

 

[fresh_42]

I understand the transformation in general is not homomorphic but what about the transformation minus the splices, that is, contort it all the way up to and not including splicing the edges? Isn't that a homomorphism? Can't we define a bijective function (rotation matrices) to map the two spaces? Seems also to be diffeomorphism as well or no?

If you don't clue the edges, then it is only a "bending", which is both, homeomorph and diffeomorph (as long as you don't fold it somewhere).