A decomposition theorem for topological branched coverings
Shahryar Ghaed Sharaf
TL;DR
The paper develops a topological analogue of the decomposition theorem for branched coverings by first establishing a canonical decomposition for unramified coverings via transition data, then extending to branched maps with a codimension-2 branch locus. It shows that, for branched coverings between closed topological manifolds, the direct image of the constant sheaf splits as a sum of the constant sheaf and a constructible part encoded by a local system reflecting monodromy around the branch locus; this extends to singular targets by refining stratifications and formulating a derived-category statement involving intersection cohomology. The results connect topological branched coverings with stratified and perverse-structure concepts, providing explicit descriptions of stalks, monodromy, and restriction maps, and yielding a framework for computing direct images in both smooth and singular settings.
Abstract
In the context of complex algebraic varieties, the decomposition theorem for semi-small maps provides a decomposition of the direct image of the constant sheaf. In this work, we develop a decomposition theorem for branched coverings of topological spaces. To achieve this, we start by constructing a decomposition theorem for unramified covering maps using transition functions. For a given branched covering of closed topological manifolds, we use the previous result to establish a decomposition of the direct image of the constant sheaf on the covering space. In the next step, we generalize our discussion to the case where the target space is not necessarily a topological manifold.