Composites of open maps

A 4-fold covering of a surface of genus 2 by a surface of genus 5 is constructed that cannot be represented as a composite of two non-trivial open maps. This demonstrates the incompleteness of Baildon's obstruction. Various results on the decomposability of a regular covering into a composite of (regular) coverings of various multiplicities arranged in various orders are established. A new obstruction to the decomposability of branched coverings is proposed, the system of branch data at the branch points.