Branched coverings over manifolds
In the classical theory of Riemannian surfaces, it is well known that every oriented closed 2--dimensional manifold can be represented as a branched covering of the sphere. J. W. Alexander proved that this result is true also for manifolds of higher dimensions. The author of this paper notes that Alexander's proof is also valid for pseudomanifolds and gives a proof of this result in maximal generality and maximal possible detail. In the case of branched coverings of arbitrary dimension, the branching set has a complicate topological structure. This paper gives a presentation of the author's main results obtained in constructing an algebraic theory of branched coverings over manifolds and previuosly published in a series of papers. This theory is based on the concept of a ``spline algebra'', associated to any finite simplicial complex and defined as the set of all continuous functions that are polynomial in the barycentric coordinates of each simplex. In particular, the author makes use of the technique of algebraic extensions, obtaining a relation between the spline algebras of the covering space and the base. Then the automorphism group of a branched covering is isomorphic to the automorphism group of the corresponding extension of the spline algebra.
scientific article; zbMATH DE number 2170267