Significance of topology in topologically ordered systems

The topology on which a lattice is placed plays an important role in topologically ordered systems, for example in toric code the degeneracy in the ground states is given by $4^{g}$ where $g$ is the genus of the manifold on which we place our lattice and thus this degeneracy is a signature of topological order. But if somehow we change the interaction of the spins on the lattice, we can move from a torus to a cylinder and thus because of the change in topology we expect a phase transition (something on the lines of toric code to double semion model where the underlying manifold is held the same with change in interaction) here we change the interaction so that the effective manifold itself changes. Is such change in underlying topology valid ? Since in defining a topologically ordered system we first fix the manifold and then talk about its properties, so how strong is the condition on the manifold ?

• The topology of the manifold is used a probe for the physics. The phase of matter formed by the system is independent of the topology of the manifold. Also, you can't smoothly tune the topology of the manifold. – Ruben Verresen May 14 '18 at 20:01
• Thanks.But in this case the phase of the matter is dependent solely on the topology, for e.g, in toric code, changing the interaction strength of the plaquette and vertex terms on the torus we can interpolate to a cylinder with open boundary conditions whose $GSD_{cylinder}$ = 1 or 2 (different boundary conditions) when compared to $GSD_{torus}$ = 4. It would be really helpful to know about what smoothly tuning in this context means ? – esornep May 15 '18 at 8:54
• Atleast my understanding is that there is a phase transition because of the break in degeneracy and we can definitely interpolate from one topology to other by changing interaction strength – esornep May 15 '18 at 8:56