Lifting Branched Covers to Braided Embeddings

TL;DR

This paper investigates when branched covers can be lifted to braided embeddings, providing criteria for liftability in various categories and dimensions, with applications to surfaces, knots, and higher-dimensional manifolds.

Contribution

It generalizes the Borsuk-Ulam problem by establishing conditions for lifting branched covers to braided embeddings across different categories and dimensions.

Findings

All branched covers over orientable surfaces lift in the PL category.

Existence of non-liftable branched covers in the smooth category.

Infinite families of coverings over knots where liftability varies.

Abstract

Braided embeddings are embeddings to a product disc bundle so the projection to the first factor is a branched cover. In this paper, we study which branched covers lift to braided embeddings, which is a generalization of the Borsuk-Ulam problem. We determine when a braided embedding in the complement of branch locus can be extended over the branch locus in smoothly (or locally flat piecewise linearly), and use it in conjunction with Hansen's criterion for lifting covers. We show that every branched cover over an orientable surface lifts to a codimension two braided embedding in the piecewise linear category, but there are non-liftable branched coverings in the smooth category. We explore the liftability question for covers over the Klein bottle. In dimension three, we consider simple branched coverings over the three sphere, branched over two-bridge, torus and pretzel knots, obtaining infinite families of examples where the coverings do and do not lift. Finally, we also discuss some examples of non-liftable branched covers in higher dimensions.