A Decomposition Theorem for Topological Branched Coverings

TL;DR

This paper develops a new decomposition theorem for topological branched coverings, extending classical results from algebraic geometry to more general topological settings.

Contribution

It introduces a decomposition theorem for branched coverings of topological spaces, generalizing known results for semi-small maps and unramified covers.

Findings

Decomposition theorem for unramified covering maps established

Decomposition of direct image sheaves for branched coverings proved

Generalization to non-manifold target spaces achieved

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 via an explicit gluing construction 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.