My question is the following: let $f:\Omega\to \mathbb{R}^n$ be a branched covering, namely $f$ is continuous, discrete (each fiber is a discrete subset of $\Omega$) and open (open sets get mapped onto open set), where $\Omega\subset \mathbb{R}^n$ is a domain. I wonder wether there is any "topological" concept defined on $\Omega$ that is invariant under the branched covering $f$.

If we look at the case where $f:\Omega\to \mathbb{R}^n$ is a homeomorphism, then the linking number (from Knot theory) is a topological invariant for $f$ (Say in $\mathbb{R}^3$, I have a circle linked with circle with link number +1, then its image under a sense-preserving homeomorphism will still have linking number +1 and cannot be -1). I wonder whether there are analogous concept that is invariant for a branched covering instead of a homeomorphism.

Motivation for this question:

It is well-known due to Hencl and Maly that for a Sobolev homeomorphism $f:\Omega\to \mathbb{R}^3$, the Jacobian of $f$ cannot change sign a.e. in $\Omega$, namely either $J_f\geq 0$ a.e. in $\Omega$ or $J_f\leq 0$ a.e. in $\Omega$. The main tool in their proof is the linking number (of circles) that is invariant under homeomorphism. I want to prove the same result but replace the homeomorphism assumption with the condition that $f$ is a branched covering. Thus I am interested in a kind of invariant for branched coverings.

Any suggestions and comments are welcome.