Corpus ID: 119319721

Characterization of branched covers with simplicial branch sets

@article{Luisto2018CharacterizationOB,
  title={Characterization of branched covers with simplicial branch sets},
  author={Rami Luisto and Eden Prywes},
  journal={arXiv: Complex Variables},
  year={2018}
}
The image of the branch set of a PL branched cover between PL $n$-manifolds is a simplicial $(n-2)$-complex. We demonstrate that the reverse implication also holds; i.e., for a branched cover $f \colon \mathbb{S}^n \to \mathbb{S}^n$ with the image of the branch set contained in a simplicial $(n-2)$-complex the mapping can be reparametrized as a PL mapping. This extends a result by Martio and Srebro [MS79]. 
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References

SHOWING 1-10 OF 38 REFERENCES
Local monodromy of branched covers and dimension of the branch set
We show that, if the local dimension of the branch set of a discrete and open mapping $f\colon M\to N$ between $n$-manifolds is less than $(n-2)$ at a point $y$ of the image of the branch set $fB_f$,Expand
Expanding Thurston Maps
We study the dynamics of Thurston maps under iteration. These are branched covering maps $f$ of 2-spheres $S^2$ with a finite set $\mathop{post}(f)$ of postcritical points. We also assume that theExpand
Quasiregular maps S3 → S3 with wild branch sets
Abstract Two examples of quasiregular maps S3 → S3 that branch on a wild Cantor set are constructed. As an application it is shown that certain interesting 3-dimensional metric spaces recentlyExpand
The Branch Set of a Quasiregular Mapping
We discuss the issue of branching in quasiregular mapping, and in par­ ticular the relation between branching and the problem of finding geometric parametrizations for topological manifolds. OtherExpand
Geometric branched covers between generalized manifolds
We develop a theory of geometrically controlled branched covering maps between metric spaces that are generalized cohomology manifolds. Our notion extends that of maps of bounded length distortion,Expand
EIGHT FACES OF THE POINCARE HOMOLOGY 3-SPHERE
Publisher Summary This chapter presents eight different descriptions of the Poincare homology sphere, and presents that they define the same 3-manifold. The dodecahedral space of Poincare wasExpand
Stoïlow’s theorem revisited
Sto\"ilow's theorem from 1928 states that a continuous, light, and open mapping between surfaces is a discrete map with a discrete branch set. This result implies that such mappings betweenExpand
Generalized Poincare's Conjecture in Dimensions Greater Than Four
Poincare has posed the problem as to whether every simply connected closed 3-manifold (triangulated) is homeomorphic to the 3-sphere, see [18] for example. This problem, still open, is usually calledExpand
Metric Spaces of Non-Positive Curvature
This book describes the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces byExpand
Introduction to Piecewise-Linear Topology
1. Polyhedra and P.L. Maps.- Basic Notation.- Joins and Cones.- Polyhedra.- Piecewise-Linear Maps.- The Standard Mistake.- P. L. Embeddings.- Manifolds.- Balls and Spheres.- The Poincare ConjectureExpand
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