Birman-Hilden theory for 3-manifolds

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.