  Topological order, the hallmark of fractional quantum Hall states, is
primarily defined in terms of ground-state degeneracy on higher-genus
manifolds, e.g. the torus. We investigate analytically and numerically the
smooth crossover between this topological regime and the Tao-Thouless thin
torus quasi-1D limit. Using the wire-construction approach, we analyze an
emergent charge density wave (CDW) signifying the break-down of topological
order, and relate its phase shifts to Wilson loop operators. The CDW amplitude
decreases exponentially with the torus circumference once it exceeds the
transverse correlation length controllable by the inter-wire coupling. By means
of numerical simulations based on the matrix product states (MPS) formalism, we
explore the extreme quasi-1D limit in a two-leg flux ladder and present a
simple recipe for probing fractional charge excitations in the $\nu=1/2$
Laughlin-like state of hard-core bosons. We discuss the possibility of
realizing this construction in cold-atom experiments. We also address the
implications of our findings to the possibility of producing non-Abelian zero
modes. As known from rigorous no-go theorems, topological protection for exotic
zero modes such as parafermions cannot exist in 1D fermionic systems and the
associated degeneracy cannot be robust. Our theory of the 1D-2D crossover
allows to calculate the splitting of the degeneracy, which vanishes
exponentially with the number of wires, similarly to the CDW amplitude.
