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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.

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    $\begingroup$ Just a couple of remarks: (1) what you described is not what is usually called a branched covering (e.g., $\exp\colon\mathbb C\to\mathbb C$ restricted to $0<\Im<4\pi$ is not a covering); (2) what do you mean by $S^1$ linked with $S^2$ in $\Bbb R^3$ and what exactly is preserved? (3) Invariants of what are you looking for? $\endgroup$ – Alex Degtyarev Apr 10 '15 at 7:47
  • $\begingroup$ @AlexDegtyarev: (1) I noticed that there are different definitions of branched coverings used in different areas. In geometric function theory, one usually defines branched coverings as above and it is not necessarily to be a real covering mapping. (2) I corrected it in 3-dim and in 4 dim, one needs to use circle and sphere. (3). I want some concept defined topologically on $\Omega$ and it still makes senses in the target under the branched covering $f$ so that this concept are kept. $\endgroup$ – Changyu Guo Apr 10 '15 at 7:55
  • $\begingroup$ @AlexDegtyarev: Take the example as in the homeomorphism case, the linking number of circle and sphere are topological invariant, it can not change from +1 to -1 under a sense-preserving homeomorphism. I want a similar kind of concept, so that they are kept by branched coverings as defined above. $\endgroup$ – Changyu Guo Apr 10 '15 at 7:58
  • $\begingroup$ Looking at your motivation, this seems an overkill. You are speaking about the orientation, and the latter is preserved (or reversed everywhere) under what is usually called a branched covering. E.g., for your purpose, it would suffice to know that the critical locus has codimension at least $2$. $\endgroup$ – Alex Degtyarev Apr 10 '15 at 8:25
  • $\begingroup$ Any progress on your question? $\endgroup$ – Piotr Hajlasz Dec 18 '18 at 17:30

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