Basic definitions and properties of topological branched coverings

Abstract

The aim of this paper is to examine topological branched coverings which were introduced in [5]. They appear naturally in algebraic and analytic geometry and they have been considered mostly in PL category (see for example [7], [2] and [4]). We introduce new notions which may be useful not only for ex- amining topological branched coverings: a strong mapping at a point, a locally strong mapping and a spreading mapping. After proving that, under certain assumptions, the only branched coverings with the Absolute Covering Homotopy Property are unbranched coverings, we give two sufficient conditions for the Arc Lifting Property. We also characterize finite and locally finite nondegenerate graphs as branched coverings over the unit circle S1 with one-point singular set.