  We examine a bulk-edge correspondence of branching MERA networks at finite
temperatures in terms of algebraic and differential topology. By using
homeomorphic mapping, we derive that the networks are nonorientable manifolds
such as a M$\ddot{\rm o}$bius strip and a Klein bottle. We also examine the
stability of the branch in connection with the second law of black hole
thermodynamics. Then, we prove that the MERA network for one-dimensional
quantum critical systems spontaneously separates into multiple branches in the
IR region of the network. On the other hand, the branch does not occur in more
than two dimensions. The result illustrates dimensionality dependence of
spin-charge separation / coupling. We point out a role of twist of the surfaces
on the phase string between spinon and holon excitations.
