In "Complex Analysis 2: Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions" and many other books is described a lifting of triangulations for branched covers between surfaces.
Lifting process (note that here we are lifting along $f:Y \to X$)
May we generalise this branched covers (open, discrete maps) between PL $n$-manifolds? A branch set $B_{f}$ is defined as the points where $f$ fails to be a local homeomorphism, by P.T.Church the branch sets and their image are of dimension $n-1$.