The short answer is, yes, the chiral edge state is determined by bulk topological property. It is known as bulk-edge correspondence.
The paper you should read is: Protected edge modes without symmetry - 1301.7355.
Generically, to determine whether the edge states are robust is by determining whether the edge states are ``protected'' by any of the three mechanism:
1. Symmetry (-protected)
2. Chirality (-protected)
3. Statistics (-protected), i.e. nontrivial (fractional) Statistics protected.
This is explained in the Popular Summary of this paper.
You may also have strong interests to learn when the non-chiral edge states can be gapped, which you can find a concise discussion in this paper: Boundary Degeneracy of Topological Order-1212.4863, they apply the so-called "boundary fully gapping rules" (or equivalent to a "Lagrangian subgroup" discussed in some other papers) to find the conditions of gapped edge, and further count the (topological) ground state degeneracy of a system with gapped boundaries.