Characterization of Branched Covers with Simplicial Branch Sets

Eden Prywes, Princeton University

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A branched covering $f \colon \mathbb R^n \to \mathbb R^n$ is an open and discrete map.  Branched coverings are topological generalizations of quasiregular and holomorphic mappings. The branch set of $f$ is the set where $f$ fails to be locally injective.  It is well known that the image of the branch set of a PL branched covering between PL $n$-manifolds is a simplicial $(n-2)$-complex. I will discuss a recent result that the reverse implication also holds. More precisely, a branched covering with the image of the branch set contained in a simplicial $(n-2)$-complex is equivalent up to homeomorphism to a PL mapping. This result is classical for $n=2$ and was shown by Martio and Srebro for $n = 3$.  This is joint work with Rami Luisto.