Birman-Hilden theory for 3-manifolds

TL;DR

This paper investigates the properties of lifting maps of homeomorphisms in branched covers of 3-manifolds, revealing that unlike surfaces, these maps are generally not injective, with explicit kernel descriptions for certain covers.

Contribution

It demonstrates that the lifting map is typically non-injective for 3-manifolds and provides a finite generating set for its kernel in specific cases, extending Birman-Hilden theory.

Findings

Lifting maps are generally not injective for 3-manifolds.

Explicit kernel generators are identified for certain branched covers.

Contrasts with the surface case where injectivity often holds.

Abstract

Given a branched cover of manifolds, one can lift homeomorphisms along the cover to obtain a (virtual) homomorphism between mapping class groups. Following a question of Margalit-Winarski, we study the injectivity of this lifting map in the case of $3$-manifolds. We show that in contrast to the case of surfaces, the lifting map is generally not injective for most regular branched covers of $3$-manifolds. This includes the double cover of $S^3$ branched over the unlink, which generalizes the hyperelliptic branched cover of $S^2$. In this case, we find a finite normal generating set for the kernel of the lifting map.