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arXiv · 2408.07798

Birman-Hilden theory for 3-manifolds

Trent Lucas
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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.

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Trent Lucas. 2025-03-09. Birman-Hilden theory for 3-manifolds. https://arxiv.org/abs/2408.07798

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